Solved: Investment Growth A large corporation starts at time to invest part of its

In Exercises 1-4, determine whether the differential equation is linear. Explain your reasoning. \(x^{3} y^{\prime}+x y=e^{x}+1\) Text Transcription: x^3 y^prime+x y=e^x+1

In Exercises 1-4, determine whether the differential equation is linear. Explain your reasoning. \(2 x y-y^{\prime} \ln x=y\) Text Transcription: 2 x y-y^prime ln x=y

In Exercises 1-4, determine whether the differential equation is linear. Explain your reasoning. \(y^{\prime}-y \sin x=x y^{2}\) Text Transcription: y^prime-y sin x=x y^2

In Exercises 1-4, determine whether the differential equation is linear. Explain your reasoning. \(\frac{2-y^{\prime}}{y}=5 x\) Text Transcription: frac 2-y^prime y=5 x

In Exercises 5-14, solve the first-order linear differential equation. \(\frac{d y}{d x}+\left(\frac{1}{x}\right) y=6 x+2\) Text Transcription: frac d y d x+\(frac 1 x) y=6 x+2

In Exercises 5-14, solve the first-order linear differential equation. \(\frac{d y}{d x}+\left(\frac{2}{x}\right) y=3 x-5\) Text Transcription: frac d y d x+(frac 2 x) y=3 x-5

In Exercises 5-14, solve the first-order linear differential equation. \(y^{\prime}-y=16\) Text Transcription: y^prime-y=16

In Exercises 5-14, solve the first-order linear differential equation. \(y^{\prime}+2 x y=10 x\) Text Transcription: y^prime+2 x y=10 x

In Exercises 5-14, solve the first-order linear differential equation. (y + 1) cos x dx - dy = 0

In Exercises 5-14, solve the first-order linear differential equation. (y - 1) sin x dx - dy = 0

In Exercises 5-14, solve the first-order linear differential equation. \((x-1) y^{\prime}+y=x^{2}-1\) Text Transcription: (x - 1) y^prime+y = x^2 - 1

In Exercises 5-14, solve the first-order linear differential equation. \(y^{\prime}+3 y=e^{3 x}\) Text Transcription: y^prime+3 y=e^3 x

In Exercises 5-14, solve the first-order linear differential equation. \(y^{\prime}-3 x^{2} y=e^{x^{3}}\) Text Transcription: y^prime-3 x^2 y=e^x^3

In Exercises 5-14, solve the first-order linear differential equation. \(y^{\prime}+y \tan x=\sec x\) Text Transcription: y^prime+y tan x=sec x

In Exercises 15 and 16, (a) sketch an approximate solution of the differential equation satisfying the given initial condition by hand on the slope field, (b) find the particular solution that satisfies the given initial condition, and (c) use a graphing utility to graph the particular solution. Compare the graph with the hand-drawn graph in part (a). To print an enlarged copy of the graph, go to MathGraphs.com. \(\frac{d y}{d x}=e^{x}-y\) (0, 1) Text Transcription: frac d y d x=e^x-y

In Exercises 15 and 16, (a) sketch an approximate solution of the differential equation satisfying the given initial condition by hand on the slope field, (b) find the particular solution that satisfies the given initial condition, and (c) use a graphing utility to graph the particular solution. Compare the graph with the hand-drawn graph in part (a). To print an enlarged copy of the graph, go to MathGraphs.com. \(y^{\prime}+\left(\frac{1}{x}\right) y=\sin x^{2}\) \((\sqrt{\pi}, 0)\) Text Transcription: y^prime+(frac 1 x) y=sin x^2 (sqrt pi, 0)

In Exercises 17-24, find the particular solution of the differential equation that satisfies the initial condition. Differential Equation Initial Condition \(y^{\prime} \cos ^{2} x+y-1=0\) y(0) = 5 Text Transcription: y^prime cos ^2 x+y-1=0

In Exercises 17-24, find the particular solution of the differential equation that satisfies the initial condition. Differential Equation Initial Condition \(x^{3} y^{\prime}+2 y=e^{1 / x^{2}}\) y(1) = e Text Transcription: x^3 y^prime+2 y=e^1 / x^2

In Exercises 17-24, find the particular solution of the differential equation that satisfies the initial condition. Differential Equation Initial Condition \(y^{\prime}+y \tan x=\sec x+\cos x\) y(0) = 1 Text Transcription: y^prime+y tan x=sec x+cos x

In Exercises 17-24, find the particular solution of the differential equation that satisfies the initial condition. Differential Equation Initial Condition \(y^{\prime}+y \sec x=\sec x\) y(0) = 4 Text Transcription: y^prime+y sec x=sec x

In Exercises 17-24, find the particular solution of the differential equation that satisfies the initial condition. Differential Equation Initial Condition \(y^{\prime}+\left(\frac{1}{x}\right) y=0\) y(2) = 2 Text Transcription: y^prime+(frac 1 x) y=0

In Exercises 17-24, find the particular solution of the differential equation that satisfies the initial condition. Differential Equation Initial Condition \(y^{\prime}+(2 x-1) y=0\) y(1) = 2 Text Transcription: y^prime+(2 x-1) y=0

In Exercises 17-24, find the particular solution of the differential equation that satisfies the initial condition. Differential Equation Initial Condition x dy = (x + y + 2) dx y(1) = 10

In Exercises 17-24, find the particular solution of the differential equation that satisfies the initial condition. Differential Equation Initial Condition \(2 x y^{\prime}-y=x^{3}-x\) y(4) = 2 Text Transcription: 2 x y^prime-y=x^3-x

When predicting population growth, demographers must consider birth and death rates as well as the net change caused by the difference between the rates of immigration and emigration. Let P be the population at time t and let N be the net increase per unit time resulting from the difference between immigration and emigration. So, the rate of growth of the population is given by \(\frac{d P}{d t}=k P+N\) where N is constant. Solve this differential equation to find P as a function of time, when at time t = 0 the size of the population is \(P_{0}\). Text Transcription: frac{d P}{d t}=k P+N P_0

A large corporation starts at time t = 0 to invest part of its receipts continuously at a rate of P dollars per year in a fund for future corporate expansion. Assume that the fund earns r percent interest per year compounded continuously. So, the rate of growth of the amount A in the fund is given by \(\frac{d A}{d t}=r A+P\) where A = 0 when t = 0. Solve this differential equation for A as a function of t. Text Transcription: frac d A d t=r A+P

In Exercises 27 and 28, use the result of Exercise 26. Find A for the following. (a) P = $275,000, r = 8%, t = 10 years (b) P = $550,000, r = 5.9%, t = 25 years

In Exercises 27 and 28, use the result of Exercise 26. Find t if the corporation needs $1,000,000 and it can invest $125,000 per year in a fund earning 8% interest compounded continuously.

The management at a certain factory has found that the maximum number of units a worker can produce in a day is 75. The rate of increase in the number of units N produced with respect to time t in days by a new employee is proportional to 75 - N. (a) Determine the differential equation describing the rate of change of performance with respect to time. (b) Solve the differential equation from part (a). (c) Find the particular solution for a new employee who produced 20 units on the first day at the factory and 35 units on the twentieth day.

Glucose is added intravenously to the bloodstream at the rate of q units per minute, and the body removes glucose from the bloodstream at a rate proportional to the amount present. Assume that Q(t) is the amount of glucose in the bloodstream at time t. (a) Determine the differential equation describing the rate of change of glucose in the bloodstream with respect to time. (b) Solve the differential equation from part (a), letting \(Q=Q_{0}\) when t = 0. (c) Find the limit of Q(t) as \(t \rightarrow \infty\). Text Transcription: Q=Q_0 t rightarrow infty

In Exercises 31 and 32, consider an eight-pound object dropped from a height of 5000 feet, where the air resistance is proportional to the velocity. Write the velocity of the object as a function of time when the velocity after 5 seconds is approximately -101 feet per second. What is the limiting value of the velocity function?

In Exercises 31 and 32, consider an eight-pound object dropped from a height of 5000 feet, where the air resistance is proportional to the velocity. Use the result of Exercise 31 to write the position of the object as a function of time. Approximate the velocity of the object when it reaches ground level.

In Exercises 33 and 34, use the differential equation for electric circuits given by \(L \frac{d I}{d t}+R I=E\) In this equation, I is the current, R is the resistance, L is the inductance, and E is the electromotive force (voltage). Solve the differential equation for the current given a constant voltage \(E_{0}\). Text Transcription: L frac d I d t+R I=E E_0

In Exercises 33 and 34, use the differential equation for electric circuits given by \(L \frac{d I}{d t}+R I=E\) In this equation, I is the current, R is the resistance, L is the inductance, and E is the electromotive force (voltage). Use the result of Exercise 33 to find the equation for the current when \(I(0)=0, E_{0}=120\) volts, R = 600 ohms, and L= 4 henrys. When does the current reach 90% of its limiting value? Text Transcription: L frac d I d t+R I=E I(0)=0, E_0=120

In Exercises 35-38, consider a tank that at time t = 0 contains \(v_{0}\) gallons of a solution of which, by weight, \(q_{0}\) pounds is soluble concentrate. Another solution containing \(q_{1}\) pounds of the concentrate per gallon is running into the tank at the rate of \(r_{1}\) gallons per minute. The solution in the tank is kept well stirred and is withdrawn at the rate of \(r_{2}\) gallons per minute. Let Q be the amount of concentrate in the solution at any time t. Show that \(\frac{d Q}{d t}+\frac{r_{2} Q}{v_{0}+\left(r_{1}-r_{2}\right) t}=q_{1} r_{1}\) Text Transcription: v_0 q_0 q_1 r_1 r_2 frac d Q d t+frac r_2 Q v_0+(r_1-r_2) t=q_1 r_1

In Exercises 35-38, consider a tank that at time t = 0 contains \(v_{0}\) gallons of a solution of which, by weight, \(q_{0}\) pounds is soluble concentrate. Another solution containing \(q_{1}\) pounds of the concentrate per gallon is running into the tank at the rate of \(r_{1}\) gallons per minute. The solution in the tank is kept well stirred and is withdrawn at the rate of \(r_{2}\) gallons per minute. Let Q be the amount of concentrate in the solution at any time t. Write the differential equation for the rate of change of Q with respect to t when \(r_{1}=r_{2}=r\). Text Transcription: v_0 q_0 q_1 r_1 r_2 r_1=r_2=r

In Exercises 35-38, consider a tank that at time t = 0 contains \(v_{0}\) gallons of a solution of which, by weight, \(q_{0}\) pounds is soluble concentrate. Another solution containing \(q_{1}\) pounds of the concentrate per gallon is running into the tank at the rate of \(r_{1}\) gallons per minute. The solution in the tank is kept well stirred and is withdrawn at the rate of \(r_{2}\) gallons per minute. A 200-gallon tank is full of a solution containing 25 pounds of concentrate. Starting at time t = 0, distilled water is admitted to the tank at a rate of 10 gallons per minute, and the well-stirred solution is withdrawn at the same rate. (a) Find the amount of concentrate Q in the solution as a function of t. (b) Find the time at which the amount of concentrate in the tank reaches 15 pounds. (c) Find the quantity of the concentrate in the solution as \(t \rightarrow \infty\). Text Transcription: v_0 q_0 q_1 r_1 r_2 t rightarrow infty

In Exercises 35-38, consider a tank that at time t = 0 contains \(v_{0}\) gallons of a solution of which, by weight, \(q_{0}\) pounds is soluble concentrate. Another solution containing \(q_{1}\) pounds of the concentrate per gallon is running into the tank at the rate of \(r_{1}\) gallons per minute. The solution in the tank is kept well stirred and is withdrawn at the rate of \(r_{2}\) gallons per minute. A 200-gallon tank is half full of distilled water. At time t = 0, a solution containing 0.5 pound of concentrate per gallon enters the tank at the rate of 5 gallons per minute, and the well-stirred mixture is withdrawn at the rate of 3 gallons per minute. (a) At what time will the tank be full? (b) At the time the tank is full, how many pounds of concentrate will it contain? (c) Repeat parts (a) and (b), assuming that the solution entering the tank contains 1 pound of concentrate per gallon. Text Transcription: v_0 q_0 q_1 r_1 r_2

The expression u(x) is an integrating factor for \(y^{\prime}+P(x) y=Q(x)\). Which of the following is equal to \(u^{\prime}(x)\)? Verify your answer. (a) P(x)u(x) (b) \(P^{\prime}(x) u(x)\) (c) Q(x)u(x) (d) \(Q^{\prime}(x) u(x)\) Text Transcription: y^prime+P(x) y=Q(x) u^prime(x) P^prime(x) u(x) Q^prime(x) u(x)

The graph shows the amount of concentrate Q (in pounds) in a solution in a tank at time t (in minutes) as a solution with concentrate enters the tank, is well stirred, and is withdrawn from the tank. (a) How much concentrate is in the tank at time t = 0? (b) Which is greater, the rate of solution into the tank, or the rate of solution withdrawn from the tank? Explain. (c) At what time is there no concentrate in the tank? What does this mean?

Standard Form Give the standard form of a first-order linear differential equation. What is its integrating factor?

What does the term “first-order” refer to in a first-order linear differential equation?

In Exercises 43-46, match the differential equation with its solution. Differential Equation Solution \(y^{\prime}-2 x=0\) (a) \(y=C e^{x^{2}}\) (b) \(y=-\frac{1}{2}+C e^{x^{2}}\) (c) \(y=x^{2}+C\) (d) \(y=C e^{2 x}\) Text Transcription: y^prime-2 x=0 y=C e^x^2 y=-frac 1 2+C e^x^2 y=x^2+C y=C e^2 x

In Exercises 43-46, match the differential equation with its solution. Differential Equation Solution \(y^{\prime}-2 y=0\) (a) \(y=C e^{x^{2}}\) (b) \(y=-\frac{1}{2}+C e^{x^{2}}\) (c) \(y=x^{2}+C\) (d) \(y=C e^{2 x}\) Text Transcription: y^prime-2 y=0 y=C e^x^2 y=-frac 1 2+C e^x^2 y=x^2+C y=C e^2 x

In Exercises 43-46, match the differential equation with its solution. Differential Equation Solution \(y^{\prime}-2 x y=0\) (a) \(y=C e^{x^{2}}\) (b) \(y=-\frac{1}{2}+C e^{x^{2}}\) (c) \(y=x^{2}+C\) (d) \(y=C e^{2 x}\) Text Transcription: y^prime-2 x y=0 y=C e^x^2 y=-frac 1 2+C e^x^2 y=x^2+C y=C e^2 x

In Exercises 43-46, match the differential equation with its solution. Differential Equation Solution \(y^{\prime}-2 x y=x\) (a) \(y=C e^{x^{2}}\) (b) \(y=-\frac{1}{2}+C e^{x^{2}}\) (c) \(y=x^{2}+C\) (d) \(y=C e^{2 x}\) Text Transcription: y^prime-2 x y=x y=C e^x^2 y=-frac 1 2+C e^x^2 y=x^2+C y=C e^2 x

In Exercises 47-50, (a) use a graphing utility to graph the slope field for the differential equation, (b) find the particular solutions of the differential equation passing through the given points, and (c) use a graphing utility to graph the particular solutions on the slope field. Differential Equation Points \(\frac{d y}{d x}-\frac{1}{x} y=x^{2}\) (-2, 4), (2, 8) Text Transcription: frac d y d x-frac 1 x y=x^2

In Exercises 47-50, (a) use a graphing utility to graph the slope field for the differential equation, (b) find the particular solutions of the differential equation passing through the given points, and (c) use a graphing utility to graph the particular solutions on the slope field. Differential Equation Points \(\frac{d y}{d x}+4 x^{3} y=x^{3}\) \(\left(0, \frac{7}{2}\right),\left(0,-\frac{1}{2}\right)\) Text Transcription: frac d y d x+4 x^3 y=x^3 (0, frac 7 2),(0,-frac 1 2)

In Exercises 47-50, (a) use a graphing utility to graph the slope field for the differential equation, (b) find the particular solutions of the differential equation passing through the given points, and (c) use a graphing utility to graph the particular solutions on the slope field. Differential Equation Points \(\frac{d y}{d x}+(\cot x) y=2\) (1, 1), (3, -1) Text Transcription: frac d y d x+(cot x) y=2

In Exercises 47-50, (a) use a graphing utility to graph the slope field for the differential equation, (b) find the particular solutions of the differential equation passing through the given points, and (c) use a graphing utility to graph the particular solutions on the slope field. Differential Equation Points \(\frac{d y}{d x}+2 x y=x y^{2}\) (0, 3), (0, 1) Text Transcription: frac d y d x+2 x y=x y^2

In Exercises 51-58, solve the first-order differential equation by any appropriate method. \(\frac{d y}{d x}=\frac{e^{2 x+y}}{e^{x-y}}\) Text Transcription: frac d y d x=frac e^2 x+y e^x-y

In Exercises 51-58, solve the first-order differential equation by any appropriate method. \(\frac{d y}{d x}=\frac{x-3}{y(y+4)}\) Text Transcription: frac d y d x=frac x-3 y(y+4)

In Exercises 51-58, solve the first-order differential equation by any appropriate method. \(y \cos x-\cos x+\frac{d y}{d x}=0\) Text Transcription: y cos x-cos x+frac d y d x=0

In Exercises 51-58, solve the first-order differential equation by any appropriate method. \(y^{\prime}=2 x \sqrt{1-y^{2}}\) Text Transcription: y^prime=2 x sqrt 1-y^2

In Exercises 51-58, solve the first-order differential equation by any appropriate method. \(\left(2 y-e^{x}\right) d x+x \ d y=0\) Text Transcription: (2 y-e^x) d x+x d y=0

In Exercises 51-58, solve the first-order differential equation by any appropriate method. \((x+y) d x-x \ d y=0\) Text Transcription: (x+y) d x-x d y=0

In Exercises 51-58, solve the first-order differential equation by any appropriate method. \(3\left(y-4 x^{2}\right) d x+x \ d y=0\) Text Transcription: 3(y-4 x^2) d x+x d y=0

In Exercises 51-58, solve the first-order differential equation by any appropriate method. \(x d x+\left(y+e^{y}\right)\left(x^{2}+1\right) d y=0\) Text Transcription: x d x+(y+e^y)(x^2+1) d y=0

In Exercises 59-66, solve the Bernoulli differential equation. The Bernoulli equation is a well-known nonlinear equation of the form \(y^{\prime}+P(x) y=Q(x) y^{n}\) that can be reduced to a linear form by a substitution. The general solution of a Bernoulli equation is \(y^{1-n} e^{\int(1-n) P(x) d x}=\int(1-n) Q(x) e^{\int(1-n) P(x) d x} d x+C\). \(y^{\prime}+3 x^{2} y=x^{2} y^{3}\) Text Transcription: y^prime+P(x) y=Q(x) y^n y^1-n e^int(1-n) P(x) d x=int(1-n) Q(x) e^int(1-n) P(x) d x d x+C y^prime+3 x^2 y=x^2 y^3

In Exercises 59-66, solve the Bernoulli differential equation. The Bernoulli equation is a well-known nonlinear equation of the form \(y^{\prime}+P(x) y=Q(x) y^{n}\) that can be reduced to a linear form by a substitution. The general solution of a Bernoulli equation is \(y^{1-n} e^{\int(1-n) P(x) d x}=\int(1-n) Q(x) e^{\int(1-n) P(x) d x} d x+C\). \(y^{\prime}+x y=x y^{-1}\) Text Transcription: y^prime+P(x) y=Q(x) y^n y^1-n e^int(1-n) P(x) d x=int(1-n) Q(x) e^int(1-n) P(x) d x d x+C y^prime+x y=x y^-1

In Exercises 59-66, solve the Bernoulli differential equation. The Bernoulli equation is a well-known nonlinear equation of the form \(y^{\prime}+P(x) y=Q(x) y^{n}\) that can be reduced to a linear form by a substitution. The general solution of a Bernoulli equation is \(y^{1-n} e^{\int(1-n) P(x) d x}=\int(1-n) Q(x) e^{\int(1-n) P(x) d x} d x+C\). \(y^{\prime}+\left(\frac{1}{x}\right) y=x y^{2}\) Text Transcription: y^prime+P(x) y=Q(x) y^n y^1-n e^int(1-n) P(x) d x=int(1-n) Q(x) e^int(1-n) P(x) d x d x+C y^prime+(frac 1 x) y=x y^2

In Exercises 59-66, solve the Bernoulli differential equation. The Bernoulli equation is a well-known nonlinear equation of the form \(y^{\prime}+P(x) y=Q(x) y^{n}\) that can be reduced to a linear form by a substitution. The general solution of a Bernoulli equation is \(y^{1-n} e^{\int(1-n) P(x) d x}=\int(1-n) Q(x) e^{\int(1-n) P(x) d x} d x+C\). \(y^{\prime}+\left(\frac{1}{x}\right) y=x \sqrt{y}\) Text Transcription: y^prime+P(x) y=Q(x) y^n y^1-n e^int(1-n) P(x) d x=int(1-n) Q(x) e^int(1-n) P(x) d x d x+C y^prime+(frac 1 x) y=x sqrt y

In Exercises 59-66, solve the Bernoulli differential equation. The Bernoulli equation is a well-known nonlinear equation of the form \(y^{\prime}+P(x) y=Q(x) y^{n}\) that can be reduced to a linear form by a substitution. The general solution of a Bernoulli equation is \(y^{1-n} e^{\int(1-n) P(x) d x}=\int(1-n) Q(x) e^{\int(1-n) P(x) d x} d x+C\). \(x y^{\prime}+y=x y^{3}\) Text Transcription: y^prime+P(x) y=Q(x) y^n y^1-n e^int(1-n) P(x) d x=int(1-n) Q(x) e^int(1-n) P(x) d x d x+C x y^prime+y=x y^3

In Exercises 59-66, solve the Bernoulli differential equation. The Bernoulli equation is a well-known nonlinear equation of the form \(y^{\prime}+P(x) y=Q(x) y^{n}\) that can be reduced to a linear form by a substitution. The general solution of a Bernoulli equation is \(y^{1-n} e^{\int(1-n) P(x) d x}=\int(1-n) Q(x) e^{\int(1-n) P(x) d x} d x+C\). \(y^{\prime}-y=y^{3}\) Text Transcription: y^prime+P(x) y=Q(x) y^n y^1-n e^int(1-n) P(x) d x=int(1-n) Q(x) e^int(1-n) P(x) d x d x+C y^prime-y=y^3

In Exercises 59-66, solve the Bernoulli differential equation. The Bernoulli equation is a well-known nonlinear equation of the form \(y^{\prime}+P(x) y=Q(x) y^{n}\) that can be reduced to a linear form by a substitution. The general solution of a Bernoulli equation is \(y^{1-n} e^{\int(1-n) P(x) d x}=\int(1-n) Q(x) e^{\int(1-n) P(x) d x} d x+C\). \(y^{\prime}-y=e^{x} \sqrt[3]{y}\) Text Transcription: y^prime+P(x) y=Q(x) y^n y^1-n e^int(1-n) P(x) d x=int(1-n) Q(x) e^int(1-n) P(x) d x d x+C y^prime-y=e^x sqrt[3]y

In Exercises 59-66, solve the Bernoulli differential equation. The Bernoulli equation is a well-known nonlinear equation of the form \(y^{\prime}+P(x) y=Q(x) y^{n}\) that can be reduced to a linear form by a substitution. The general solution of a Bernoulli equation is \(y^{1-n} e^{\int(1-n) P(x) d x}=\int(1-n) Q(x) e^{\int(1-n) P(x) d x} d x+C\). \(y y^{\prime}-2 y^{2}=e^{x}\) Text Transcription: y^prime+P(x) y=Q(x) y^n y^1-n e^int(1-n) P(x) d x=int(1-n) Q(x) e^int(1-n) P(x) d x d x+C y y^prime-2 y^2=e^x

In Exercises 67 and 68, determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. \(y^{\prime}+x \sqrt{y}=x^{2}\) is a first-order linear differential equation. Text Transcription: y^prime+x sqrt y=x^2

In Exercises 67 and 68, determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. \(y^{\prime}+x y=e^{x} y\) is a first-order linear differential equation. Text Transcription: y^prime+x y=e^x y