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# 8R: Chapter 8 Review Exercises

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True or False? Justify your answer with a proof or a counterexample.

1) The differential equation $$y'=3x^2y−\cos(x)y''$$ is linear.

2) The differential equation $$y'=x−y$$ is separable.

$$F$$

3) You can explicitly solve all first-order differential equations by separation or by the method of integrating factors.

4) You can determine the behavior of all first-order differential equations using directional fields or Euler’s method.

$$T$$

For the following problems, find the general solution to the differential equations.

5) $$y′=x^2+3e^x−2x$$

6) $$y'=2^x+\cos^{−1}x$$

$$y(x)=\frac{2^x}{ln(2)}+xcos^{−1}x−\sqrt{1−x^2}+C$$

7) $$y'=y(x^2+1)$$

8) $$y'=e^{−y}\sin x$$

$$y(x)=\ln(C−\cos x)$$

9) $$y'=3x−2y$$

10) $$y'=y\ln y$$

$$y(x)=e^{e^{C+x}}$$

For the following problems, find the solution to the initial value problem.

11) $$y'=8x−\ln x−3x^4, \quad y(1)=5$$

12) $$y'=3x−\cos x+2, \quad y(0)=4$$

$$y(x)=4+\frac{3}{2}x^2+2x−\sin x$$

13) $$xy'=y(x−2), \quad y(1)=3$$

14) $$y'=3y^2(x+\cos x), \quad y(0)=−2$$

$$y(x)=−\dfrac{2}{1+3(x^2+2\sin x)}$$

15) $$(x−1)y'=y−2, \quad y(0)=0$$

16) $$y'=3y−x+6x^2, \quad y(0)=−1$$

$$y(x)=−2x^2−2x−\frac{1}{3}−\frac{2}{3}e^{3x}$$

For the following problems, draw the directional field associated with the differential equation, then solve the differential equation. Draw a sample solution on the directional field.

17) $$y'=2y−y^2$$

18) $$y'=\dfrac{1}{x}+\ln x−y,$$ for $$x>0$$

$$y(x)=Ce^{−x}+\ln x$$

For the following problems, use Euler’s Method with $$n=5$$ steps over the interval $$t=[0,1].$$ Then solve the initial-value problem exactly. How close is your Euler’s Method estimate?

19) $$y'=−4yx, \quad y(0)=1$$

20) $$y'=3^x−2y, \quad y(0)=0$$

Euler: $$0.6939$$,
Exact solution: $$y(x)=\dfrac{3^x−e^{−2x}}{2+\ln(3)}$$

For the following problems, set up and solve the differential equations.

21) A car drives along a freeway, accelerating according to $$a=5\sin(πt),$$ where $$t$$ represents time in minutes. Find the velocity at any time $$t$$, assuming the car starts with an initial speed of $$60$$ mph.

22) You throw a ball of mass $$2$$ kilograms into the air with an upward velocity of $$8$$ m/s. Find exactly the time the ball will remain in the air, assuming that gravity is given by $$g=9.8\,\text{m/s}^2$$.

$$\frac{40}{49}$$ second

23) You drop a ball with a mass of $$5$$ kilograms out an airplane window at a height of $$5000$$ m. How long does it take for the ball to reach the ground?

24) You drop the same ball of mass $$5$$ kilograms out of the same airplane window at the same height, except this time you assume a drag force proportional to the ball’s velocity, using a proportionality constant of $$3$$ and the ball reaches terminal velocity. Solve for the distance fallen as a function of time. How long does it take the ball to reach the ground?

$$x(t)=5000+\frac{245}{9}−\frac{49}{3}t−\frac{245}{9}e^{−5/3t}, \quad t=307.8$$ seconds

25) A drug is administered to a patient every $$24$$ hours and is cleared at a rate proportional to the amount of drug left in the body, with proportionality constant $$0.2$$. If the patient needs a baseline level of $$5$$ mg to be in the bloodstream at all times, how large should the dose be?

26) A $$1000$$-liter tank contains pure water and a solution of $$0.2$$ kg salt/L is pumped into the tank at a rate of $$1$$ L/min and is drained at the same rate. Solve for total amount of salt in the tank at time $$t$$.

$$T(t)=200\left(1−e^{−t/1000}\right)$$

27) You boil water to make tea. When you pour the water into your teapot, the temperature is $$100°C.$$ After $$5$$ minutes in your $$15°C$$ room, the temperature of the tea is $$85°C$$. Solve the equation to determine the temperatures of the tea at time $$t$$. How long must you wait until the tea is at a drinkable temperature ($$72°C$$)?

28) The human population (in thousands) of Nevada in $$1950$$ was roughly $$160$$. If the carrying capacity is estimated at $$10$$ million individuals, and assuming a growth rate of $$2\%$$ per year, develop a logistic growth model and solve for the population in Nevada at any time (use $$1950$$ as time = 0). What population does your model predict for $$2000$$? How close is your prediction to the true value of $$1,998,257$$?

$$P(t)=\dfrac{1600000e^{0.02t}}{9840+160e^{0.02t}}$$