Solved: Torricellis law An open cylindrical tank initially

What is a separable first-order differential equation?

Is the equation t2y_1t2 = t + 4 y2 separable?

Is the equation y_1t2 = 2y - t separable?

Explain how to solve a separable differential equation of the form g1y2y_1t2 = h1t2.

516. Solving separable equations Find the general solution of the following equations. Express the solution explicitly as a function of the independent variable. t-3y_1t2 = 1

516. Solving separable equations Find the general solution of the following equations. Express the solution explicitly as a function of the independent variable. e4t y_1t2 = 5

516. Solving separable equations Find the general solution of the following equations. Express the solution explicitly as a function of the independent variable. dy dt = 3t2 y

516. Solving separable equations Find the general solution of the following equations. Express the solution explicitly as a function of the independent variable. dy dx = y1x2 + 12

516. Solving separable equations Find the general solution of the following equations. Express the solution explicitly as a function of the independent variable. y_1t2 = ey>2 sin t

516. Solving separable equations Find the general solution of the following equations. Express the solution explicitly as a function of the independent variable. x2 dw dx = 2w13x + 12, x 7 0

516. Solving separable equations Find the general solution of the following equations. Express the solution explicitly as a function of the independent variable. x2y_1x2 = y2, x 7 0

516. Solving separable equations Find the general solution of the following equations. Express the solution explicitly as a function of the independent variable. 1t2 + 123yy_1t2 = t1y2 + 42

516. Solving separable equations Find the general solution of the following equations. Express the solution explicitly as a function of the independent variable. y_1t2 csc t = -y3 2

516. Solving separable equations Find the general solution of the following equations. Express the solution explicitly as a function of the independent variable. y_1t2et>2 = y2 + 4

516. Solving separable equations Find the general solution of the following equations. Express the solution explicitly as a function of the independent variable. u_1x2 = e2x-u

516. Solving separable equations Find the general solution of the following equations. Express the solution explicitly as a function of the independent variable. xu_1x2 = u2 - 4, x 7 0

1726. Solving initial value problems Determine whether the following equations are separable. If so, solve the initial value problem. ty_1t2 = 1, y112 = 2, t 7 0

1726. Solving initial value problems Determine whether the following equations are separable. If so, solve the initial value problem. sec t y_1t2 = 1, y102 = 1

1726. Solving initial value problems Determine whether the following equations are separable. If so, solve the initial value problem. 2yy_1t2 = 3t2, y102 = 9

1726. Solving initial value problems Determine whether the following equations are separable. If so, solve the initial value problem. y_1t2 = ety, y102 = 1

1726. Solving initial value problems Determine whether the following equations are separable. If so, solve the initial value problem. dy dt = ty + 2, y112 = 2

1726. Solving initial value problems Determine whether the following equations are separable. If so, solve the initial value problem. y_1t2 = y14t3 + 12, y102 = 4

1726. Solving initial value problems Determine whether the following equations are separable. If so, solve the initial value problem. y_1t2 = et 2y , y1ln 22 = 1

1726. Solving initial value problems Determine whether the following equations are separable. If so, solve the initial value problem. sec x y_1x2 = y3, y102 = 3

1726. Solving initial value problems Determine whether the following equations are separable. If so, solve the initial value problem. dy dx = ex-y, y102 = ln 3

1726. Solving initial value problems Determine whether the following equations are separable. If so, solve the initial value problem. y_1t2 = cos2 y, y112 = p 4

2732. Solutions in implicit form Solve the following initial value problems and leave the solution in implicit form. Use graphing software to plot the solution. If the implicit solution describes more than one curve, be sure to indicate which curve corresponds to the solution of the initial value problem. y_1t2 = t y , y112 = 2

2732. Solutions in implicit form Solve the following initial value problems and leave the solution in implicit form. Use graphing software to plot the solution. If the implicit solution describes more than one curve, be sure to indicate which curve corresponds to the solution of the initial value problem. y_1x2 = 1 + x 2 - y , y112 = 1

2732. Solutions in implicit form Solve the following initial value problems and leave the solution in implicit form. Use graphing software to plot the solution. If the implicit solution describes more than one curve, be sure to indicate which curve corresponds to the solution of the initial value problem. u_1x2 = csc u cos x 2 , u1p2 = p 2

2732. Solutions in implicit form Solve the following initial value problems and leave the solution in implicit form. Use graphing software to plot the solution. If the implicit solution describes more than one curve, be sure to indicate which curve corresponds to the solution of the initial value problem. yy_1x2 = 2x 12 + y222, y112 = -1

2732. Solutions in implicit form Solve the following initial value problems and leave the solution in implicit form. Use graphing software to plot the solution. If the implicit solution describes more than one curve, be sure to indicate which curve corresponds to the solution of the initial value problem. y_1x2 = A x + 1 y + 4 , y132 = 5

2732. Solutions in implicit form Solve the following initial value problems and leave the solution in implicit form. Use graphing software to plot the solution. If the implicit solution describes more than one curve, be sure to indicate which curve corresponds to the solution of the initial value problem. z_1x2 = z 2 + 4 x2 + 16 , z142 = 2

Logistic equation for a population A community of hares on an island has a population of 50 when observations begin (at t = 0). The population is modeled by the initial value problem dP dt = 0.08Pa1 - P 200 b, P102 = 50. a. Find and graph the solution of the initial value problem, for t 0. b. What is the steady-state population?

Logistic equation for an epidemic When an infected person is introduced into a closed and otherwise healthy community, the number of people who contract the disease (in the absence of any intervention) may be modeled by the logistic equation dP dt = kPa1 - P A b, P102 = P0, where k is a positive infection rate, A is the number of people in the community, and P0 is the number of infected people at t = 0. The model also assumes no recovery. a. Find the solution of the initial value problem, for t 0, in terms of k, A, and P0. b. Graph the solution in the case that k = 0.025, A = 300, and P0 = 1. c. For a fixed value of k and A, describe the long-term behavior of the solutions, for any P0 with 0 6 P0 6 A.

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample. a. The equation u_1x2 = 1x2u72-1 is separable. b. The general solution of the separable equation y_1t2 = t y7 + 10y4 can be expressed explicitly with y in terms of t. c. The general solution of the equation yy_1x2 = xe-y can be found using integration by parts. T T T

3639. Solutions of separable equations Solve the following initial value problems. When possible, give the solution as an explicit function of t. eyy_1t2 = ln2 t t , y112 = ln 2

3639. Solutions of separable equations Solve the following initial value problems. When possible, give the solution as an explicit function of t. y_1t2 = 3y1y + 12 t , y112 = 1

3639. Solutions of separable equations Solve the following initial value problems. When possible, give the solution as an explicit function of t. y_1t2 = cos2 t 2y , y102 = -2

3639. Solutions of separable equations Solve the following initial value problems. When possible, give the solution as an explicit function of t. y_1t2 = y + 3 5t + 6 , y122 = 0

4041. Implicit solutions for separable equations For the following separable equations, carry out the indicated analysis. a. Find the general solution of the equation. b. Find the value of the arbitrary constant associated with each initial condition. (Each initial condition requires a different constant.) c. Use the graph of the general solution that is provided to sketch the solution curve for each initial condition. y2y_1t2 = t2 + 2 3 t; y1-12 = 1, y112 = 0, y1-12 = -1 y t 1 _1 _1 1

4041. Implicit solutions for separable equations For the following separable equations, carry out the indicated analysis. a. Find the general solution of the equation. b. Find the value of the arbitrary constant associated with each initial condition. (Each initial condition requires a different constant.) c. Use the graph of the general solution that is provided to sketch the solution curve for each initial condition. e-y>2y_1x2 = 4x sin x2 - x; y102 = 0, y102 = ln a 1 4 b, ya A p 2 b = 0 y x 1 _1 1 _1

Orthogonal trajectories Two curves are orthogonal to each other if their tangent lines are perpendicular at each point of intersection. A family of curves forms orthogonal trajectories with another family of curves if each curve in one family is orthogonal to each curve in the other family. Use the following steps to find the orthogonal trajectories of the family of ellipses 2x2 + y2 = a2. a. Apply implicit differentiation to 2x2 + y2 = a2 to show that dy dx = -2x y . b. The family of trajectories orthogonal to 2x2 + y2 = a2 satisfies the differential equation dy dx = y 2x . Why? c. Solve the differential equation in part (b) to verify that y2 = eC_x_ and then explain why it follows that y2 = kx. Therefore, the family of parabolas y2 = kx forms the orthogonal trajectories of the family of ellipses 2x2 + y2 = a2.

Orthogonal trajectories Use the method in Exercise 42 to find the orthogonal trajectories for the family of circles x2 + y2 = a2.

Logistic equation for spread of rumors Sociologists model the spread of rumors using logistic equations. The key assumption is that at any given time, a fraction y of the population, where 0 y 1, knows the rumor, while the remaining fraction 1 - y does not. Furthermore, the rumor spreads by interactions between those who know the rumor and those who do not. The number of such interactions is proportional to y11 - y2. Therefore, the equation that describes the spread of the rumor is y_1t2 = ky11 - y2, where k is a positive real number. The number of people who initially know the rumor is y102 = y0, where 0 y0 1. a. Solve this initial value problem and give the solution in terms of k and y0. b. Assume k = 0.3 weeks-1 and graph the solution for y0 = 0.1 and y0 = 0.7. c. Describe and interpret the long-term behavior of the rumor function, for any 0 y0 1.

Free fall An object in free fall may be modeled by assuming that the only forces at work are the gravitational force and air resistance. By Newtons Second Law of Motion 1mass * acceleration = the sum of the external forces2, the velocity of the object satisfies the differential equation m # v_1t2 = mg + f 1v2, 6 ()* (++)++* mass acceleration external forces where f is a function that models the air resistance (assuming the positive direction is downward). One common assumption (often used for motion in air) is that f 1v2 = -kv2, where k 7 0 is a drag coefficient. a. Show that the equation can be written in the form v_1t2 = g - av2, where a = k>m. b. For what (positive) value of v is v_1t2 = 0? (This equilibrium solution is called the terminal velocity.) c. Find the solution of this separable equation assuming v102 = 0 and 0 6 v2 6 g>a. d. Graph the solution found in part (c) with g = 9.8 m>s2, m = 1, and k = 0.1, and verify that the terminal velocity agrees with the value found in part (b).

Free fall Using the background given in Exercise 45, assume the resistance is given by f 1v2 = -Rv, where R 7 0 is a drag coefficient (an assumption often made for a heavy medium such as water or oil). a. Show that the equation can be written in the form v_1t2 = g - bv, where b = R>m. b. For what value of v is v_1t2 = 0? (This equilibrium solution is called the terminal velocity.) c. Find the solution of this separable equation assuming v102 = 0 and 0 6 v 6 g>b. T T T d. Graph the solution found in part (c) with g = 9.8 m>s2, m = 1, and R = 0.1, and verify that the terminal velocity agrees with the value found in part (b).

Torricellis law An open cylindrical tank initially filled with water drains through a hole in the bottom of the tank according to Torricellis Law (see figure). If h1t2 is the depth of water in the tank, for t 0, then Torricellis Law implies h_1t2 = -2k1h, where k is a constant that includes g = 9.8 m>s2, the radius of the tank, and the radius of the drain. Assume that the initial depth of the water is h102 = H. a. Find the solution of the initial value problem. b. Find the solution in the case that k = 0.1 and H = 0.5 m. c. In part (b), how long does it take for the tank to drain? d. Graph the solution in part (b) and check that it is consistent with part (c). h(t) A a

Chemical rate equations Let y1t2 be the concentration of a substance in a chemical reaction (typical units are moles>liter). The change in the concentration, under appropriate conditions, is modeled by the equation dy dt = -kyn, where k 7 0 is a rate constant and the positive integer n is the order of the reaction. a. Show that for a first-order reaction 1n = 12, the concentration obeys an exponential decay law. b. Solve the initial value problem for a second-order reaction 1n = 22 assuming y102 = y0. c. Graph the concentration for a first-order and second-order reaction with k = 0.1 and y0 = 1.

Tumor growth The Gompertz growth equation is often used to model the growth of tumors. Let M1t2 be the mass of a tumor at time t 0. The relevant initial value problem is dM dt = -rM ln a M K b, M102 = M0, where r and K are positive constants and 0 6 M0 6 K. a. Graph the growth rate function R1M2 = -rM ln a M K b (which equals M_1t2) assuming r = 1 and K = 4. For what values of M is the growth rate positive? For what value of M is the growth rate a maximum? b. Solve the initial value problem and graph the solution for r = 1, K = 4, and M0 = 1. Describe the growth pattern of the tumor. Is the growth unbounded? If not, what is the limiting size of the tumor? c. In the general solution, what is the meaning of K?

Technology for an initial value problem Solve y_1t2 = yet cos3 4t, y102 = 1, and plot the solution for 0 t p.

Blowup in finite time Consider the initial value problem y_1t2 = yn+1, y102 = y0, where n is a positive integer. a. Solve the initial value problem with n = 1 and y0 = 1. b. Solve the initial value problem with n = 2 and y0 = 1 12 . c. Solve the problem for positive integers n and y0 = n-1>n. How do solutions behave as tS 1-?

Analysis of a separable equation Consider the differential equation y_1t2 = y1y + 12 t1t + 22 and carry out the following analysis. a. Show that the general solution of the equation can be written in the form y1t2 = 1t C1t + 2 - 1t . b. Now consider the initial value problem y112 = A, where A is a real number. Show that the solution of the initial value problem is y1t2 = 1t a 1 + A 13A b1t + 2 - 1t . T T c. Find and graph the solution that satisfies the initial condition y112 = 1. d. Describe the behavior of the solution in part (c) as t increases. e. Find and graph the solution that satisfies the initial condition y112 = 2. f. Describe the behavior of the solution in part (e) as t increases. g. In the cases in which the solution is bounded for t 7 0, what is the value of lim tS_ y1t2?

Analysis of a separable equation Consider the differential equation yy_1t2 = 12 et + t and carry out the following analysis. a. Find the general solution of the equation and express it explicitly as a function of t in two cases: y 7 0 and y 6 0. b. Find the solutions that satisfy the initial conditions y1-12 = 1 and y1-12 = 2. c. Graph the solutions in part (b) and describe their behavior as t increases. d. Find the solutions that satisfy the initial conditions y1-12 = -1 and y1-12 = -2. e. Graph the solutions in part (d) and describe their behavior as t increases.