Use the method of to fit the logistic equation to the actual U.S. population data (Fig

Elementary Differential Equations | 6th Edition | ISBN: 9780132397308 | Authors: C. Henry Edwards David E. Penney

Chapter 1: Problem 1 Elementary Differential Equations 6

In Problems I through 12, verify by substitution that each given function is a solution of the given differential equation. Throughout these problems, primes denote derivatives with respect to x. y' = 3x2 ; y = x3 + 7

Chapter 1: Problem 1 Elementary Differential Equations 6

In Problems I through 12, verify by substitution that each given function is a solution of the given differential equation. Throughout these problems, primes denote derivatives with respect to x. y' + 2y = 0; Y = 3e-2x

Chapter 1: Problem 1 Elementary Differential Equations 6

In Problems I through 12, verify by substitution that each given function is a solution of the given differential equation. Throughout these problems, primes denote derivatives with respect to x. y" + 4y = 0; YI = cos 2x, Y2 = sin 2x

Chapter 1: Problem 1 Elementary Differential Equations 6

In Problems I through 12, verify by substitution that each given function is a solution of the given differential equation. Throughout these problems, primes denote derivatives with respect to x. y" = 9y; YI = e3x , Y2 = e-3x

Chapter 1: Problem 1 Elementary Differential Equations 6

In Problems I through 12, verify by substitution that each given function is a solution of the given differential equation. Throughout these problems, primes denote derivatives with respect to x. y' = y + 2e-x ; y = eX - e-X

Chapter 1: Problem 1 Elementary Differential Equations 6

In Problems I through 12, verify by substitution that each given function is a solution of the given differential equation. Throughout these problems, primes denote derivatives with respect to x. y" + 4y' + 4y = 0; YI = e-2x , Y2 = xe -2x

Chapter 1: Problem 1 Elementary Differential Equations 6

In Problems I through 12, verify by substitution that each given function is a solution of the given differential equation. Throughout these problems, primes denote derivatives with respect to x. Y" - 2y' + 2y = 0; YI = eX cosx, Y2 = eX sinx

Chapter 1: Problem 1 Elementary Differential Equations 6

In Problems I through 12, verify by substitution that each given function is a solution of the given differential equation. Throughout these problems, primes denote derivatives with respect to x. y"+y = 3 cos 2x, YI = cosx-cos 2x, Y2 = sinx-cos 2x

Chapter 1: Problem 1 Elementary Differential Equations 6

In Problems I through 12, verify by substitution that each given function is a solution of the given differential equation. Throughout these problems, primes denote derivatives with respect to x. y' + 2xy 2 = 0; Y = -I +x2

Chapter 1: Problem 1 Elementary Differential Equations 6

In Problems I through 12, verify by substitution that each given function is a solution of the given differential equation. Throughout these problems, primes denote derivatives with respect to x. x2y" + xy' - Y = lnx; Yl = x - lnx, Y2 = - - lnx

Chapter 1: Problem 1 Elementary Differential Equations 6

In Problems I through 12, verify by substitution that each given function is a solution of the given differential equation. Throughout these problems, primes denote derivatives with respect to x. x2y" +5xy' +4y = 0; Yt = 2' Y2 = - X X 2

Chapter 1: Problem 1 Elementary Differential Equations 6

In Problems I through 12, verify by substitution that each given function is a solution of the given differential equation. Throughout these problems, primes denote derivatives with respect to x. x2y" - xy' + 2y = 0; Yt = x cos(lnx), Y2 = x sin (In x)

Chapter 1: Problem 1 Elementary Differential Equations 6

In Problems 13 through 16, substitute y = erx into the given differential equation to determine all values of the constant r for which y = erx is a solution of the equation. 3y' = 2y

Chapter 1: Problem 1 Elementary Differential Equations 6

In Problems 13 through 16, substitute y = erx into the given differential equation to determine all values of the constant r for which y = erx is a solution of the equation. 4y" = y

Chapter 1: Problem 1 Elementary Differential Equations 6

In Problems 13 through 16, substitute y = erx into the given differential equation to determine all values of the constant r for which y = erx is a solution of the equation. y" + y' - 2y = 0

Chapter 1: Problem 1 Elementary Differential Equations 6

In Problems 13 through 16, substitute y = erx into the given differential equation to determine all values of the constant r for which y = erx is a solution of the equation. 3y" + 3y' - 4y = 0

Chapter 1: Problem 1 Elementary Differential Equations 6

In Problems 17 through 26, first verify that y(x) satisfies the given differential equation. Then determine a value of the constant C so that y(x) satisfies the given initial condition. Use a computer or graphing calculator (if desired) to sketch several typical solutions of the given differential equation, and highlight the one that satisfies the given initial condition. y' + y = 0; y(x) = Ce-X, y(O) = 2

Chapter 1: Problem 1 Elementary Differential Equations 6

In Problems 17 through 26, first verify that y(x) satisfies the given differential equation. Then determine a value of the constant C so that y(x) satisfies the given initial condition. Use a computer or graphing calculator (if desired) to sketch several typical solutions of the given differential equation, and highlight the one that satisfies the given initial condition. y' = 2y; y(x) = Ce2x, y(O) = 3

Chapter 1: Problem 1 Elementary Differential Equations 6

In Problems 17 through 26, first verify that y(x) satisfies the given differential equation. Then determine a value of the constant C so that y(x) satisfies the given initial condition. Use a computer or graphing calculator (if desired) to sketch several typical solutions of the given differential equation, and highlight the one that satisfies the given initial condition. y' = y + 1 ; y(x) = Cex - I, y(O) = 5

Chapter 1: Problem 1 Elementary Differential Equations 6

In Problems 17 through 26, first verify that y(x) satisfies the given differential equation. Then determine a value of the constant C so that y(x) satisfies the given initial condition. Use a computer or graphing calculator (if desired) to sketch several typical solutions of the given differential equation, and highlight the one that satisfies the given initial condition. y' = x - y; y(x) = Ce-X + x - I, y(O) = 10

Chapter 1: Problem 1 Elementary Differential Equations 6

In Problems 17 through 26, first verify that y(x) satisfies the given differential equation. Then determine a value of the constant C so that y(x) satisfies the given initial condition. Use a computer or graphing calculator (if desired) to sketch several typical solutions of the given differential equation, and highlight the one that satisfies the given initial condition. y' + 3x2y = 0; y(x) = Ce-x 3 , y(O) = 7

Chapter 1: Problem 1 Elementary Differential Equations 6

In Problems 17 through 26, first verify that y(x) satisfies the given differential equation. Then determine a value of the constant C so that y(x) satisfies the given initial condition. Use a computer or graphing calculator (if desired) to sketch several typical solutions of the given differential equation, and highlight the one that satisfies the given initial condition. eYy' = 1 ; y(x) = In(x + C), y(O) = 0

Chapter 1: Problem 1 Elementary Differential Equations 6

In Problems 17 through 26, first verify that y(x) satisfies the given differential equation. Then determine a value of the constant C so that y(x) satisfies the given initial condition. Use a computer or graphing calculator (if desired) to sketch several typical solutions of the given differential equation, and highlight the one that satisfies the given initial condition. x dx + 3y = 2x5; y(x) = ix5 + Cx-3, y(2) = 1

Chapter 1: Problem 1 Elementary Differential Equations 6

In Problems 17 through 26, first verify that y(x) satisfies the given differential equation. Then determine a value of the constant C so that y(x) satisfies the given initial condition. Use a computer or graphing calculator (if desired) to sketch several typical solutions of the given differential equation, and highlight the one that satisfies the given initial condition. xy' - 3y = x3 ; y(x) = x3 (C + lnx), y(l) = 1 7

Chapter 1: Problem 1 Elementary Differential Equations 6

In Problems 17 through 26, first verify that y(x) satisfies the given differential equation. Then determine a value of the constant C so that y(x) satisfies the given initial condition. Use a computer or graphing calculator (if desired) to sketch several typical solutions of the given differential equation, and highlight the one that satisfies the given initial condition. y' = 3x2(y2 + 1 ); y(x) = tan(x3 + C), y(O) = 1

Chapter 1: Problem 1 Elementary Differential Equations 6

In Problems 17 through 26, first verify that y(x) satisfies the given differential equation. Then determine a value of the constant C so that y(x) satisfies the given initial condition. Use a computer or graphing calculator (if desired) to sketch several typical solutions of the given differential equation, and highlight the one that satisfies the given initial condition. y' + y tanx = cosx; y(x) = (x + C)cosx, y(rr) = 0

Chapter 1: Problem 1 Elementary Differential Equations 6

In Problems 27 through 31, a function y = g (x) is described by some geometric property of its graph. Write a differential equation of the form dyjdx = f(x, y) having the function g as its solution (or as one of its solutions). The slope of the graph of g at the point (x, y) is the sum of x and y.

Chapter 1: Problem 1 Elementary Differential Equations 6

In Problems 27 through 31, a function y = g (x) is described by some geometric property of its graph. Write a differential equation of the form dyjdx = f(x, y) having the function g as its solution (or as one of its solutions). The line tangent to the graph of g at the point (x, y) intersects the x-axis at the point (xj2, 0).

Chapter 1: Problem 1 Elementary Differential Equations 6

In Problems 27 through 31, a function y = g (x) is described by some geometric property of its graph. Write a differential equation of the form dyjdx = f(x, y) having the function g as its solution (or as one of its solutions). Every straight line normal to the graph of g passes through the point (0, 1). Can you guess what the graph of such a function g might look like?

Chapter 1: Problem 1 Elementary Differential Equations 6

In Problems 27 through 31, a function y = g (x) is described by some geometric property of its graph. Write a differential equation of the form dyjdx = f(x, y) having the function g as its solution (or as one of its solutions). The graph of g is normal to every curve of the form y = x2 + k (k is a constant) where they meet.

Chapter 1: Problem 1 Elementary Differential Equations 6

In Problems 27 through 31, a function y = g (x) is described by some geometric property of its graph. Write a differential equation of the form dyjdx = f(x, y) having the function g as its solution (or as one of its solutions). The line tangent to the graph of g at (x, y) passes through the point (-y, x).

Chapter 1: Problem 1 Elementary Differential Equations 6

In Problems 32 through 36, write-in the manner of Eqs. (3) through (6) of this section-a differential equation that is a mathematical model of the situation described. The time rate of change of a population P is proportional to the square root of P.

Chapter 1: Problem 1 Elementary Differential Equations 6

In Problems 32 through 36, write-in the manner of Eqs. (3) through (6) of this section-a differential equation that is a mathematical model of the situation described. The time rate of change of the velocity v of a coasting motorboat is proportional'to the square of v.

Chapter 1: Problem 1 Elementary Differential Equations 6

In Problems 32 through 36, write-in the manner of Eqs. (3) through (6) of this section-a differential equation that is a mathematical model of the situation described. The acceleration dvjdt of a Lamborghini is proportional to the difference between 250 km/h and the velocity of the car.

Chapter 1: Problem 1 Elementary Differential Equations 6

In Problems 32 through 36, write-in the manner of Eqs. (3) through (6) of this section-a differential equation that is a mathematical model of the situation described. In a city having a fixed population of P persons, the time rate of change of the number N of those persons who have heard a certain rumor is proportional to the number of those who have not yet heard the rumor.

Chapter 1: Problem 1 Elementary Differential Equations 6

In Problems 32 through 36, write-in the manner of Eqs. (3) through (6) of this section-a differential equation that is a mathematical model of the situation described. In a city with a fixed population of P persons, the time rate of change of the number N of those persons infected with a certain contagious disease is proportional to the product of the number who have the disease and the number who do not.

Chapter 1: Problem 1 Elementary Differential Equations 6

In Problems 37 through 42, determine by inspection at least one solution of the given differential equation. That is, use your knowledge of derivatives to make an intelligent guess. Then test your hypothesis. y" = 0

Chapter 1: Problem 1 Elementary Differential Equations 6

In Problems 37 through 42, determine by inspection at least one solution of the given differential equation. That is, use your knowledge of derivatives to make an intelligent guess. Then test your hypothesis. y' = y

Chapter 1: Problem 1 Elementary Differential Equations 6

In Problems 37 through 42, determine by inspection at least one solution of the given differential equation. That is, use your knowledge of derivatives to make an intelligent guess. Then test your hypothesis. xy' + y = 3x2

Chapter 1: Problem 1 Elementary Differential Equations 6

In Problems 37 through 42, determine by inspection at least one solution of the given differential equation. That is, use your knowledge of derivatives to make an intelligent guess. Then test your hypothesis.. (y')2 + y2 = 1

Chapter 1: Problem 1 Elementary Differential Equations 6

In Problems 37 through 42, determine by inspection at least one solution of the given differential equation. That is, use your knowledge of derivatives to make an intelligent guess. Then test your hypothesis. y' + y = eX

Chapter 1: Problem 1 Elementary Differential Equations 6

In Problems 37 through 42, determine by inspection at least one solution of the given differential equation. That is, use your knowledge of derivatives to make an intelligent guess. Then test your hypothesis. y" + y = 0

Chapter 1: Problem 1 Elementary Differential Equations 6

(a) If k is a constant, show that a general (one-parameter) solution of the differential equation dx = kx2 dt is given by x(t) = lj(C -kt), where C is an arbitrary constant. (b) Determine by inspection a solution of the initial value problem x' = kx2, x (0) = O.

Chapter 1: Problem 1 Elementary Differential Equations 6

(a) Continuing Problem 43, assume that k is positive, and then sketch graphs of solutions of x' = kx2 with several typical positive values of x(O). (b) How would these solutions differ if the constant k were negative?

Chapter 1: Problem 1 Elementary Differential Equations 6

Suppose a population P of rodents satisfies the differential equation dPjdt = kp2. Initially, there are P (O) = 2 rodents, and their number is increasing at the rate of dPjdt = 1 rodent per month when there are P = 10 rodents. How long will it take for this population to grow to a hundred rodents? To a thousand? What's happening here?

Chapter 1: Problem 1 Elementary Differential Equations 6

Suppose the velocity v of a motorboat coasting in water satisfies the differential equation dvjdt = kv2. The initial speed of the motorboat is v(O) = 10 meters per second (mls), and v is decreasing at the rate of 1 mls2 when v = 5 mls. How long does it take for the velocity of the boat to decrease to 1 mls? To kmls? When does the boat come to a stop?

Chapter 1: Problem 1 Elementary Differential Equations 6

In Example 7 we saw that y(x) = lj(C - x) defines a one-parameter family of solutions of the differential equation dyjdx = y2. (a) Determine a value of C so that y(lO) = 1 0. (b) Is there a value of C such that y(O) = O? Can you nevertheless find by inspection a solution of dyjdx = y2 such that y(O) = O? (c) Figure 1.1.8 shows typical graphs of solutions of the form y(x) = lj(C -x). Does it appear that these solution curves fill the entire xyplane? Can you conclude that, given any point (a, b) in the plane, the differential equation dyjdx = y2 has exactly one solution y(x) satisfying the condition y(a) = b?

Chapter 1: Problem 1 Elementary Differential Equations 6

(a) Show that y(x) = CX4 defines a one-parameter family of differentiable solutions of the differential equation xy' = 4y (Fig. 1.1.9). (b) Show that y(x) = { _x4 if x < 0, X4 if x 0 defines a differentiable solution of xy' = 4y for all x, but is not of the form y(x) = CX4. (c) Given any two real numbers a and h, explain why-in contrast to the situation in part (c) of Problem 47-there exist infinitely many differentiable solutions of xy' = 4y that all satisfy the condition yea) = h.

Chapter 1: Problem 1 Elementary Differential Equations 6

In Problems 1 through 10, find a function y = f(x) satisfying the given differential equation and the prescribed initial condition. dy 1. - =2x+l ; y(0) =3 d

Chapter 1: Problem 1 Elementary Differential Equations 6

In Problems 1 through 10, find a function y = f(x) satisfying the given differential equation and the prescribed initial condition. dy = (x -2)2; y(2) = 1

Chapter 1: Problem 1 Elementary Differential Equations 6

In Problems 1 through 10, find a function y = f(x) satisfying the given differential equation and the prescribed initial condition. - = ..jX; y(4) =

Chapter 1: Problem 1 Elementary Differential Equations 6

In Problems 1 through 10, find a function y = f(x) satisfying the given differential equation and the prescribed initial condition. dx = x2 ; y(1) = 5

Chapter 1: Problem 1 Elementary Differential Equations 6

In Problems 1 through 10, find a function y = f(x) satisfying the given differential equation and the prescribed initial condition. y 1 5. - = ;::---;--;; ; y(2) = -1 d

Chapter 1: Problem 1 Elementary Differential Equations 6

In Problems 1 through 10, find a function y = f(x) satisfying the given differential equation and the prescribed initial condition. dx = X'\lX2 + 9; y(-4) = 0

Chapter 1: Problem 1 Elementary Differential Equations 6

In Problems 1 through 10, find a function y = f(x) satisfying the given differential equation and the prescribed initial condition. dx x2 + 1 y(O) =0

Chapter 1: Problem 1 Elementary Differential Equations 6

In Problems 1 through 10, find a function y = f(x) satisfying the given differential equation and the prescribed initial condition. dy 8. dx = cos 2x; y(O) = 1

Chapter 1: Problem 1 Elementary Differential Equations 6

In Problems 1 through 10, find a function y = f(x) satisfying the given differential equation and the prescribed initial condition. - = ; y(O) = 0 dx

Chapter 1: Problem 1 Elementary Differential Equations 6

In Problems 1 through 10, find a function y = f(x) satisfying the given differential equation and the prescribed initial condition. - = xe-x; y(O) = 1

Chapter 1: Problem 1 Elementary Differential Equations 6

In Problems 11 through 18, find the position function x(t) ofa moving particle with the given acceleration a(t), initial position Xo = x(O), and initial velocity Vo = v(O). a(t) = 50, Vo = 10, Xo = 20

Chapter 1: Problem 1 Elementary Differential Equations 6

In Problems 11 through 18, find the position function x(t) ofa moving particle with the given acceleration a(t), initial position Xo = x(O), and initial velocity Vo = v(O). a(t) = -20, Vo = -15, Xo = 5

Chapter 1: Problem 1 Elementary Differential Equations 6

In Problems 11 through 18, find the position function x(t) ofa moving particle with the given acceleration a(t), initial position Xo = x(O), and initial velocity Vo = v(O). a(t) = 3t, Vo = 5, xo = 0

Chapter 1: Problem 1 Elementary Differential Equations 6

In Problems 11 through 18, find the position function x(t) ofa moving particle with the given acceleration a(t), initial position Xo = x(O), and initial velocity Vo = v(O). a(t) = 2t + l, vo = -7, xo =4

Chapter 1: Problem 1 Elementary Differential Equations 6

In Problems 11 through 18, find the position function x(t) ofa moving particle with the given acceleration a(t), initial position Xo = x(O), and initial velocity Vo = v(O). a(t) = 4(t + 3)2, Vo = -1, Xo = 1

Chapter 1: Problem 1 Elementary Differential Equations 6

In Problems 11 through 18, find the position function x(t) ofa moving particle with the given acceleration a(t), initial position Xo = x(O), and initial velocity Vo = v(O). a(t) = ' Vo = -1, xo = 1 '\It +4

Chapter 1: Problem 1 Elementary Differential Equations 6

In Problems 11 through 18, find the position function x(t) ofa moving particle with the given acceleration a(t), initial position Xo = x(O), and initial velocity Vo = v(O). a(t) = 3 ' Vo = 0, Xo = 0

Chapter 1: Problem 1 Elementary Differential Equations 6

In Problems 11 through 18, find the position function x(t) ofa moving particle with the given acceleration a(t), initial position Xo = x(O), and initial velocity Vo = v(O). a(t) = 50sin5t, Vo = -10, Xo = 8

Chapter 1: Problem 1 Elementary Differential Equations 6

In Problems 19 through 22, a particle starts at the origin and travels along the x-axis with the velocity function v(t) whose graph is shown in Figs. 1.2.6 through 1.2. 9. Sketch the graph of the resulting position function x (t) for 0 :;;; t :;;; 10.

Chapter 1: Problem 1 Elementary Differential Equations 6

In Problems 19 through 22, a particle starts at the origin and travels along the x-axis with the velocity function v(t) whose graph is shown in Figs. 1.2.6 through 1.2. 9. Sketch the graph of the resulting position function x (t) for 0 :;;; t :;;; 10.

Chapter 1: Problem 1 Elementary Differential Equations 6

In Problems 19 through 22, a particle starts at the origin and travels along the x-axis with the velocity function v(t) whose graph is shown in Figs. 1.2.6 through 1.2. 9. Sketch the graph of the resulting position function x (t) for 0 :;;; t :;;; 10.

Chapter 1: Problem 1 Elementary Differential Equations 6

In Problems 19 through 22, a particle starts at the origin and travels along the x-axis with the velocity function v(t) whose graph is shown in Figs. 1.2.6 through 1.2. 9. Sketch the graph of the resulting position function x (t) for 0 :;;; t :;;; 10.

Chapter 1: Problem 1 Elementary Differential Equations 6

What is the maximum height attained by the arrow of part (b) of Example 3?

Chapter 1: Problem 1 Elementary Differential Equations 6

A ball is dropped from the top of a building 400 ft high. How long does it take to reach the ground? With what speed does the ball strike the ground?

Chapter 1: Problem 1 Elementary Differential Equations 6

The brakes of a car are applied when it is moving at 1 00 km/h and provide a constant deceleration of 10 meters per second per second (m/s2 ). How far does the car travel before coming to a stop?

Chapter 1: Problem 1 Elementary Differential Equations 6

A projectile is fired straight upward with an initial velocity of 1 00 m/s from the top of a building 20 m high and falls to the ground at the base of the building. Find (a) its maximum height above the ground; (b) when it passes the top of the building; (c) its total time in the air.

Chapter 1: Problem 1 Elementary Differential Equations 6

A ball is thrown straight downward from the top of a tall building. The initial speed of the ball is 10 m/s. It strikes the ground with a speed of 60 m/s. How tall is the building?

Chapter 1: Problem 1 Elementary Differential Equations 6

A baseball is thrown straight downward with an initial speed of 40 ft/s from the top of the Washington Monument (555 ft high). How long does it take to reach the ground, and with what speed does the baseball strike the ground?

Chapter 1: Problem 1 Elementary Differential Equations 6

A diesel car gradually speeds up so that for the first 10 s its acceleration is given by dv dt = (0. 1 2) t2 + (0.6)t (ft/s2 ). If the car starts from rest (xo = 0, Vo = 0), find the distance it has traveled at the end of the first 10 s and its velocity at that time.

Chapter 1: Problem 1 Elementary Differential Equations 6

A car traveling at 60 mi/h (88 ft/s) skids 1 76 ft after its brakes are suddenly applied. Under the assumption that the braking system provides constant deceleration, what is that deceleration? For how long does the skid continue?

Chapter 1: Problem 1 Elementary Differential Equations 6

The skid marks made by an automobile indicated that its brakes were fully applied for a distance of 75 m before it came to a stop. The car in question is known to have a constant deceleration of 20 m/s2 under these conditions. How fast-in km/h-was the car traveling when the brakes were first applied?

Chapter 1: Problem 1 Elementary Differential Equations 6

Suppose that a car skids 15 m if it is moving at 50 km/h when the brakes are applied. Assuming that the car has the same constant deceleration, how far will it skid if it is moving at 1 00 km/h when the brakes are applied?

Chapter 1: Problem 1 Elementary Differential Equations 6

On the planet Gzyx, a ball dropped from a height of 20 ft hits the ground in 2 s. If a ball is dropped from the top of a 200-ft-tall building on Gzyx, how long will it take to hit the ground? With what speed will it hit?

Chapter 1: Problem 1 Elementary Differential Equations 6

A person can throw a ball straight upward from the surface of the earth to a maximum height of 144 ft. How high could this person throw the ball on the planet Gzyx of Problem 29?

Chapter 1: Problem 1 Elementary Differential Equations 6

A stone is dropped from rest at an initial height h above the surface of the earth. Show that the speed with which it strikes the ground is v = J2gh.

Chapter 1: Problem 1 Elementary Differential Equations 6

Suppose a woman has enough "spring" in her legs to jump (on earth) from the ground to a height of 2.25 feet. If she jumps straight upward with the same initial velocity on the moon-where the surface gravitational acceleration is (approximately) 5.3 ftls2-how high above the surface will she rise?

Chapter 1: Problem 1 Elementary Differential Equations 6

At noon a car starts from rest at point A and proceeds at constant acceleration along a straight road toward point B. If the car reaches B at 1 2:50 P.M. with a velocity of 60 mi/h, what is the distance from A to B?

Chapter 1: Problem 1 Elementary Differential Equations 6

At noon a car starts from rest at point A and proceeds with constant acceleration along a straight road toward point C, 35 miles away. If the constantly accelerated car arrives at C with a velocity of 60 mi/h, at what time does it arrive at C?

Chapter 1: Problem 1 Elementary Differential Equations 6

If a = 0.5 mi and Vo = 9 mi/h as in Example 4, what must the swimmer's speed Vs be in order that he drifts only 1 mile downstream as he crosses the river?

Chapter 1: Problem 1 Elementary Differential Equations 6

Suppose that a = 0.5 mi, Vo = 9 mi/h, and Vs = 3 mi/h as in Example 4, but that the velocity of the river is given by the fourth-degree function rather than the quadratic function in Eq. (18). Now find how far downstream the swimmer drifts as he crosses the river.

Chapter 1: Problem 1 Elementary Differential Equations 6

A bomb is dropped from a helicopter hovering at an altitude of 800 feet above the ground. From the ground directly beneath the helicopter, a projectile is fired straight upward toward the bomb, exactly 2 seconds after the bomb is released. With what initial velocity should the projectile be fired, in order to hit the bomb at an altitude of exactly 400 feet?

Chapter 1: Problem 1 Elementary Differential Equations 6

A spacecraft is in free fall toward the surface of the moon at a speed of 1000 mph (mi/h). Its retrorockets, when fired, provide a constant deceleration of 20,000 mi/h2 . At what height above the lunar surface should the astronauts fire the retrorockets to insure a soft touchdown? (As in Example 2, ignore the moon's gravitational field.)

Chapter 1: Problem 1 Elementary Differential Equations 6

Arthur Clarke's The WindJrom the Sun ( 1963) describes Diana, a spacecraft propelled by the solar wind. Its aluminized sail provides it with a constant acceleration of O.oolg = 0.0098 m/s2 Suppose this spacecraft starts from rest at time t = 0 and simultaneously fires a projectile (straight ahead in the same direction) that travels at one-tenth of the speed c = 3 X 1 08 m/s oflight. How long will it take the spacecraft to catch up with the projectile, and how far will it have traveled by then?

Chapter 1: Problem 1 Elementary Differential Equations 6

A driver involved in an accident claims he was going only 25 mph. When police tested his car, they found that when its brakes were applied at 25 mph, the car skidded only 45 feet before coming to a stop. But the driver's skid marks at the accident scene measured 210 feet. Assuming the same (constant) deceleration, determine the speed he was actually traveling just prior to the accident.

Chapter 1: Problem 1 Elementary Differential Equations 6

In Problems 1 through 10, we have provided the slope field of the indicated differential equation, together with one or more solution curves. Sketch likely solution curves through the additional points marked in each slope field. . - = -y-smx dx

Chapter 1: Problem 1 Elementary Differential Equations 6

In Problems 1 through 10, we have provided the slope field of the indicated differential equation, together with one or more solution curves. Sketch likely solution curves through the additional points marked in each slope field. dy dx = x + y

Chapter 1: Problem 1 Elementary Differential Equations 6

In Problems 1 through 10, we have provided the slope field of the indicated differential equation, together with one or more solution curves. Sketch likely solution curves through the additional points marked in each slope field. . - =y-smx

Chapter 1: Problem 1 Elementary Differential Equations 6

In Problems 1 through 10, we have provided the slope field of the indicated differential equation, together with one or more solution curves. Sketch likely solution curves through the additional points marked in each slope field. dy - =x-y

Chapter 1: Problem 1 Elementary Differential Equations 6

In Problems 1 through 10, we have provided the slope field of the indicated differential equation, together with one or more solution curves. Sketch likely solution curves through the additional points marked in each slope field. dy - = Y -x+

Chapter 1: Problem 1 Elementary Differential Equations 6

In Problems 1 through 10, we have provided the slope field of the indicated differential equation, together with one or more solution curves. Sketch likely solution curves through the additional points marked in each slope field. dy - =x -y+l

Chapter 1: Problem 1 Elementary Differential Equations 6

In Problems 1 through 10, we have provided the slope field of the indicated differential equation, together with one or more solution curves. Sketch likely solution curves through the additional points marked in each slope field. dx - = smx+smy

Chapter 1: Problem 1 Elementary Differential Equations 6

In Problems 1 through 10, we have provided the slope field of the indicated differential equation, together with one or more solution curves. Sketch likely solution curves through the additional points marked in each slope field. dy - = x2 _y dx

Chapter 1: Problem 1 Elementary Differential Equations 6

In Problems 1 through 10, we have provided the slope field of the indicated differential equation, together with one or more solution curves. Sketch likely solution curves through the additional points marked in each slope field. dy - =x 2-y-2 dx

Chapter 1: Problem 1 Elementary Differential Equations 6

In Problems 1 through 10, we have provided the slope field of the indicated differential equation, together with one or more solution curves. Sketch likely solution curves through the additional points marked in each slope field. - = -x +smy dx

Chapter 1: Problem 1 Elementary Differential Equations 6

In Problems I I through 20, determine whether Theorem I does or does not guarantee existence of a solution of the given initial value problem. If existence is guaranteed, determine whether Theorem I does or does not guarantee uniqueness of that solution. dy = 2x2y2; dx y(l) = -1

Chapter 1: Problem 1 Elementary Differential Equations 6

In Problems I I through 20, determine whether Theorem I does or does not guarantee existence of a solution of the given initial value problem. If existence is guaranteed, determine whether Theorem I does or does not guarantee uniqueness of that solution. dy - = xlny; y(1) = 1 dx

Chapter 1: Problem 1 Elementary Differential Equations 6

In Problems I I through 20, determine whether Theorem I does or does not guarantee existence of a solution of the given initial value problem. If existence is guaranteed, determine whether Theorem I does or does not guarantee uniqueness of that solution. dy - = -W; y(O) =

Chapter 1: Problem 1 Elementary Differential Equations 6

In Problems I I through 20, determine whether Theorem I does or does not guarantee existence of a solution of the given initial value problem. If existence is guaranteed, determine whether Theorem I does or does not guarantee uniqueness of that solution. y 14. dx = -W; y(O) = 0

Chapter 1: Problem 1 Elementary Differential Equations 6

In Problems I I through 20, determine whether Theorem I does or does not guarantee existence of a solution of the given initial value problem. If existence is guaranteed, determine whether Theorem I does or does not guarantee uniqueness of that solution. dy - = "';x-y; y(2) = 2

Chapter 1: Problem 1 Elementary Differential Equations 6

In Problems I I through 20, determine whether Theorem I does or does not guarantee existence of a solution of the given initial value problem. If existence is guaranteed, determine whether Theorem I does or does not guarantee uniqueness of that solution. y dx = "';x -y; y(2) = 1

Chapter 1: Problem 1 Elementary Differential Equations 6

In Problems I I through 20, determine whether Theorem I does or does not guarantee existence of a solution of the given initial value problem. If existence is guaranteed, determine whether Theorem I does or does not guarantee uniqueness of that solution. y- =x - 1 ; y(O) =

Chapter 1: Problem 1 Elementary Differential Equations 6

In Problems I I through 20, determine whether Theorem I does or does not guarantee existence of a solution of the given initial value problem. If existence is guaranteed, determine whether Theorem I does or does not guarantee uniqueness of that solution. y- =x- l; y(1) =

Chapter 1: Problem 1 Elementary Differential Equations 6

In Problems I I through 20, determine whether Theorem I does or does not guarantee existence of a solution of the given initial value problem. If existence is guaranteed, determine whether Theorem I does or does not guarantee uniqueness of that solution. dy _ = In(1 + y2) ; y(O) = 0

Chapter 1: Problem 1 Elementary Differential Equations 6

In Problems I I through 20, determine whether Theorem I does or does not guarantee existence of a solution of the given initial value problem. If existence is guaranteed, determine whether Theorem I does or does not guarantee uniqueness of that solution. dy _ =x2 _ y2 y(O) =

Chapter 1: Problem 1 Elementary Differential Equations 6

In Problems 21 and 22, first use the method of Example 2 to construct a slope field for the given differential equation. Then sketch the solution curve corresponding to the given initial condition. Finally, use this solution curve to estimate the desired value of the solution y(x). y' =x+y, y(O) = 0; y(-4) =?

Chapter 1: Problem 1 Elementary Differential Equations 6

In Problems 21 and 22, first use the method of Example 2 to construct a slope field for the given differential equation. Then sketch the solution curve corresponding to the given initial condition. Finally, use this solution curve to estimate the desired value of the solution y(x). y' = y -x, y(4) = 0; y(-4) =?

Chapter 1: Problem 1 Elementary Differential Equations 6

Problems 23 and 24 are like Problems 21 and 22, but now use a computer algebra system to plot and print out a slope fieldfor the given differential equation. lfyou wish (and know how), you can check your manually sketched solution curve by plotting it with the computer. y' = x2 + y2 - 1, yeO) = 0; y(2) = ?

Chapter 1: Problem 1 Elementary Differential Equations 6

Problems 23 and 24 are like Problems 21 and 22, but now use a computer algebra system to plot and print out a slope fieldfor the given differential equation. lfyou wish (and know how), you can check your manually sketched solution curve by plotting it with the computer. y' = x + "2?2, y(-2) = 0; y(2) =?

Chapter 1: Problem 1 Elementary Differential Equations 6

You bail out of the helicopter of Example 3 and pull the ripcord of your parachute. Now k = 1 .6 in Eq. (3), so your downward velocity satisfies the initial value problem dv dt = 32 - 1 .6v, v(O) = O. In order to investigate your chances of survival, construct a slope field for this differential equation and sketch the appropriate solution curve. What will your limiting velocity be? Will a strategically located haystack do any good? How long will it take you to reach 95% of your limiting velocity?

Chapter 1: Problem 1 Elementary Differential Equations 6

Suppose the deer population P(t)in a small forest satisfies the logistic equation dP dt = 0.0225P -0.0003P2. Construct a slope field and appropriate solution curve to answer the following questions: If there are 25 deer at time t = 0 and t is measured in months, how long will it take the number of deer to double? What will be the limiting deer population? The next seven problems illustrate the fact that, if the hypotheses of Theorem 1 are not satisfied, then the initial value problem y' = f(x, y), yea) = b may have either no solutions, finitely many solutions, or infinitely many solutions.

Chapter 1: Problem 1 Elementary Differential Equations 6

(a) Verify that if c is a constant, then the function defined piecewise by y(x) = {x _ C)2 for x c, for x > c satisfies the differential equation y' = 2..jY for all x (including the point x = c). Construct a figure illustrating the fact that the initial value problem y' = 2..jY, yeO) = 0 has infinitely many different solutions. (b) For what values of b does the initial value problem y' = 2..jY, yeO) = b have (i) no solution, (ii) a unique solution that is defined for all x?

Chapter 1: Problem 1 Elementary Differential Equations 6

Verify that if k is a constant, then the function y(x) == kx satisfies the differential equation xy' = y for all x. Construct a slope field and several of these straight line solution curves. Then determine (in terms of a and b) how many different solutions the initial value problem xy' = y, yea) = b has-one, none, or infinitely many.

Chapter 1: Problem 1 Elementary Differential Equations 6

Verify that if c is a constant, then the function defined piecewise by y(x) = { o (x - c) 3 for x c, for x > c satisfies the differential equation y' = 3y2 /3 for all x. Can you also use the "left half" of the cubic y = (x -c)3 in piecing together a solution curve of the differential equation? (See Fig. 1.3.25.) Sketch a variety of such solution curves. Is there a point (a, b) of the xy-plane such that the initial value problem y' = 3y2/3, yea) = b has either no solution or a unique solution that is defined for all x? Reconcile your answer with Theorem 1.

Chapter 1: Problem 1 Elementary Differential Equations 6

Verify that if c is a constant, then the function defined piecewise by 1 +1 y(x) = cos(x - c) -1 if x c, ifc < x < c+Jl', if x c + Jl' satisfies the differential equation y' = -.Ji=Y2 for all x. (Perhaps a preliminary sketch with c = 0 will be helpful.) Sketch a variety of such solution curves. Then determine (in terms of a and b) how many different solutions the initial value problem y' = -, yea) = b has.

Chapter 1: Problem 1 Elementary Differential Equations 6

Carry out an investigation similar to that in Problem 30, except with the differential equation y' = +. Does it suffice simply to replace cos(x -c) with sin (x - c) in piecing together a solution that is defined for all x?

Chapter 1: Problem 1 Elementary Differential Equations 6

Verify that if c > 0, then the function defined piecewise by { O f 2 < ( ) 1 X = c, y x = (x2 -C)2 if x2 > c satisfies the differential equation y' = 4x..jY for all x. Sketch a variety of such solution curves for different values of c. Then determine (in terms of a and b) how many different solutions the initial value problem y' = 4x.JY, yea) = b has.

Chapter 1: Problem 1 Elementary Differential Equations 6

If c oft 0, verify that the function defined by y(x) = xl(cx - 1) (with graph illustrated in Fig. 1 .3.26) satisfies the differential equation x2y' + y2 = if x oft 1/c. Sketch a variety of such solution curves for different values of c. Also, note the constant-valued function y(x) == that does not result from any choice of the constant c. Finally, determine (in terms of a and b) how many different solutions the initial value problem x2y' + y2 = 0, y(a) = b has.

Chapter 1: Problem 1 Elementary Differential Equations 6

(a) Use the direction field of Problem 5 to estimate the values at x = 1 of the two solutions of the differential equation y' = y - x + 1 with initial values y(-l) = -1.2 and y(-l) = -0.8. (b) Use a computer algebra system to estimate the values at x = 3 of the two solutions of this differential equation with initial values y( -3) = -3.01 and y( -3) = -2.99. The lesson of this problem is that small changes in initial conditions can make big differences in results.

Chapter 1: Problem 1 Elementary Differential Equations 6

(a) Use the direction field of Problem 6 to estimate the values at x = 2 of the two solutions of the differential equation y' = x - y + 1 with initial values y( -3) = -0.2 and y( -3) = +0.2. (b) Use a computer algebra system to estimate the values at x = 3 of the two solutions of this differential equation with initial values y( -3) = -0.5 and y(-3) = +0.5. The lesson of this problem is that big changes in initial conditions may make only small differences in results.

Chapter 1: Problem 1 Elementary Differential Equations 6

Find general solutions (implicit ifnecessary, explicit if convenient) of the differential equations in Problems 1 through 18. Primes denote derivatives with respect to x. dx + 2xy = 0

Chapter 1: Problem 1 Elementary Differential Equations 6

Find general solutions (implicit ifnecessary, explicit if convenient) of the differential equations in Problems 1 through 18. Primes denote derivatives with respect to x. dy - + 2xy2 =

Chapter 1: Problem 1 Elementary Differential Equations 6

Find general solutions (implicit ifnecessary, explicit if convenient) of the differential equations in Problems 1 through 18. Primes denote derivatives with respect to x. dy . 3. dx = y sm x

Chapter 1: Problem 1 Elementary Differential Equations 6

Find general solutions (implicit ifnecessary, explicit if convenient) of the differential equations in Problems 1 through 18. Primes denote derivatives with respect to x. dy (l +x)- = 4y dx

Chapter 1: Problem 1 Elementary Differential Equations 6

Find general solutions (implicit ifnecessary, explicit if convenient) of the differential equations in Problems 1 through 18. Primes denote derivatives with respect to x. ..[X dy = j

Chapter 1: Problem 1 Elementary Differential Equations 6

Find general solutions (implicit ifnecessary, explicit if convenient) of the differential equations in Problems 1 through 18. Primes denote derivatives with respect to x. dx = 3.

Chapter 1: Problem 1 Elementary Differential Equations 6

Find general solutions (implicit ifnecessary, explicit if convenient) of the differential equations in Problems 1 through 18. Primes denote derivatives with respect to x. dy 7. - = (64xy) I/3

Chapter 1: Problem 1 Elementary Differential Equations 6

Find general solutions (implicit ifnecessary, explicit if convenient) of the differential equations in Problems 1 through 18. Primes denote derivatives with respect to x. dx = 2x sec y

Chapter 1: Problem 1 Elementary Differential Equations 6

Find general solutions (implicit ifnecessary, explicit if convenient) of the differential equations in Problems 1 through 18. Primes denote derivatives with respect to x. (l - x2) dy = 2y dx

Chapter 1: Problem 1 Elementary Differential Equations 6

Find general solutions (implicit ifnecessary, explicit if convenient) of the differential equations in Problems 1 through 18. Primes denote derivatives with respect to x. l + X)2 dy = (l + y)

Chapter 1: Problem 1 Elementary Differential Equations 6

Find general solutions (implicit ifnecessary, explicit if convenient) of the differential equations in Problems 1 through 18. Primes denote derivatives with respect to x. y' = xy3

Chapter 1: Problem 1 Elementary Differential Equations 6

Find general solutions (implicit ifnecessary, explicit if convenient) of the differential equations in Problems 1 through 18. Primes denote derivatives with respect to x. yy' = X(y2 + 1)

Chapter 1: Problem 1 Elementary Differential Equations 6

Find general solutions (implicit ifnecessary, explicit if convenient) of the differential equations in Problems 1 through 18. Primes denote derivatives with respect to x. y3 - = (y4 + 1) cos x

Chapter 1: Problem 1 Elementary Differential Equations 6

Find general solutions (implicit ifnecessary, explicit if convenient) of the differential equations in Problems 1 through 18. Primes denote derivatives with respect to x. dy 1 +..[X dx 1 +,

Chapter 1: Problem 1 Elementary Differential Equations 6

Find general solutions (implicit ifnecessary, explicit if convenient) of the differential equations in Problems 1 through 18. Primes denote derivatives with respect to x. dy = (x - 1)y5 dx x2(2y3 -y)

Chapter 1: Problem 1 Elementary Differential Equations 6

Find general solutions (implicit ifnecessary, explicit if convenient) of the differential equations in Problems 1 through 18. Primes denote derivatives with respect to x. (x2 + l) (tan y)y' = x

Chapter 1: Problem 1 Elementary Differential Equations 6

Find general solutions (implicit ifnecessary, explicit if convenient) of the differential equations in Problems 1 through 18. Primes denote derivatives with respect to x. y' = l+x+y+xy (Suggestion: Factor the right-hand side.)

Chapter 1: Problem 1 Elementary Differential Equations 6

Find general solutions (implicit ifnecessary, explicit if convenient) of the differential equations in Problems 1 through 18. Primes denote derivatives with respect to x. x2y' = 1 -x2 + y2 _ x2y2

Chapter 1: Problem 1 Elementary Differential Equations 6

Find explicit particular solutions of the initial value problems in Problems 19 through 28. dy 19. dx = yeX, yeO) = 2e

Chapter 1: Problem 1 Elementary Differential Equations 6

Find explicit particular solutions of the initial value problems in Problems 19 through 28. dx = 3x2(y2 + 1), yeO) = 1

Chapter 1: Problem 1 Elementary Differential Equations 6

Find explicit particular solutions of the initial value problems in Problems 19 through 28. dy x 21. 2y- = y(5) = 2 dx v'x2 - 16

Chapter 1: Problem 1 Elementary Differential Equations 6

Find explicit particular solutions of the initial value problems in Problems 19 through 28. dy 22. dx = 4x3y - y, y(l) = -3

Chapter 1: Problem 1 Elementary Differential Equations 6

Find explicit particular solutions of the initial value problems in Problems 19 through 28. 23. - + 1 = 2y, y(l) = 1 dx

Chapter 1: Problem 1 Elementary Differential Equations 6

Find explicit particular solutions of the initial value problems in Problems 19 through 28. dy 24. (tan x) dx = y, y On) = n

Chapter 1: Problem 1 Elementary Differential Equations 6

Find explicit particular solutions of the initial value problems in Problems 19 through 28. dy 25. x- - y = 2x2y, y(l) = 1 dx

Chapter 1: Problem 1 Elementary Differential Equations 6

Find explicit particular solutions of the initial value problems in Problems 19 through 28. dx = 2xy2 + 3x2y2 , y(l) = -1

Chapter 1: Problem 1 Elementary Differential Equations 6

Find explicit particular solutions of the initial value problems in Problems 19 through 28. 27. = 6e2x-y, yeO) = 0

Chapter 1: Problem 1 Elementary Differential Equations 6

Find explicit particular solutions of the initial value problems in Problems 19 through 28. dy 28. 2..[X - = cos2 y, y(4) = nj4

Chapter 1: Problem 1 Elementary Differential Equations 6

(a) Find a general solution of the differential equation dyjdx = y2. (b) Find a singular solution that is not included in the general solution. (c) Inspect a sketch of typical solution curves to determine the points (a, b) for which the initial value problem y' = y2 , y(a) = b has a unique solution.

Chapter 1: Problem 1 Elementary Differential Equations 6

Solve the differential equation (dyjdx)2 = 4y to verify the general solution curves and singular solution curve that are illustrated in Fig. 1 .4.5. Then determine the points (a, b) in the plane for which the initial value problem (y') 2 = 4y, yea) = b has (a) no solution, (b) infinitely many solutions that are defined for all x, (c) on some neighborhood of the point x = a, only finitely many solutions.

Chapter 1: Problem 1 Elementary Differential Equations 6

Discuss the difference between the differential equations (dyjdx) 2 = 4y and dyjdx = 2,JY. Do they have the same solution curves? Why or why not? Determine the points (a, b) in the plane for which the initial value problem y' = 2,JY, yea) = b has (a) no solution, (b) a unique solution, (c) infinitely many solutions.

Chapter 1: Problem 1 Elementary Differential Equations 6

Find a general solution and any singular solutions of the differential equation dymyslashdx = yJY2=1. Determine the points (a, b) in the plane for which the initial value problem y' = yJY2="1, yea) = b has (a) no solution, (b) a unique solution, (c) infinitely many solutions.

Chapter 1: Problem 1 Elementary Differential Equations 6

(Population growth) A certain city had a population of 25000 in 1960 and a population of 30000 in 1970. Assume that its population will continue to grow exponentially at a constant rate. What population can its city planners expect in the year 2000?

Chapter 1: Problem 1 Elementary Differential Equations 6

(Population growth) In a certain culture of bacteria, the number of bacteria increased sixfold in 10 h. How long did it take for the population to double?

Chapter 1: Problem 1 Elementary Differential Equations 6

(Radiocarbon dating) Carbon extracted from an ancient skull contained only one-sixth as much 14C as carbon extracted from present-day bone. How old is the skull?

Chapter 1: Problem 1 Elementary Differential Equations 6

(Radiocarbon dating) Carbon taken from a purported relic of the time of Christ contained 4.6 x 1010 atoms of 14C per gram. Carbon extracted from a present-day specimen of the same substance contained 5.0 x 1010 atoms of 14C per gram. Compute the approximate age of the relic. What is your opinion as to its authenticity?

Chapter 1: Problem 1 Elementary Differential Equations 6

(Continuously compounded interest) Upon the birth of their first child, a couple deposited $5000 in an account that pays 8% interest compounded continuously. The interest payments are allowed to accumulate. How much will the account contain on the child's eighteenth birthday?

Chapter 1: Problem 1 Elementary Differential Equations 6

Continuously compounded interest) Suppose that you discover in your attic an overdue library book on which your grandfather owed a fine of 30 cents 100 years ago. If an overdue fine grows exponentially at a 5% annual rate compounded continuously, how much would you have to pay if you returned the book today?

Chapter 1: Problem 1 Elementary Differential Equations 6

(Drug elimination) Suppose that sodium pentobarbital is used to anesthetize a dog. The dog is anesthetized when its bloodstream contains at least 45 milligrams (mg) of sodium pentobarbitol per kilogram of the dog's body weight. Suppose also that sodium pentobarbitol is eliminated exponentially from the dog's bloodstream, with a half-life of 5 h. What single dose should be administered in order to anesthetize a 50-kg dog for 1 h?

Chapter 1: Problem 1 Elementary Differential Equations 6

The half-life of radioactive cobalt is 5.27 years. Suppose that a nuclear accident has left the level of cobalt radiation in a certain region at 100 times the level acceptable for human habitation. How long will it be until the region is again habitable? (Ignore the probable presence of other radioactive isotopes.)

Chapter 1: Problem 1 Elementary Differential Equations 6

Suppose that a mineral body formed in an ancient cataclysm-perhaps the formation of the earth itselforiginally contained the uranium isotope 238U (which has a half-life of 4.51 x 109 years) but no lead, the end product of the radioactive decay of 238U. If today the ratio of 238U atoms to lead atoms in the mineral body is 0.9, when did the cataclysm occur?

Chapter 1: Problem 1 Elementary Differential Equations 6

A certain moon rock was found to contain equal numbers of potassium and argon atoms. Assume that all the argon is the result of radioactive decay of potassium (its half-life is about 1 .28 x 109 years) and that one of every nine potassium atom disintegrations yields an argon atom. What is the age of the rock, measured from the time it contained only potassium?

Chapter 1: Problem 1 Elementary Differential Equations 6

A pitcher of buttermilk initially at 25C is to be cooled by setting it on the front porch, where the temperature is 0 C. Suppose that the temperature of the buttermilk has dropped to 15C after 20 min. When will it be at 5C?

Chapter 1: Problem 1 Elementary Differential Equations 6

When sugar is dissolved in water, the amount A that remains undissolved after t minutes satisfies'the differential equation dAfdt = -kA (k > 0). If 25% of the sugar dissolves after 1 min, how long does it take for half of the sugar to dissolve?

Chapter 1: Problem 1 Elementary Differential Equations 6

The intensity 1 of light at a depth of x meters below the surface of a lake satisfies the differential equation d I/dx = (- 1 .4) I. (a) At what depth is the intensity half the intensity 10 at the surface (where x = O)? (b) What is the intensity at a depth of 10 m (as a fraction of lo)? (c) At what depth will the intensity be 1 % of that at the surface?

Chapter 1: Problem 1 Elementary Differential Equations 6

The barometric pressure p (in inches of mercury) at an altitude x miles above sea level satisfies the initial value problem dp/dx = (-0.2) p, p(O) = 29.92. (a) Calculate the barometric pressure at 10,000 ft and again at 30,000 ft. (b) Without prior conditioning, few people can survive when the pressure drops to less than 15 in. of mercury. How high is that?

Chapter 1: Problem 1 Elementary Differential Equations 6

A certain piece of dubious information about phenylethylamine in the drinking water began to spread one day in a city with a population of 100,000. Within a week, 10,000 people had heard this rumor. Assume that the rate of increase of the number who have heard the rumor is proportional to the number who have not yet heard it. How long will it be until half the population of the city has heard the rumor?

Chapter 1: Problem 1 Elementary Differential Equations 6

According to one cosmological theory, there were equal amounts of the two uranium isotopes 235U and 238U at the creation of the universe in the "big bang." At present there are 137.7 atoms of 238U for each atom of 235U. Using the half-lives 4.51 x 109 years for 238U and 7.10 x 108 years for 235U, calculate the age of the universe.

Chapter 1: Problem 1 Elementary Differential Equations 6

A cake is removed from an oven at 210F and left to cool at room temperature, which is 70F. After 30 min the temperature of the cake is 140 F. When will it be 1 00F?

Chapter 1: Problem 1 Elementary Differential Equations 6

The amount A(t) of atmospheric pollutants in a certain mountain valley grows naturally and is tripling every 7.5 years. (a) If the initial amount is 10 pu (pollutant units), write a formula for A(t) giving the amount (in pu) present after t years. (b) What will be the amount (in pu) of pollutants present in the valley atmosphere after 5 years? (c) If it will be dangerous to stay in the valley when the amount of pollutants reaches 100 pu, how long will this take?

Chapter 1: Problem 1 Elementary Differential Equations 6

An accident at a nuclear power plant has left the surrounding area polluted with radioactive material that decays naturally. The initial amount of radioactive material present is 15 su (safe units), and 5 months later it is still 10 suo (a) Write a formula giving the amount A (t) of radioactive material (in su) remaining after t months. (b) What amount of radioactive material will remain after 8 months? (c) How long-total number of months or fraction thereof-will it be until A = 1 su, so it is safe for people to return to the area?

Chapter 1: Problem 1 Elementary Differential Equations 6

There are now about 3300 different human "language families" in the whole world. Assume that all these are derived from a single original language, and that a language family develops into 1 .5 language families every 6 thousand years. About how long ago was the single original human language spoken?

Chapter 1: Problem 1 Elementary Differential Equations 6

Thousands of years ago ancestors ofthe Native Americans crossed the Bering Strait from Asia and entered the western hemisphere. Since then, they have fanned out across North and South America. The single language that the original Native Americans spoke has since split into many Indian "language families." Assume (as in Problem 52) that the number of these language families has been multiplied by 1 .5 every 6000 years. There are now 150 Native American language families in the western hemisphere. About when did the ancestors of today's Native Americans arrive?

Chapter 1: Problem 1 Elementary Differential Equations 6

A tank is shaped like a vertical cylinder; it initially contains water to a depth of9 ft, and a bottom plug is removed at time t = 0 (hours). After 1 h the depth of the water has dropped to 4 ft. How long does it take for all the water to drain from the tank?

Chapter 1: Problem 1 Elementary Differential Equations 6

Suppose that the tank of Problem 48 has a radius of 3 ft and that its bottom hole is circular with radius 1 in. How long will it take the water (initially 9 ft deep) to drain completely?

Chapter 1: Problem 1 Elementary Differential Equations 6

At time t = 0 the bottom plug (at the vertex) of a full conical water tank 16 ft high is removed. After 1 h the water in the tank is 9 ft deep. When will the tank be empty?

Chapter 1: Problem 1 Elementary Differential Equations 6

Suppose that a cylindrical tank initially containing Vo gallons of water drains (through a bottom hole) in T minutes. Use Torricelli's law to show that the volume of water in the tank after t T minutes is V = Vo [1 - (t/T)]2 .

Chapter 1: Problem 1 Elementary Differential Equations 6

A water tank has the shape obtained by revolving the curve y = X4/3 around the y-axis. A plug at the bottom is removed at 12 noon, when the depth of water in the tank is 12 ft. At 1 P.M. the depth of the water is 6 ft. When will the tank be empty?

Chapter 1: Problem 1 Elementary Differential Equations 6

A water tank has the shape obtained by revolving the parabola x2 = by around the y-axis. The water depth is 4 ft at 12 noon, when a circular plug in the bottom of the tank is removed. At 1 P.M. the depth of the water is 1 ft. (a) Find the depth y(t) of water remaining after t hours. (b) When will the tank be empty? (c) If the initial radius of the top surface of the water is 2 ft, what is the radius of the circular hole in the bottom?

Chapter 1: Problem 1 Elementary Differential Equations 6

A cylindrical tank with length 5 ft and radius 3 ft is situated with its axis horizontal. If a circular bottom hole with a radius of 1 in. is opened and the tank is initially half full of xylene, how long will it take for the liquid to drain completely?

Chapter 1: Problem 1 Elementary Differential Equations 6

A spherical tank of radius 4 ft is full of gasoline when a circular bottom hole with radius 1 in. is opened. How long will be required for all the gasoline to drain from the tank?

Chapter 1: Problem 1 Elementary Differential Equations 6

Suppose that an initially full hemispherical water tank of radius 1 m has its flat side as its bottom. It has a bottom hole of radius 1 cm. If this bottom hole is opened at 1 P.M., when will the tank be empty?

Chapter 1: Problem 1 Elementary Differential Equations 6

Consider the initially full hemispherical water tank of Example 8, except that the radius r of its circular bottom hole is now unknown. At 1 P.M. the bottom hole is opened and at 1 :30 P.M. the depth of water in the tank is 2 ft. (a) Use Torricelli's law in the form dV /dt = - (0.6)Jrr2 J2gy (taking constriction into account) to determine when the tank will be empty. (b) What is the radius of the bottom hole?

Chapter 1: Problem 1 Elementary Differential Equations 6

(The clepsydra, or water clock) A 12-h water clock is to be designed with the dimensions shown in Fig. 1 .4. 10, shaped like the surface obtained by revolving the curve y = I(x) around the y-axis. What should be this curve, and what should be the radius of the circular bottom hole, in order that the water level will fall at the constant rate of 4 inches per hour (in./h)?

Chapter 1: Problem 1 Elementary Differential Equations 6

Just before midday the body of an apparent homicide victim is found in a room that is kept at a constant temperature of 70F. At 12 noon the temperature of the body is 80F and at 1 P.M. it is 75F. Assume that the temperature of the body at the time of death was 98.6F and that it has cooled in accord with Newton's law. What was the time of death?

Chapter 1: Problem 1 Elementary Differential Equations 6

Early one morning it began to snow at a constant rate. At 7 A.M. a snowplow set off to clear a road. By 8 A.M. it had traveled 2 miles, but it took two more hours (until 10 A.M.) for the snowplow to go an additional 2 miles. (a) Let t = 0 when it began to snow and let x denote the distance traveled by the snowplow at time t. Assuming that the snowplow clears snow from the road at a constant rate (in cubic feet per hour, say), show that dx 1 k- = - dt t where k is a constant. (b) What time did it start snowing? (Answer: 6 A.M.)

Chapter 1: Problem 1 Elementary Differential Equations 6

A snowplow sets off at 7 A.M. as in Problem 66. Suppose now that by 8 A.M. it had traveled 4 miles and that by 9 A.M. it had moved an additional 3 miles. What time did it start snowing? This is a more difficult snowplow problem because now a transcendental equation must be solved numerically to find the value of k. (Answer: 4:27 A.M.)

Chapter 1: Problem 1 Elementary Differential Equations 6

Figure 1 .4.11 shows a bead sliding down a frictionless wire from point P to point Q. The brachistochrone problem asks what shape the wire should be in order to minimize the bead's time of descent from P to Q. In June of 1696, John Bernoulli proposed this problem as a public challenge, with a 6-month deadline (later extended to Easter 1697 at George Leibniz's request). Isaac Newton, then retired from academic life and serving as Warden of the Mint in London, received Bernoulli's challenge on January 29, 1697. The very next day he communicated his own solution-the curve of minimal descent time is an arc of an inverted cycloid-to the Royal Society of London. For a modem derivation of this result, suppose the bead starts from rest at the origin P and let y = y(x) be the equation of the desired curve in a coordinate system with the y-axis pointing downward. Then a mechanical analogue of Snell's law in optics implies that --sin a = constant, v (i) where a denotes the angle of deflection (from the vertical) of the tangent line to the curve-so cot a = y' (x) (why?)-and v = .,J2gy is the bead's velocity when it has descended a distance y vertically (from KE = mv2 mgy = -PE). p Q FIGURE 1.4.11. A bead sliding down a wire-the brachistochrone problem. (a) First derive from Eq. (i) the differential equation dy _ J 2a - y dx - y where a is an appropriate positive constant. (ii) (b) Substitute y = 2a sin2 t, dy = 4a sint cos tdt in (ii) to derive the solution x = a(2t - sin 2t) , y = a(l - cos 2t) (iii) for which t = Y = 0 when x = O. Finally, the substitution of e = 2a in (iii) yields the standard parametric equations x = a(e - sin e), y = a(l - cos e) I11III Linear First-Order E9ation 69. of the cycloid that is generated by a point on the rim of a circular wheel of radius a as it rolls along the xaxis. [See Example 5 in Section 9.4 of Edwards and Penney, Calculus: Early Transcendentals, 7th edition (Upper Saddle River, NJ: Prentice Hall, 2008).]

Chapter 1: Problem 1 Elementary Differential Equations 6

Suppose a uniform flexible cable is suspended between two points (L, H) at equal heights located symmetrically on either side of the x-axis (Fig. 1 .4. 12). Principles of physics can be used to show that the shape y = y(x) of the hanging cable satisfies the differential equation (dy) 2 1 + dx ' where the constant a = T/p is the ratio of the cable's tension T at its lowest point x = 0 (where y'(0) = 0 ) and its (constant) linear density p. If we substitute v = dymyslashdx, dv/dx = d2y/dx2 in this secondorder differential equation, we get the first-order equation dv a- = v 1 + v2 dx Solve this differential equation for y'(x) = vex) sinh(x/a). Then integrate to get the shape function y(x) = a cosh () + C of the hanging cable. This curve is called a catenary, from the Latin word for chain. Y (-L, H) (L. H) Yo x FIGURE 1.4.12. The catenary.

Chapter 1: Problem 1 Elementary Differential Equations 6

Find general solutions of the differential equations in Problems 1 through 25. If an initial condition is given, find the corresponding particular solution. Throughout, primes denote derivatives with respect to x. y' + y = 2, y(O) = 0

Chapter 1: Problem 1 Elementary Differential Equations 6

Find general solutions of the differential equations in Problems 1 through 25. If an initial condition is given, find the corresponding particular solution. Throughout, primes denote derivatives with respect to x. y' - 2y = 3e2x , y(O) = 0

Chapter 1: Problem 1 Elementary Differential Equations 6

Find general solutions of the differential equations in Problems 1 through 25. If an initial condition is given, find the corresponding particular solution. Throughout, primes denote derivatives with respect to x. y' + 3y = 2xe-3x

Chapter 1: Problem 1 Elementary Differential Equations 6

Find general solutions of the differential equations in Problems 1 through 25. If an initial condition is given, find the corresponding particular solution. Throughout, primes denote derivatives with respect to x. y' - 2xy = ex2

Chapter 1: Problem 1 Elementary Differential Equations 6

Find general solutions of the differential equations in Problems 1 through 25. If an initial condition is given, find the corresponding particular solution. Throughout, primes denote derivatives with respect to x. xy' + 2y = 3x, y(1) = 5

Chapter 1: Problem 1 Elementary Differential Equations 6

Find general solutions of the differential equations in Problems 1 through 25. If an initial condition is given, find the corresponding particular solution. Throughout, primes denote derivatives with respect to x. xy' + 5y = 7x2, y(2) = 5

Chapter 1: Problem 1 Elementary Differential Equations 6

Find general solutions of the differential equations in Problems 1 through 25. If an initial condition is given, find the corresponding particular solution. Throughout, primes denote derivatives with respect to x. 2xy' + y = 1O.JX

Chapter 1: Problem 1 Elementary Differential Equations 6

Find general solutions of the differential equations in Problems 1 through 25. If an initial condition is given, find the corresponding particular solution. Throughout, primes denote derivatives with respect to x. 3xy' + y = 12x

Chapter 1: Problem 1 Elementary Differential Equations 6

Find general solutions of the differential equations in Problems 1 through 25. If an initial condition is given, find the corresponding particular solution. Throughout, primes denote derivatives with respect to x. xy' - y = x, y(1) = 7

Chapter 1: Problem 1 Elementary Differential Equations 6

Find general solutions of the differential equations in Problems 1 through 25. If an initial condition is given, find the corresponding particular solution. Throughout, primes denote derivatives with respect to x. 2xy' - 3y = 9x3

Chapter 1: Problem 1 Elementary Differential Equations 6

Find general solutions of the differential equations in Problems 1 through 25. If an initial condition is given, find the corresponding particular solution. Throughout, primes denote derivatives with respect to x. xy' + y = 3xy, y(1) = 0

Chapter 1: Problem 1 Elementary Differential Equations 6

Find general solutions of the differential equations in Problems 1 through 25. If an initial condition is given, find the corresponding particular solution. Throughout, primes denote derivatives with respect to x. xy' + 3y = 2x5, y(2) = 1

Chapter 1: Problem 1 Elementary Differential Equations 6

Find general solutions of the differential equations in Problems 1 through 25. If an initial condition is given, find the corresponding particular solution. Throughout, primes denote derivatives with respect to x. y' + y = eX, y(O) = 1

Chapter 1: Problem 1 Elementary Differential Equations 6

Find general solutions of the differential equations in Problems 1 through 25. If an initial condition is given, find the corresponding particular solution. Throughout, primes denote derivatives with respect to x. xy' - 3y = x3 , y(1) = 10

Chapter 1: Problem 1 Elementary Differential Equations 6

Find general solutions of the differential equations in Problems 1 through 25. If an initial condition is given, find the corresponding particular solution. Throughout, primes denote derivatives with respect to x. y' + 2xy = x, y(O) = -2

Chapter 1: Problem 1 Elementary Differential Equations 6

Find general solutions of the differential equations in Problems 1 through 25. If an initial condition is given, find the corresponding particular solution. Throughout, primes denote derivatives with respect to x. y' = (1 - y) cos x, y(n) = 2

Chapter 1: Problem 1 Elementary Differential Equations 6

Find general solutions of the differential equations in Problems 1 through 25. If an initial condition is given, find the corresponding particular solution. Throughout, primes denote derivatives with respect to x. (1 + x)y' + y = cos x, y(O) = 1

Chapter 1: Problem 1 Elementary Differential Equations 6

Find general solutions of the differential equations in Problems 1 through 25. If an initial condition is given, find the corresponding particular solution. Throughout, primes denote derivatives with respect to x. xy' = 2y + x3 cos x

Chapter 1: Problem 1 Elementary Differential Equations 6

Find general solutions of the differential equations in Problems 1 through 25. If an initial condition is given, find the corresponding particular solution. Throughout, primes denote derivatives with respect to x. y' + y cot x = cos x

Chapter 1: Problem 1 Elementary Differential Equations 6

Find general solutions of the differential equations in Problems 1 through 25. If an initial condition is given, find the corresponding particular solution. Throughout, primes denote derivatives with respect to x. y' = 1 + x + y + xy, y(O) = 0

Chapter 1: Problem 1 Elementary Differential Equations 6

Find general solutions of the differential equations in Problems 1 through 25. If an initial condition is given, find the corresponding particular solution. Throughout, primes denote derivatives with respect to x. xy' = 3y + x4 cos x, y(2n) = 0

Chapter 1: Problem 1 Elementary Differential Equations 6

Find general solutions of the differential equations in Problems 1 through 25. If an initial condition is given, find the corresponding particular solution. Throughout, primes denote derivatives with respect to x. y' = 2xy + 3x2 exp(x2), y(O) = 5

Chapter 1: Problem 1 Elementary Differential Equations 6

Find general solutions of the differential equations in Problems 1 through 25. If an initial condition is given, find the corresponding particular solution. Throughout, primes denote derivatives with respect to x. xy' + (2x - 3)y = 4X4

Chapter 1: Problem 1 Elementary Differential Equations 6

Find general solutions of the differential equations in Problems 1 through 25. If an initial condition is given, find the corresponding particular solution. Throughout, primes denote derivatives with respect to x. (x2 + 4)y' + 3xy = x, y(O) = 1

Chapter 1: Problem 1 Elementary Differential Equations 6

Find general solutions of the differential equations in Problems 1 through 25. If an initial condition is given, find the corresponding particular solution. Throughout, primes denote derivatives with respect to x. x2 + 1) - + 3x3y=6x exp (- x2 ), y(O) = 1

Chapter 1: Problem 1 Elementary Differential Equations 6

Solve the differential equations in Problems 26 through 28 by regarding y as the independent variable rather than x. (1 _ 4xy2) dy = y3 dx

Chapter 1: Problem 1 Elementary Differential Equations 6

Solve the differential equations in Problems 26 through 28 by regarding y as the independent variable rather than x. (x + yeY) dx = 1

Chapter 1: Problem 1 Elementary Differential Equations 6

Solve the differential equations in Problems 26 through 28 by regarding y as the independent variable rather than x. (1 + 2xy) dy= 1 + y2

Chapter 1: Problem 1 Elementary Differential Equations 6

Express the general solution of dyjdx = 1 + 2xy in terms of the error function 2 r 2 erf(x) = .;rr 10 e-I dt.

Chapter 1: Problem 1 Elementary Differential Equations 6

Express the solution of the initial value problem dy 2x- = y + 2x cos x, y(1) = 0 dx as an integral as in Example 3 of this section.

Chapter 1: Problem 1 Elementary Differential Equations 6

Problems 31 and 32 illustrate-Jor the special case oj firstorder linear equations-techniques that will be important when we study higher-order linear equations in Chapter 3. (a) Show that yc (x) = Ce- J P(x) dx is a general solution of dy/dx + P (x)y = O. (b) Show that yp (x) = e- J P(x) dx [f (Q (x)eJ P(X) dX ) dx J is a particular solution of dy/dx + P(x)y = Q (x). (c) Suppose that Yc(x) is any general solution of dy/dx + P (x) y = 0 and that y p (x) is any particular solution of dy/dx + P(x)y = Q (x). Show that y(x) = Yc(x) + yp (x) is a general solution of dy/dx + P(x)y = Q (x).

Chapter 1: Problem 1 Elementary Differential Equations 6

Problems 31 and 32 illustrate-Jor the special case oj firstorder linear equations-techniques that will be important when we study higher-order linear equations in Chapter 3. a) Find constants A and B such that yp (x) = A sin x + B cos x is a solution of dy/dx + y = 2 sin x. (b) Use the result of part (a) and the method of Problem 31 to find the general solution of dy/dx + y = 2 sin x. (c) Solve the initial value problem dy/dx + y = 2 sin x, y (O) = 1.

Chapter 1: Problem 1 Elementary Differential Equations 6

A tank contains 1 000 liters (L) of a solution consisting of 1 00 kg of salt dissolved in water. Pure water is pumped into the tank at the rate of 5 L/s, and the mixture-kept uniform by stirring- is pumped out at the same rate. How long will it be until only 10 kg of salt remains in the tank?

Chapter 1: Problem 1 Elementary Differential Equations 6

Consider a reservoir with a volume of 8 billion cubic feet (ft3 ) and an initial pollutant concentration of 0.25%. There is a daily inflow of 500 million ft3 of water with a pollutant concentration of 0.05% and an equal daily outflow of the well-mixed water in the reservoir. How long will it take to reduce the pollutant concentration in the reservoir to 0. 1 O%?

Chapter 1: Problem 1 Elementary Differential Equations 6

Rework Example 4 for the case of Lake Ontario, which empties into the St. Lawrence River and receives inflow from Lake Erie (via the Niagara River). The only differences are that this lake has a volume of 1 640 krn3 and an inflow-outflow rate of 410 krn3 /year.

Chapter 1: Problem 1 Elementary Differential Equations 6

A tank initially contains 60 gal of pure water. Brine containing 1 Ib of salt per gallon enters the tank at 2 gal/min, and the (perfectly mixed) solution leaves the tank at 3 gal/min; thus the tank is empty after exactly 1 h. (a) Find the amount of salt in the tank after t minutes. (b) What is the maximum amount of salt ever in the tank?

Chapter 1: Problem 1 Elementary Differential Equations 6

A 400-gal tank initially contains 1 00 gal of brine containing 50 Ib of salt. Brine containing 1 Ib of salt per gallon enters the tank at the rate of 5 gal/s, and the well-mixed brine in the tank flows out at the rate of 3 gal/s. How much salt will the tank contain when it is full of brine?

Chapter 1: Problem 1 Elementary Differential Equations 6

Consider the cascade of two tanks shown in Fig. 1 .5.5, with VI = 1 00 (gal) and V2 = 200 (gal) the volumes of brine in the two tanks. Each tank also initially contains 50 Ib of salt. The three flow rates indicated in the figure are each 5 gal/min, with pure water flowing into tank 1. (a) Find the amount x(t) of salt in tank 1 at time t. (b) Suppose that y(t) is the amount of salt in tank 2 at 1 .5 Linear First-Order Equations 55 time t. Show first that dy dt 5x 5y 1 00 - 200 ' and then solve for y(t), using the function x(t) found in part (a). (c) Finally, find the maximum amount of salt ever in tank 2.

Chapter 1: Problem 1 Elementary Differential Equations 6

Suppose that in the cascade shown in Fig. 1 .5.5, tank 1 initially contains 1 00 gal of pure ethanol and tank 2 initially contains 1 00 gal of pure water. Pure water flows into tank 1 at 10 gal/min, and the other two flow rates are also 10 gal/min. (a) Find the amounts x(t) and y(t) of ethanol in the two tanks at time t :::::: O. (b) Find the maximum amount of ethanol ever in tank 2.

Chapter 1: Problem 1 Elementary Differential Equations 6

A multiple cascade is shown in Fig. 1 .5.6. At time t = 0, tank 0 contains 1 gal of ethanol and 1 gal of water; all the remaining tanks contain 2 gal of pure water each. Pure water is pumped into tank 0 at 1 gal/min, and the varying mixture in each tank is pumped into the one below it at the same rate. Assume, as usual, that the mixtures are kept perfectly uniform by stirring. Let Xn (t) denote the amount of ethanol in tank n at time t. (a) Show that xo(t) = e-t/2 (b) Show by induction on n that t n e-t/2 xn(t) = --- for n > O. n! 2 n (c) Show that the maximum value of xn (t) for n > 0 is Mn = xn(2n) = n n en /nL (d) Conclude from Stirling's approximation n! :::::: n n e-n.J2rrn that Mn :::::: (2rrn)-1/2 .

Chapter 1: Problem 1 Elementary Differential Equations 6

A 30-year-old woman accepts an engineering position with a starting salary of $30,000 per year. Her salary S(t) increases exponentially, with S(t) = 30et /20 thousand dollars after t years. Meanwhile, 1 2% of her salary is deposited continuously in a retirement account, which accumulates interest at a continuous annual rate of 6%. (a) Estimate LlA in terms of Llt to derive the differential equation satisfied by the amount A(t) in her retirement account after t years. (b) Compute A (40), the amount available for her retirement at age 70.

Chapter 1: Problem 1 Elementary Differential Equations 6

Suppose that a falling hailstone with density 8 = 1 starts from rest with negligible radius r = O. Thereafter its radius is r = kt (k is a constant) as it grows by accretion during its fall. Use Newton's second law-according to which the net force F acting on a possibly variable mass m equals the time rate of change dp/dt of its momentum p = m v-to set up and solve the initial value problem d -(mv) = mg, v(O) = 0, dt where m is the variable mass of the hailstone, v = dy/dt is its velocity, and the positive y-axis points downward. Then show that dv/dt = g/4. Thus the hailstone falls as though it were under one-fourth the influence of gravity.

Chapter 1: Problem 1 Elementary Differential Equations 6

Figure I .S.7 shows a slope field and typical solution curves for the equation y' = x - y. 10 8 6 4 2 O rlrl -2 -4 -6 -8 - 10 WL_-L_-L-L__W 5 x FIGURE 1.5.7. Slope field and solution curves for y' = x - y. (a) Show that every solution curve approaches the straight line y = x - I as x --+ +00. (b) For each of the five values Yl = 3.998, 3.999, 4.000, 4.001, and 4.002, determine the initial value Yo (accurate to four decimal places) such that y(S) = Yl for the solution satisfying the initial condition y( -S) = Yo.

Chapter 1: Problem 1 Elementary Differential Equations 6

Figure I .S.8 shows a slope field and typical solution curves for the equation y' = x + y. (a) Show that every solution curve approaches the straight line y = -x - I as x --+ -00. (b) For each of the five values Yl = - 1 0, -S, 0, S, and 1 0, determine the initial value Yo (accurate to five decimal places) such that y(S) = Yl for the solution satisfying the initial condition y( -S) = Yo. 10 8 6 4 2 0 44fJ. -2 -4 -6 -8 - I O __ -L __ x FIGURE 1.5.8. Slope field and solution curves for y' = x + y.

Chapter 1: Problem 1 Elementary Differential Equations 6

The incoming water has a pollutant concentration of c(t) = 10 liters per cubic meter (Um3 ). Verify that the graph of x(t) resembles the steadily rising curve in Fig. I .S.9, which approaches asymptotically the graph of the equilibrium solution x(t) == 20 that corresponds to the reservoir's long-term pollutant content. How long does it take the pollutant concentration in the reservoir to reach S Um3 ?

Chapter 1: Problem 1 Elementary Differential Equations 6

The incoming water has pollutant concentration c(t) 10( 1 + cos t) Um3 that varies between 0 and 20, with an average concentration of 10 Um3 and a period of oscillation of slightly over 6i months. Does it seem predictable that the lake's polutant content should ultimately oscillate periodically about an average level of 20 million liters?

Chapter 1: Problem 1 Elementary Differential Equations 6

Find general solutions of the differential equations in Problems 1 through 30. Primes denote derivatives with respect to x throughout. (x + y)y' = x - Y

Chapter 1: Problem 1 Elementary Differential Equations 6

Find general solutions of the differential equations in Problems 1 through 30. Primes denote derivatives with respect to x throughout. 2xyy' = x2 + 2y2

Chapter 1: Problem 1 Elementary Differential Equations 6

Find general solutions of the differential equations in Problems 1 through 30. Primes denote derivatives with respect to x throughout. xy' = y + 2.jXY

Chapter 1: Problem 1 Elementary Differential Equations 6

Find general solutions of the differential equations in Problems 1 through 30. Primes denote derivatives with respect to x throughout. . (x - y)y' = x + y

Chapter 1: Problem 1 Elementary Differential Equations 6

Find general solutions of the differential equations in Problems 1 through 30. Primes denote derivatives with respect to x throughout. x(x + y)y' = y(x - y)

Chapter 1: Problem 1 Elementary Differential Equations 6

Find general solutions of the differential equations in Problems 1 through 30. Primes denote derivatives with respect to x throughout. (x + 2y)y' = Y

Chapter 1: Problem 1 Elementary Differential Equations 6

Find general solutions of the differential equations in Problems 1 through 30. Primes denote derivatives with respect to x throughout. xyzy' = x3 + y3

Chapter 1: Problem 1 Elementary Differential Equations 6

Find general solutions of the differential equations in Problems 1 through 30. Primes denote derivatives with respect to x throughout. x2y' = xy + x2eY/x

Chapter 1: Problem 1 Elementary Differential Equations 6

Find general solutions of the differential equations in Problems 1 through 30. Primes denote derivatives with respect to x throughout. xZy' = xy + yZ

Chapter 1: Problem 1 Elementary Differential Equations 6

Find general solutions of the differential equations in Problems 1 through 30. Primes denote derivatives with respect to x throughout. xyy' = x2 + 3y2

Chapter 1: Problem 1 Elementary Differential Equations 6

Find general solutions of the differential equations in Problems 1 through 30. Primes denote derivatives with respect to x throughout. (xz - yZ)y' = 2xy

Chapter 1: Problem 1 Elementary Differential Equations 6

Find general solutions of the differential equations in Problems 1 through 30. Primes denote derivatives with respect to x throughout. xyy' = yZ + x..jr4xZ + yz

Chapter 1: Problem 1 Elementary Differential Equations 6

Find general solutions of the differential equations in Problems 1 through 30. Primes denote derivatives with respect to x throughout. xy' = y + J x2 + y2

Chapter 1: Problem 1 Elementary Differential Equations 6

Find general solutions of the differential equations in Problems 1 through 30. Primes denote derivatives with respect to x throughout. yy' +x = Jx2 + y2

Chapter 1: Problem 1 Elementary Differential Equations 6

Find general solutions of the differential equations in Problems 1 through 30. Primes denote derivatives with respect to x throughout. x(x + y)y' + y(3x + y) = 0

Chapter 1: Problem 1 Elementary Differential Equations 6

Find general solutions of the differential equations in Problems 1 through 30. Primes denote derivatives with respect to x throughout. y' = Jx + y + 1

Chapter 1: Problem 1 Elementary Differential Equations 6

Find general solutions of the differential equations in Problems 1 through 30. Primes denote derivatives with respect to x throughout. y' = (4x + y)z

Chapter 1: Problem 1 Elementary Differential Equations 6

Find general solutions of the differential equations in Problems 1 through 30. Primes denote derivatives with respect to x throughout. (x + y)y' = 1

Chapter 1: Problem 1 Elementary Differential Equations 6

Find general solutions of the differential equations in Problems 1 through 30. Primes denote derivatives with respect to x throughout. xZy' + 2xy = 5y3

Chapter 1: Problem 1 Elementary Differential Equations 6

Find general solutions of the differential equations in Problems 1 through 30. Primes denote derivatives with respect to x throughout. y2y' + 2xy3 = 6x

Chapter 1: Problem 1 Elementary Differential Equations 6

Find general solutions of the differential equations in Problems 1 through 30. Primes denote derivatives with respect to x throughout. y' = Y + y3

Chapter 1: Problem 1 Elementary Differential Equations 6

Find general solutions of the differential equations in Problems 1 through 30. Primes denote derivatives with respect to x throughout. xZy' + 2xy = 5y4

Chapter 1: Problem 1 Elementary Differential Equations 6

Find general solutions of the differential equations in Problems 1 through 30. Primes denote derivatives with respect to x throughout. xy' + 6y = 3xy4/3

Chapter 1: Problem 1 Elementary Differential Equations 6

Find general solutions of the differential equations in Problems 1 through 30. Primes denote derivatives with respect to x throughout. 2xy' + y3e -2x = 2xy

Chapter 1: Problem 1 Elementary Differential Equations 6

Find general solutions of the differential equations in Problems 1 through 30. Primes denote derivatives with respect to x throughout. y2(xy' + y)(l + X4)1/2 = x

Chapter 1: Problem 1 Elementary Differential Equations 6

Find general solutions of the differential equations in Problems 1 through 30. Primes denote derivatives with respect to x throughout. 3y2y' + y3 = e-X

Chapter 1: Problem 1 Elementary Differential Equations 6

Find general solutions of the differential equations in Problems 1 through 30. Primes denote derivatives with respect to x throughout. 3xy2y' = 3x4 + y3

Chapter 1: Problem 1 Elementary Differential Equations 6

Find general solutions of the differential equations in Problems 1 through 30. Primes denote derivatives with respect to x throughout. xeYy' = 2(eY + x3e2x)

Chapter 1: Problem 1 Elementary Differential Equations 6

Find general solutions of the differential equations in Problems 1 through 30. Primes denote derivatives with respect to x throughout. (2x sin y cos y)y' = 4x2 + sin2 y

Chapter 1: Problem 1 Elementary Differential Equations 6

Find general solutions of the differential equations in Problems 1 through 30. Primes denote derivatives with respect to x throughout. (x + eY)y' = xe - Y - 1

Chapter 1: Problem 1 Elementary Differential Equations 6

In Problems 31 through 42, verify that the given differential equation is exact; then solve it. (2x + 3y)dx + (3x + 2y)dy = 0

Chapter 1: Problem 1 Elementary Differential Equations 6

In Problems 31 through 42, verify that the given differential equation is exact; then solve it. (4x - y)dx + (6y -x)dy = 0

Chapter 1: Problem 1 Elementary Differential Equations 6

In Problems 31 through 42, verify that the given differential equation is exact; then solve it. (3x2 + 2y2) dx + (4xy + 6y2) dy = 0

Chapter 1: Problem 1 Elementary Differential Equations 6

In Problems 31 through 42, verify that the given differential equation is exact; then solve it. (2xy2 + 3x2) dx + (2x2y + 4y3) dy = 0

Chapter 1: Problem 1 Elementary Differential Equations 6

In Problems 31 through 42, verify that the given differential equation is exact; then solve it. (x3+ ) dx+ (y2+ln x)dy = o

Chapter 1: Problem 1 Elementary Differential Equations 6

In Problems 31 through 42, verify that the given differential equation is exact; then solve it. (1 + yexy) dx + (2y + xexy) dy = 0

Chapter 1: Problem 1 Elementary Differential Equations 6

In Problems 31 through 42, verify that the given differential equation is exact; then solve it. (cos x + lny ) dx + ( + eY) dy = 0

Chapter 1: Problem 1 Elementary Differential Equations 6

In Problems 31 through 42, verify that the given differential equation is exact; then solve it. (x + tan- 1 y) dx + x + Y dy = 0

Chapter 1: Problem 1 Elementary Differential Equations 6

In Problems 31 through 42, verify that the given differential equation is exact; then solve it. (3x2y3 + y4) dx + (3x3y2 + y4 + 4xy3) dy = 0

Chapter 1: Problem 1 Elementary Differential Equations 6

In Problems 31 through 42, verify that the given differential equation is exact; then solve it. (eX sin y + tan y) dx + (eX cos y +x sec2 y) dy = 0

Chapter 1: Problem 1 Elementary Differential Equations 6

In Problems 31 through 42, verify that the given differential equation is exact; then solve it. 2X _ 3y2) dx + ( 2Y _ x2+ _ 1_) dy = 0 Y x4 x3 y2

Chapter 1: Problem 1 Elementary Differential Equations 6

In Problems 31 through 42, verify that the given differential equation is exact; then solve it. 2X5/2 _ 3y5/3 3y5/3 _ 2X5/2 42. 2X5/2y2/3 dx + 3X3/2y5/3 dy = 0

Chapter 1: Problem 1 Elementary Differential Equations 6

Find a general solution of each reducible second-order differential equation in Problems 43-54. Assume x, y and/or y' positive where helpful (as in Example 11). xy" = y

Chapter 1: Problem 1 Elementary Differential Equations 6

Find a general solution of each reducible second-order differential equation in Problems 43-54. Assume x, y and/or y' positive where helpful (as in Example 11). yy" + (y')2 = 0

Chapter 1: Problem 1 Elementary Differential Equations 6

Find a general solution of each reducible second-order differential equation in Problems 43-54. Assume x, y and/or y' positive where helpful (as in Example 11). y" + 4y = 0

Chapter 1: Problem 1 Elementary Differential Equations 6

Find a general solution of each reducible second-order differential equation in Problems 43-54. Assume x, y and/or y' positive where helpful (as in Example 11). xy" + y' = 4x

Chapter 1: Problem 1 Elementary Differential Equations 6

Find a general solution of each reducible second-order differential equation in Problems 43-54. Assume x, y and/or y' positive where helpful (as in Example 11). y" = (y')2

Chapter 1: Problem 1 Elementary Differential Equations 6

Find a general solution of each reducible second-order differential equation in Problems 43-54. Assume x, y and/or y' positive where helpful (as in Example 11). x2y" + 3xy' = 2

Chapter 1: Problem 1 Elementary Differential Equations 6

Find a general solution of each reducible second-order differential equation in Problems 43-54. Assume x, y and/or y' positive where helpful (as in Example 11). yy" + (y')2 = yy'

Chapter 1: Problem 1 Elementary Differential Equations 6

Find a general solution of each reducible second-order differential equation in Problems 43-54. Assume x, y and/or y' positive where helpful (as in Example 11). y" = (x + y,)2

Chapter 1: Problem 1 Elementary Differential Equations 6

Find a general solution of each reducible second-order differential equation in Problems 43-54. Assume x, y and/or y' positive where helpful (as in Example 11). y" = 2y(y,)3

Chapter 1: Problem 1 Elementary Differential Equations 6

Find a general solution of each reducible second-order differential equation in Problems 43-54. Assume x, y and/or y' positive where helpful (as in Example 11).y3y" = 1

Chapter 1: Problem 1 Elementary Differential Equations 6

Find a general solution of each reducible second-order differential equation in Problems 43-54. Assume x, y and/or y' positive where helpful (as in Example 11). y" = 2yy

Chapter 1: Problem 1 Elementary Differential Equations 6

Find a general solution of each reducible second-order differential equation in Problems 43-54. Assume x, y and/or y' positive where helpful (as in Example 11). yy" = 3(y')2

Chapter 1: Problem 1 Elementary Differential Equations 6

Show that the substitution v = ax + by + c transforms the differential equation dy/dx = F(ax + by + c) into a separable equation.

Chapter 1: Problem 1 Elementary Differential Equations 6

Suppose that n = 0 and n = 1. Show that the substitution v = y l - n transforms the Bernoulli equation dy/dx + P(x)y = Q(x)yn into the linear equation dv dx + (1 -n)P(x)v(x) = (1 -n)Q(x).

Chapter 1: Problem 1 Elementary Differential Equations 6

Show that the substitution v = In y transforms the differential equation dy/dx + P(x)y = Q(x)(y In y) into the linear equation dv/dx + P(x) = Q(x)v(x).

Chapter 1: Problem 1 Elementary Differential Equations 6

Use the idea in Problem 57 to solve the equation dy X 2 dx -4x y + 2y In y = o.

Chapter 1: Problem 1 Elementary Differential Equations 6

Solve the differential equation dy x - y - 1 dx x + y +3 by finding h and k so that the substitutions x = u + h, y = v + k transform it into the homogeneous equation dv u - v du u + v

Chapter 1: Problem 1 Elementary Differential Equations 6

Use the method in Problem 59 to solve the differential equation dy 2y -x +7 dx 4x - 3y - 18

Chapter 1: Problem 1 Elementary Differential Equations 6

Make an appropriate substitution to find a solution of the equation dy/dx = sin (x - y). Does this general solution contain the linear solution y(x) = x - rrf2 that is readily verified by substitution in the differential equation?

Chapter 1: Problem 1 Elementary Differential Equations 6

Show that the solution curves of the differential equation dy y(2x3 _ y3) dx X(2y3 -x3) are of the form x3 + y3 = 3Cxy.

Chapter 1: Problem 1 Elementary Differential Equations 6

The equation dy/dx = A(X)y2 + B(x)y + C(x) is called a Riccati equation. Suppose that one particular solution Yl (x) of this equation is known. Show that the substitution 1 Y = Yl + V transforms the Riccati equation into the linear equation dv dx + (B + 2AYl )V = -A.

Chapter 1: Problem 1 Elementary Differential Equations 6

Use the method of Problem 63 to solve the equations in Problems 64 and 65, given that Yl (x) = x is a solution of each. - + y2 = 1 + x2

Chapter 1: Problem 1 Elementary Differential Equations 6

Use the method of Problem 63 to solve the equations in Problems 64 and 65, given that Yl (x) = x is a solution of each. dy - + 2xy = 1 + x2 + y2

Chapter 1: Problem 1 Elementary Differential Equations 6

Use the method of Problem 63 to solve the equations in Problems 64 and 65, given that Yl (x) = x is a solution of each. An equation of the form y = xy' + g(y') (37) is called a Clairaut equation. Show that the oneparameter family of straight lines described by y(x) = Cx + g(C) (38) is a general solution of Eq. (37).

Chapter 1: Problem 1 Elementary Differential Equations 6

Consider the Clairaut equation y = xy' _ (y') 2 for which g (y') = - (y')2 in Eq. (37). Show that the line y = Cx - C2 is tangent to the parabola y = x2 at the point (4C, C2). Explain why this implies that y = x2 is a singular solution of the given Clairaut equation. This singular solution and the one-parameter family of straight line solutions are illustrated in Fig. 1 .6. 10.

Chapter 1: Problem 1 Elementary Differential Equations 6

Derive Eq. (18) in this section from Eqs. (16) and (17).

Chapter 1: Problem 1 Elementary Differential Equations 6

In the situation of Example 7, suppose that a = 1 00 mi, Vo = 400 milh, and w = 40 mi/h. Now how far northward does the wind blow the airplane?

Chapter 1: Problem 1 Elementary Differential Equations 6

As in the text discussion, suppose that an airplane maintains a heading toward an airport at the origin. If Vo = 500 milh and w = 50 milh (with the wind blowing due north), and the plane begins at the point (200, 1 50), show that its trajectory is described by y + .JX2 + y2 = 2(200X9) 1/1 O .

Chapter 1: Problem 1 Elementary Differential Equations 6

A river 1 00 ft wide is flowing north at w feet per second. A dog starts at (100, 0) and swims at Vo = 4 ftls, always heading toward a tree at (0, 0) on the west bank directly across from the dog's starting point. (a) If w = 2 ftls, show that the dog reaches the tree. (b) If w = 4 ft/s, show that the dog reaches instead the point on the west bank 50 ft north of the tree. (c) If w = 6 ftls, show that the dog never reaches the west bank.

Chapter 1: Problem 1 Elementary Differential Equations 6

In the calculus of plane curves, one learns that the curvature K of the curve y = y(x) at the point (x, y) is given by 1y" (x) 1 K=- -'-'-- [1 + y'(x)2P/2 ' and that the curvature of a circle of radius r is K = 1 I r. [See Example 3 in Section 11.6 of Edwards and Penney, Calculus: Early Transcendentals, 7th edition (Upper Saddle River, NJ: Prentice Hall, 2008).] Conversely, substitute p = y' to derive a general solution of the second-order differential equation (with r constant) in the form Thus a circle of radius r (or a part thereof) is the only plane curve with constant curvature llr.

Chapter 1: Problem 1 Elementary Differential Equations 6

Separate variables and use partial fractions to solve the initial value problems in Problems 1-8. Use either the exact solution ora computer-generated slope field to sketch the graphs ofseveral solutions of the given differential equation, and highlight the indicated particular solution. dx - = x -x2 x(O) = 2 dt '

Chapter 1: Problem 1 Elementary Differential Equations 6

Separate variables and use partial fractions to solve the initial value problems in Problems 1-8. Use either the exact solution ora computer-generated slope field to sketch the graphs ofseveral solutions of the given differential equation, and highlight the indicated particular solution. -- = lOx -x2, X (0) = 1

Chapter 1: Problem 1 Elementary Differential Equations 6

Separate variables and use partial fractions to solve the initial value problems in Problems 1-8. Use either the exact solution ora computer-generated slope field to sketch the graphs ofseveral solutions of the given differential equation, and highlight the indicated particular solution. - = 1 - x2, x(O) = 3

Chapter 1: Problem 1 Elementary Differential Equations 6

Separate variables and use partial fractions to solve the initial value problems in Problems 1-8. Use either the exact solution ora computer-generated slope field to sketch the graphs ofseveral solutions of the given differential equation, and highlight the indicated particular solution. dx - =9-4x 2 , x (0) =0

Chapter 1: Problem 1 Elementary Differential Equations 6

Separate variables and use partial fractions to solve the initial value problems in Problems 1-8. Use either the exact solution ora computer-generated slope field to sketch the graphs ofseveral solutions of the given differential equation, and highlight the indicated particular solution. x - = 3x(5 - x), x(O) = 8 d

Chapter 1: Problem 1 Elementary Differential Equations 6

Separate variables and use partial fractions to solve the initial value problems in Problems 1-8. Use either the exact solution ora computer-generated slope field to sketch the graphs ofseveral solutions of the given differential equation, and highlight the indicated particular solution. dx - = 3x(x - 5), x(O) = 2

Chapter 1: Problem 1 Elementary Differential Equations 6

Separate variables and use partial fractions to solve the initial value problems in Problems 1-8. Use either the exact solution ora computer-generated slope field to sketch the graphs ofseveral solutions of the given differential equation, and highlight the indicated particular solution. dx - = 4x(7 -x), x(O) = 11

Chapter 1: Problem 1 Elementary Differential Equations 6

Separate variables and use partial fractions to solve the initial value problems in Problems 1-8. Use either the exact solution ora computer-generated slope field to sketch the graphs ofseveral solutions of the given differential equation, and highlight the indicated particular solution. dx - = 7x(x - 13), x(O) = 17

Chapter 1: Problem 1 Elementary Differential Equations 6

The time rate of change of a rabbit popUlation P is proportional to the square root of P. At time t = 0 (months) the population numbers 100 rabbits and is increasing at the rate of 20 rabbits per month. How many rabbits will there be one year later?

Chapter 1: Problem 1 Elementary Differential Equations 6

Suppose that the fish population P(t) in a lake is attacked by a disease at time t = 0, with the result that the fish cease to reproduce (so that the birth rate is f3 = 0) and the death rate 8 (deaths per week per fish) is thereafter proportional to 1/.../P. If there were initially 900 fish in the lake and 441 were left after 6 weeks, how long did it take all the fish in the lake to die?

Chapter 1: Problem 1 Elementary Differential Equations 6

Suppose that when a certain lake is stocked with fish, the birth and death rates f3 and 8 are both inversely proportional to .../P. (a) Show that where k is a constant. (b) If Po = 100 and after 6 months there are 1 69 fish in the lake, how many will there be after 1 year?

Chapter 1: Problem 1 Elementary Differential Equations 6

The time rate of change of an alligator population P in a swamp is proportional to the square of P. The swamp contained a dozen alligators in 1988, two dozen in 1998. When will there be four dozen alligators in the swamp? What happens thereafter?

Chapter 1: Problem 1 Elementary Differential Equations 6

Consider a prolific breed of rabbits whose birth and death rates, f3 and 8, are each proportional to the rabbit population P = P (t) , with f3 > 8. (a) Show that Po pet) - k constant. - 1 - kPot ' Note that P (t) --+ +00 as t --+ liCk Po). This is doomsday. (b) Suppose that Po = 6 and that there are nine rabbits after ten months. When does doomsday occur?

Chapter 1: Problem 1 Elementary Differential Equations 6

Repeat part (a) of Problem 13 in the case f3 < 8. What now happens to the rabbit population in the long run?

Chapter 1: Problem 1 Elementary Differential Equations 6

Consider a population pet) satisfying the logistic equation dP/dt = aP - bP2, where B = aP is the time rate at which births occur and D = bP2 is the rate at which deaths occur. If the initial population is P(O) = Po, and Bo births per month and Do deaths per month are occurring at time t = 0, show that the limiting population is M = BoPo/Do.

Chapter 1: Problem 1 Elementary Differential Equations 6

Consider a rabbit population pet) satisfying the logistic equation as in Problem 15. If the initial population is 1 20 rabbits and there are 8 births per month and 6 deaths per month occurring at time t = 0, how many months does it take for P (t) to reach 95% of the limiting population M?

Chapter 1: Problem 1 Elementary Differential Equations 6

Consider a rabbit population pet) satisfying the logistic equation as in Problem 15. If the initial population is 240 rabbits and there are 9 births per month and 12 deaths per month occurring at time t = 0, how many months does it take for pet) to reach 105% of the limiting population M?

Chapter 1: Problem 1 Elementary Differential Equations 6

Consider a population pet) satisfying the extinctionexplosion equation dP/dt = ap2 - bP, where B = aP2 is the time rate at which births occur and D = b P is the rate at which deaths occur. If the initial population is P(O) = Po and Bo births per month and Do deaths per month are occurring at time t = 0, show that the threshold population is M = DoPo/Bo.

Chapter 1: Problem 1 Elementary Differential Equations 6

Consider an alligator population p et) satisfying the extinction/explosion equation as in Problem 18. If the initial population is 100 alligators and there are 10 births per month and 9 deaths per months occurring at time t = 0, how many months does it take for pet) to reach 10 times the threshold population M?

Chapter 1: Problem 1 Elementary Differential Equations 6

Consider an alligator population pet) satisfying the extinction/explosion equation as in Problem 1 8. If the initial population is 1 1 0 alligators and there are 11 births per month and 12 deaths per month occurring at time t = 0, how many months does it take for P(t) to reach 10% of the threshold population M?

Chapter 1: Problem 1 Elementary Differential Equations 6

Suppose that the population pet) of a country satisfies the differential equation dP/dt = kP(200 - P) with k constant. Its population in 1940 was 100 million and was then 1 .7 Population Models 83 growing at the rate of 1 million per year. Predict this country's population for the year 2000.

Chapter 1: Problem 1 Elementary Differential Equations 6

Suppose that at time t = 0, half of a "logistic" population of 100, 000 persons have heard a certain rumor, and that the number of those who have heard it is then increasing at the rate of 1000 persons per day. How long will it take for this rumor to spread to 80% of the population? (Suggestion: Find the value of k by substituting P(O) and P'(O) in the logistic equation, Eq. (3).)

Chapter 1: Problem 1 Elementary Differential Equations 6

As the salt KN03 dissolves in methanol, the number x(t) of grams of the salt in a solution after t seconds satisfies the differential equation dx/dt = 0.8x - 0.004x2. (a) What is the maximum amount of the salt that will ever dissolve in the methanol? (b) If x = 50 when t = 0, how long will it take for an additional 50 g of salt to dissolve?

Chapter 1: Problem 1 Elementary Differential Equations 6

Suppose that a community contains 15,000 people who are susceptible to Michaud's syndrome, a contagious disease. At time t = 0 the number N (t) of people who have developed Michaud's syndrome is 5000 and is increasing at the rate of 500 per day. Assume that N'(t) is proportional to the product of the numbers of those who have caught the disease and of those who have not. How long will it take for another 5000 people to develop Michaud' s syndrome?

Chapter 1: Problem 1 Elementary Differential Equations 6

The data in the table in Fig. 1 .7.7 are given for a certain population P(t) that satisfies the logistic equation in (3). (a) What is the limiting population M? (Suggestion: Use the approximation , pet + h) - P(t - h) P (t) 2h with h = 1 to estimate the values of P'(t) when P 25.00 and when P = 47.54. Then substitute these values in the logistic equation and solve for k and M.) (b) Use the values of k and M found in part (a) to determine when P = 75. (Suggestion: Take t = 0 to correspond to the year 1925.)

Chapter 1: Problem 1 Elementary Differential Equations 6

A population pet) of small rodents has birth rate f3 (0.001 ) P (births per month per rodent) and constant death rate 8. If P(O) = 100 and P'(O) = 8, how long (in months) will it take this population to double to 200 rodents? (Suggestion: First find the value of 8.)

Chapter 1: Problem 1 Elementary Differential Equations 6

Consider an animal population P (t) with constant death rate 8 = 0.01 (deaths per animal per month) and with birth rate f3 proportional to P. Suppose that P (0) = 200 and PI(O) = 2. (a) When is P = 1000? (b) When does doomsday occur?

Chapter 1: Problem 1 Elementary Differential Equations 6

Suppose that the number x (t) (with t in months) of alligators in a swamp satisfies the differential equation dx/dt = 0.0001x2 -O.Olx. (a) If initially there are 25 alligators in the swamp, solve this differential equation to determine what happens to the alligator population in the long run. (b) Repeat part (a), except with 150 alligators initially.

Chapter 1: Problem 1 Elementary Differential Equations 6

During the period from 1790 to 1930, the u.S. population P(t) (t in years) grew from 3.9 million to 123.2 million. Throughout this period, P (t) remained close to the solution of the initial value problem dP dt = 0.03135P -0.0001489P2 , P(O) = 3.9. (a) What 1930 population does this logistic equation predict? (b) What limiting population does it predict? (c) Has this logistic equation continued since 1930 to accurately model the U.S. population? [This problem is based on a computation by Verhulst, who in 1845 used the 1790-1840 U.S. population data to predict accurately the U.S. population through the year 1930 (long after his own death, of course).]

Chapter 1: Problem 1 Elementary Differential Equations 6

A tumor may be regarded as a population of multiplying cells. It is found empirically that the "birth rate" of the cells in a tumor decreases exponentially with time, so that f3(t) = f3oe-at (where a and f30 are positive constants), and hence dP dt = f3oe-at P, P(O) = Po Solve this initial value problem for P(t) = Po exp ( (1 - e-OIt ) . Observe that P(t) approaches the finite limiting population Po exp (f3o/a) as t +00.

Chapter 1: Problem 1 Elementary Differential Equations 6

For the tumor of Problem 30, suppose that at time t = 0 there are Po = 106 cells and that P (t) is then increasing at the rate of 3 x 105 cells per month. After 6 months the tumor has doubled (in size and in number of cells). Solve numerically for a, and then find the limiting population of the tumor.

Chapter 1: Problem 1 Elementary Differential Equations 6

Derive the solution P(t) = MPo Po + (M - Po)e-kMt of the logistic initial value problem pI = kP(M - P), P (0) = Po. Make it clear how your derivation depends on whether 0 < Po < M or Po > M.

Chapter 1: Problem 1 Elementary Differential Equations 6

(a) Derive the solution MPo P(t) = --- Po + (M - Po -)ekMt of the extinction-explosion initial value problem P' = kP(P - M), P(O) = Po. (b) How does the behavior of P (t) as t increases depend on whether 0 < Po < M or Po > M?

Chapter 1: Problem 1 Elementary Differential Equations 6

If P(t) satisfies the logistic equation in (3), use the chain rule to show that PII(t) = 2k2 P(P - M) (P -M). Conclude that pI! > 0 if 0 < P < 1M pI! = 0 if 2 ' P = 1M pI! < 0 if 1M < P < M and P" > 0 2 ' 2 ' if P > M. In particular, it follows that any solution curve that crosses the line P = M has an inflection point where it crosses that line, and therefore resembles one of the lower S-shaped curves in Fig. 1.7.3.

Chapter 1: Problem 1 Elementary Differential Equations 6

Consider two population functions PI (t) and P2 (t), both of which satisfy the logistic equation with the same limiting population M but with different values kl and k2 of the constant k in Eq. (3). Assume that kl < k2 . Which population approaches M the most rapidly? You can reason geometrically by examining slope fields (especially if appropriate software is available), symbolically by analyzing the solution given in Eq. (7), or numerically by substituting successive values of t

Chapter 1: Problem 1 Elementary Differential Equations 6

To solve the two equations in ( 10) for the values of k and M, begin by solving the first equation for the quantity x = e-5 0kM and the second equation for x2 = e-IOOkM. Upon equating the two resulting expressions for x2 in terms of M, you get an equation that is readily solved for M. With M now known, either of the original equations is readily solved for k. This technique can be used to "fit" the logistic equation to any three population values Po, PI, and P2 corresponding to equally spaced times to = 0, tl , and t 2 = 2tl.

Chapter 1: Problem 1 Elementary Differential Equations 6

Use the method of Problem 36 to fit the logistic equation to the actual U.S. population data (Fig. 1.7.4) for the years 1850, 1900, and 1950. Solve the resulting logistic equation and compare the predicted and actual populations for the years 1990 and 2000.

Chapter 1: Problem 1 Elementary Differential Equations 6

Fit the logistic equation to the actual U.S. population data (Fig. 1.7.4) for the years 1900, 1930, and 1960. Solve the resulting logistic equation, then compare the predicted and actual populations for the years 1980, 1990, and 2000.

Chapter 1: Problem 1 Elementary Differential Equations 6

Birth and death rates of animal populations typically are not constant; instead, they vary periodically with the passage of seasons. Find P (t) if the population P satisfies the differential equation dP - = (k + b cos 2:rrt)P, where t is in years and k and b are positive constants. Thus the growth-rate function r (t) = k + b cos 2:n:t varies periodically about its mean value k. Construct a graph that contrasts the growth of this population with one that has the same initial value Po but satisfies the natural growth equation P' = kP (same constant k). How would the two populations compare after the passage of many years?

Chapter 1: Problem 1 Elementary Differential Equations 6

The acceleration of a Maserati is proportional to the difference between 250 km/h and the velocity of this sports car. If this machine can accelerate from rest to 100 km/h in 10 s, how long will it take for the car to accelerate from rest to 200 km/h?

Chapter 1: Problem 1 Elementary Differential Equations 6

Suppose that a body moves through a resisting medium with resistance proportional to its velocity v, so that dvldt = -kv. (a) Show that its velocity and position at time t are given by v(t) = voekt and x(t) = Xo + () 0 - e-kt). (b) Conclude that the body travels only a finite distance, and find that distance.

Chapter 1: Problem 1 Elementary Differential Equations 6

Suppose that a motorboat is moving at 40 ft/s when its motor suddenly quits, and that 10 s later the boat has slowed to 20 ft/s. Assume, as in Problem 2, that the resistance it encounters while coasting is proportional to its velocity. How far will the boat coast in all?

Chapter 1: Problem 1 Elementary Differential Equations 6

Consider a body that moves horizontally through a medium whose resistance is proportional to the square of the velocity v, so that dvldt = -kv2 Show that and that Vo v(t) = 1 + vokt 1 x(t) = Xo + k lnO + vokt). Note that, in contrast with the result of Problem 2, x(t) -+ +00 as t -+ +00. Which offers less resistance when the body is moving fairly slowly-the medium in this problem or the one in Problem 2? Does your answer seem consistent with the observed behaviors of x(t) as t -+ oo?

Chapter 1: Problem 1 Elementary Differential Equations 6

Assuming resistance proportional to the square of the velocity (as in Problem 4), how far does the motorboat of Problem 3 coast in the first minute after its motor quits?

Chapter 1: Problem 1 Elementary Differential Equations 6

Assume that a body moving with velocity v encounters resistance of the form dvldt = _kV3 /2 Show that and that 4vo v(t) = ( 2 ktFo + 2) x(t) = Xo + Fo(I - kt+2) . Conclude that under a -power resistance a body coasts only a finite distance before coming to a stop.

Chapter 1: Problem 1 Elementary Differential Equations 6

Suppose that a car starts from rest, its engine providing an acceleration of 10 ft/s2, while air resistance provides 0. 1 ft/S2 of deceleration for each foot per second of the car's velocity. (a) Find the car's maximum possible (limiting) velocity. (b) Find how long it takes the car to attain 90% of its limiting velocity, and how far it travels while doing so.

Chapter 1: Problem 1 Elementary Differential Equations 6

Rework both parts of Problem 7, with the sole difference that the deceleration due to air resistance now is (0.001)v2 ft/s2 when the car's velocity is v feet per second.

Chapter 1: Problem 1 Elementary Differential Equations 6

A motorboat weighs 32,000 lb and its motor provides a thrust of 5000 lb. Assume that the water resistance is 100 pounds for each foot per second of the speed v of the boat. Then dv 1000- = 5000 - 1OOv. dt If the boat starts from rest, what is the maximum velocity that it can attain?

Chapter 1: Problem 1 Elementary Differential Equations 6

A woman bails out of an airplane at an altitude of 10,000 ft, falls freely for 20 s, then opens her parachute. How long will it take her to reach the ground? Assume linear air resistance pv ft/S2, taking p = 0. 15 without the parachute and p = 1 .5 with the parachute. (Suggestion: First determine her height above the ground and velocity when the parachute opens.)

Chapter 1: Problem 1 Elementary Differential Equations 6

According to a newspaper account, a paratrooper survived a training jump from 1200 ft when his parachute failed to open but provided some resistance by flapping unopened in the wind. Allegedly he hit the ground at 100 mi/h after falling for 8 s. Test the accuracy of this account. (Suggestion: Find p in Eq. (4) by assuming a terminal velocity of 100 mi/h. Then calculate the time required to fall 1200 ft.)

Chapter 1: Problem 1 Elementary Differential Equations 6

It is proposed to dispose of nuclear wastes-in drums with weight W = 640 lb and volume 8 ft3-by dropping them into the ocean (vo = 0). The force equation for a drum falling through water is where the buoyant force B is equal to the weight (at 62.5 lb/ft3) of the volume of water displaced by the drum (Archimedes' principle) and FR is the force of water resistance, found empirically to be 1 lb for each foot per second of the velocity of a drum. If the drums are likely to burst upon an impact of more than 75 ft/s, what is the maximum depth to which they can be dropped in the ocean without likelihood of bursting?

Chapter 1: Problem 1 Elementary Differential Equations 6

Separate variables in Eq. (2) and substitute u = v"fjifg to obtain the upward-motion velocity function given in Eq. (13) with initial condition v(O) = Vo.

Chapter 1: Problem 1 Elementary Differential Equations 6

Integrate the velocity function in Eq. (13) to obtain the upward-motion position function given in Eq. (4) with initial condition y(O) = Yo.

Chapter 1: Problem 1 Elementary Differential Equations 6

Separate variables in Eq. (5) and substitute u = v"fjifg to obtain the downward-motion velocity function given in Eq. (16) with initial condition v(O) = Vo.

Chapter 1: Problem 1 Elementary Differential Equations 6

Integrate the velocity function in Eq. (16) to obtain the downward-motion position function given in Eq. (17) with initial condition y(O) = Yo.

Chapter 1: Problem 1 Elementary Differential Equations 6

Consider the crossbow bolt of Example 3, shot straight upward from the ground (y = 0) at time t = 0 with initial velocity Vo = 49 m/s. Take g = 9.8 m/s2 and p = 0.001 1 in Eq. (12). Then use Eqs. (13) and (14) to show that the bolt reaches its maximum height of about 108.47 m in about 4.61 s.

Chapter 1: Problem 1 Elementary Differential Equations 6

Continuing Problem 17, suppose that the bolt is now dropped (vo = 0) from a height of Yo = 108.47 m. Then use Eqs. (16) and (17) to show that it hits the ground about 4.80 s later with an impact speed of about 43.49 m/s.

Chapter 1: Problem 1 Elementary Differential Equations 6

A motorboat starts from rest (initial velocity v (O) = Vo = 0). Its motor provides a constant acceleration of 4 ft/S2 , but water resistance causes a deceleration of v 2 / 400 ft/ S 2 . Find v when t = 10 s, and also find the limiting velocity as t --+ +00 (that is, the maximum possible speed of the boat).

Chapter 1: Problem 1 Elementary Differential Equations 6

An arrow is shot straight upward from the ground with an initial velocity of 160 ft/s. It experiences both the deceleration of gravity and deceleration v2/800 due to air resistance. How high in the air does it go?

Chapter 1: Problem 1 Elementary Differential Equations 6

If a ball is projected upward from the ground with initial velocity Vo and resistance proportional to v 2 , deduce from Eq. (14) that the maximum height it attains is 1 ( PV5 Ymax = - In 1 + -) .

Chapter 1: Problem 1 Elementary Differential Equations 6

Suppose that p = 0.075 (in fps units, with g = 32 ft/s 2 ) in Eq. (15) for a paratrooper falling with parachute open. If he jumps from an altitude of 10,000 ft and opens his parachute immediately, what will be his terminal speed? How long will it take him to reach the ground?

Chapter 1: Problem 1 Elementary Differential Equations 6

Suppose that the paratrooper of Problem 22 falls freely for 30 s with p = 0.00075 before opening his parachute. How long will it now take him to reach the ground?

Chapter 1: Problem 1 Elementary Differential Equations 6

The mass of the sun is 329,320 times that of the earth and its radius is 109 times the radius of the earth. (a) To what radius (in meters) would the earth have to be compressed in order for it to become a black hole-the escape velocity from its surface equal to the velocity c = 3 X 108 m/s of light? (b) Repeat part (a) with the sun in place of the earth.

Chapter 1: Problem 1 Elementary Differential Equations 6

(a) Show that if a projectile is launched straight upward from the surface of the earth with initial velocity Vo less than escape velocity "j2GM/R, then the maximum distance from the center of the earth attained by the projectile is 2GMR r max - -----,;- - 2GM _ Rv2 ' o where M and R are the mass and radius of the earth, respectively. (b) With what initial velocity Vo must such a projectile be launched to yield a maximum altitude of 100 kilometers above the surface of the earth? (c) Find the maximum distance from the center of the earth, expressed in terms of earth radii, attained by a projectile launched from the surface of the earth with 90% of escape velocity.

Chapter 1: Problem 1 Elementary Differential Equations 6

Suppose that you are stranded-your rocket engine has failed-on an asteroid of diameter 3 miles, with density equal to that of the earth with radius 3960 miles. If you have enough spring in your legs to jump 4 feet straight up on earth while wearing your space suit, can you blast off from this asteroid using leg power alone?

Chapter 1: Problem 1 Elementary Differential Equations 6

(a) Suppose a projectile is launched vertically from the surface r = R of the earth with initial velocity Vo = "j2GM/R so V5 = e/R where k 2 = 2GM. Then solve the differential equation dr/dt = k/,.Jr (from Eq. (23) in this section) explicitly to deduce that ret) --+ 00 as t --+ 00. (b) If the projectile is launched vertically with initial velocity Vo > "j2GM/R, deduce that dr = jk2 +a > . dt r ,.Jr Why does it again follow that ret) --+ 00 as t --+ oo?

Chapter 1: Problem 1 Elementary Differential Equations 6

(a) Suppose that a body is dropped (vo = 0) from a distance ro > R from the earth's center, so its acceleration is dv/dt = -GM/r2 Ignoring air resistance, show that it reaches the height r < ro at time t =j ro (Jrro - r2 + rocos-1 ) . 2GM V;; (Suggestion: Substitute r ro cos2 e to evaluate f "jr/(ro - r) dr.) (b) If a body is dropped from a height of 1000 km above the earth's surface and air resistance is neglected, how long does it take to fall and with what speed will it strike the earth's surface?

Chapter 1: Problem 1 Elementary Differential Equations 6

Suppose that a projectile is fired straight upward from the surface of the earth with initial velocity Vo < "j2GM/R. Then its height yet) above the surface satisfies the initial value problem GM (y + R)2' yeO) = 0, y'(O) = vo. Substitute dv/dt = v (dv/dy) and then integrate to obtain 2GMy v 2 = V5 - -,:---'--,- R(R + y) for the velocity v of the projectile at height y. What maximum altitude does it reach if its initial velocity is 1 km/s?

Chapter 1: Problem 1 Elementary Differential Equations 6

In Jules Verne's original problem, the projectile launched from the surface of the earth is attracted by both the earth and the moon, so its distance r (t) from the center of the earth satisfies the initial value problem d2r GMe GMm dt2 = -7 + (S _ r)2 ; reO) = R, r'(O) = Vo where Me and Mm denote the masses of the earth and the moon, respectively; R is the radius of the earth and S = 384,400 km is the distance between the centers of the earth and the moon. To reach the moon, the projectile must only just pass the point between the moon and earth where its net acceleration vanishes. Thereafter it is "under the control" of the moon, and falls from there to the lunar surface. Find the minimal launch velocity Vo that suffices for the projectile to make it "From the Earth to the Moon."

Chapter 1: Problem 1 Elementary Differential Equations 6

Find general solutions of the differential equations in Problems 1 through 30. Primes denote derivatives with respect to x. x3 + 3y - xy' = 0

Chapter 1: Problem 2 Elementary Differential Equations 6

Find general solutions of the differential equations in Problems 1 through 30. Primes denote derivatives with respect to x. xy2 + 3y2 - x2y' = 0

Chapter 1: Problem 3 Elementary Differential Equations 6

Find general solutions of the differential equations in Problems 1 through 30. Primes denote derivatives with respect to x. xy + y2 - x2y' = 0

Chapter 1: Problem 4 Elementary Differential Equations 6

Find general solutions of the differential equations in Problems 1 through 30. Primes denote derivatives with respect to x. 2xy3 + eX + (3x2y2 + sin y)y' = 0

Chapter 1: Problem 5 Elementary Differential Equations 6

Find general solutions of the differential equations in Problems 1 through 30. Primes denote derivatives with respect to x. 3y + x4y' = 2xy

Chapter 1: Problem 6 Elementary Differential Equations 6

Find general solutions of the differential equations in Problems 1 through 30. Primes denote derivatives with respect to x. 2xy2 + x2y' = y2

Chapter 1: Problem 7 Elementary Differential Equations 6

Find general solutions of the differential equations in Problems 1 through 30. Primes denote derivatives with respect to x. 2x2y + x3y' = 1

Chapter 1: Problem 8 Elementary Differential Equations 6

Find general solutions of the differential equations in Problems 1 through 30. Primes denote derivatives with respect to x. 2xy + x2y' = y2

Chapter 1: Problem 9 Elementary Differential Equations 6

Find general solutions of the differential equations in Problems 1 through 30. Primes denote derivatives with respect to x. xy' + 2y = 6x2.jY

Chapter 1: Problem 10 Elementary Differential Equations 6

Find general solutions of the differential equations in Problems 1 through 30. Primes denote derivatives with respect to x. y' = 1 + x2 + y2 + X2y2

Chapter 1: Problem 11 Elementary Differential Equations 6

Find general solutions of the differential equations in Problems 1 through 30. Primes denote derivatives with respect to x. x2y' = xy + 3y2

Chapter 1: Problem 12 Elementary Differential Equations 6

Find general solutions of the differential equations in Problems 1 through 30. Primes denote derivatives with respect to x. 6xy3 + 2y4 + (9x2y2 + 8xy3)y' = 0

Chapter 1: Problem 13 Elementary Differential Equations 6

Find general solutions of the differential equations in Problems 1 through 30. Primes denote derivatives with respect to x. 4xy2 + y' = 5x4y2

Chapter 1: Problem 14 Elementary Differential Equations 6

Find general solutions of the differential equations in Problems 1 through 30. Primes denote derivatives with respect to x. x3y' = x2y _ y3

Chapter 1: Problem 15 Elementary Differential Equations 6

Find general solutions of the differential equations in Problems 1 through 30. Primes denote derivatives with respect to x. y' + 3y = 3x2e-3x

Chapter 1: Problem 16 Elementary Differential Equations 6

Find general solutions of the differential equations in Problems 1 through 30. Primes denote derivatives with respect to x. y' = x2 - 2xy + y2

Chapter 1: Problem 17 Elementary Differential Equations 6

Find general solutions of the differential equations in Problems 1 through 30. Primes denote derivatives with respect to x. eX + yexy + (eY + xeYX)y' = 0

Chapter 1: Problem 18 Elementary Differential Equations 6

Find general solutions of the differential equations in Problems 1 through 30. Primes denote derivatives with respect to x. 2x2y _ x3y' = y3

Chapter 1: Problem 19 Elementary Differential Equations 6

Find general solutions of the differential equations in Problems 1 through 30. Primes denote derivatives with respect to x. 3X5y2 + x3y' = 2y2

Chapter 1: Problem 20 Elementary Differential Equations 6

Find general solutions of the differential equations in Problems 1 through 30. Primes denote derivatives with respect to x. xy' + 3y = 3X -3/2

Chapter 1: Problem 21 Elementary Differential Equations 6

Find general solutions of the differential equations in Problems 1 through 30. Primes denote derivatives with respect to x. x2 - l)y' + (x - l)y = 1

Chapter 1: Problem 22 Elementary Differential Equations 6

Find general solutions of the differential equations in Problems 1 through 30. Primes denote derivatives with respect to x. xy' = 6y + 12x4y2/3

Chapter 1: Problem 23 Elementary Differential Equations 6

Find general solutions of the differential equations in Problems 1 through 30. Primes denote derivatives with respect to x. eY + y cos x + (xeY + sin x)y' = 0

Chapter 1: Problem 24 Elementary Differential Equations 6

Find general solutions of the differential equations in Problems 1 through 30. Primes denote derivatives with respect to x. 9x2y2 + X3/2y' = y2

Chapter 1: Problem 25 Elementary Differential Equations 6

Find general solutions of the differential equations in Problems 1 through 30. Primes denote derivatives with respect to x. 2y + (x + l)y' = 3x + 3

Chapter 1: Problem 26 Elementary Differential Equations 6

Find general solutions of the differential equations in Problems 1 through 30. Primes denote derivatives with respect to x. 9X 1/2y4/3 - 12x 1/5y3/2 + (8X3/2yl /3 - 15x6/5yl /2)y' = 0

Chapter 1: Problem 27 Elementary Differential Equations 6

Find general solutions of the differential equations in Problems 1 through 30. Primes denote derivatives with respect to x. 3y + X3y4 + 3xy' = 0

Chapter 1: Problem 28 Elementary Differential Equations 6

Find general solutions of the differential equations in Problems 1 through 30. Primes denote derivatives with respect to x. y + xy' = 2e2x

Chapter 1: Problem 29 Elementary Differential Equations 6

Find general solutions of the differential equations in Problems 1 through 30. Primes denote derivatives with respect to x. (2x + 1)y' + y = (2x + 1)3/2

Chapter 1: Problem 30 Elementary Differential Equations 6

Find general solutions of the differential equations in Problems 1 through 30. Primes denote derivatives with respect to x. y' = .Jx + y

Chapter 1: Problem 31 Elementary Differential Equations 6

Each of the differential equations in Problems 31 through 36 is of two different types considered in this chapter-separable, linear, homogeneous, Bernoulli, exact, etc. Hence, derive general solutionsfor each of these equations in two different ways; then reconcile your results. dy dy = 3(y +7)x2

Chapter 1: Problem 32 Elementary Differential Equations 6

Each of the differential equations in Problems 31 through 36 is of two different types considered in this chapter-separable, linear, homogeneous, Bernoulli, exact, etc. Hence, derive general solutionsfor each of these equations in two different ways; then reconcile your results. dy = 3(y +7)x2 32. - = xy3 -x

Chapter 1: Problem 33 Elementary Differential Equations 6

Each of the differential equations in Problems 31 through 36 is of two different types considered in this chapter-separable, linear, homogeneous, Bernoulli, exact, etc. Hence, derive general solutionsfor each of these equations in two different ways; then reconcile your results. dy 3x2 + 2y2 34. dy = x+3y = dx 4xy

Chapter 1: Problem 34 Elementary Differential Equations 6

Each of the differential equations in Problems 31 through 36 is of two different types considered in this chapter-separable, linear, homogeneous, Bernoulli, exact, etc. Hence, derive general solutionsfor each of these equations in two different ways; then reconcile your results. y = x+3y = dx 4xy dx y -3x

Chapter 1: Problem 35 Elementary Differential Equations 6

Each of the differential equations in Problems 31 through 36 is of two different types considered in this chapter-separable, linear, homogeneous, Bernoulli, exact, etc. Hence, derive general solutionsfor each of these equations in two different ways; then reconcile your results. dx x2 + 1

Chapter 1: Problem 36 Elementary Differential Equations 6

Each of the differential equations in Problems 31 through 36 is of two different types considered in this chapter-separable, linear, homogeneous, Bernoulli, exact, etc. Hence, derive general solutionsfor each of these equations in two different ways; then reconcile your results. dx tan x

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Textbook Solutions for Elementary Differential Equations

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Textbook Solutions for Elementary Differential Equations

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