8.R: Chapter 8 Review Exercises

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.

Answer

$F$

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.

Answer

$T$

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

5) $y′=x^2+3e^x−2x$

6) $y'=2^x+\cos^{−1}x$

Answer

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

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

8) $y'=e^{−y}\sin x$

Answer

$y(x)=\ln(C−\cos x)$

9) $y'=3x−2y$

10) $y'=y\ln y$

Answer

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

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

11) $y'=8x−\ln x−3x^4, \quad y(1)=5$

12) $y'=3x−\cos x+2, \quad y(0)=4$

Answer

$y(x)=4+\frac{3}{2}x^2+2x−\sin x$

13) $xy'=y(x−2), \quad y(1)=3$

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

Answer

$y(x)=−\dfrac{2}{1+3(x^2+2\sin x)}$

15) $(x−1)y'=y−2, \quad y(0)=0$

16) $y'=3y−x+6x^2, \quad y(0)=−1$

Answer

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

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 $x>0$

Answer

$y(x)=Ce^{−x}+\ln x$

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) $y'=3^x−2y, \quad y(0)=0$

Answer

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 $g=9.8\,\text{m/s}^2$.

Answer

$\frac{40}{49}$ 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?

Answer

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

Answer

$T(t)=200\left(1−e^{−t/1000}\right)$

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 $1,998,257$?

Answer

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

29) Repeat the previous problem but use Gompertz growth model. Which is more accurate?