Module 4 Review Problems

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

1. The differential equation

$y^{\prime} =3{x}^{2}y-\cos\left(x\right)y^{\prime}\prime$ is linear.

2. The differential equation $y^{\prime} =x-y$ is separable.

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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.

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

${y}^{\prime }={x}^{2}+3{e}^{x}-2x$

6. $y^{\prime} ={2}^{x}+{\cos}^{-1}x$

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$y\left(x\right)=\frac{{2}^{x}}{\text{ln}\left(2\right)}+x{\cos}^{-1}x-\sqrt{1-{x}^{2}}+C$

$y^{\prime} =y\left({x}^{2}+1\right)$

8. $y^{\prime} ={e}^{\text{-}y}\sin{x}$

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$y\left(x\right)=\text{ln}\left(C-\cos{x}\right)$

$y^{\prime} =3x - 2y$

10. $y^{\prime} =y\text{ln}y$

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

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

11.

$y^{\prime} =8x-\text{ln}x - 3{x}^{4},y\left(1\right)=5$

12. $y^{\prime} =3x-\cos{x}+2,y\left(0\right)=4$

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$y\left(x\right)=4+\frac{3}{2}{x}^{2}+2x-\sin{x}$

13.

$xy^{\prime} =y\left(x - 2\right),y\left(1\right)=3$

14. $y^{\prime} =3{y}^{2}\left(x+\cos{x}\right),y\left(0\right)=-2$

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$y\left(x\right)=-\frac{2}{1+3\left({x}^{2}+2\sin{x}\right)}$

15.

$\left(x - 1\right)y^{\prime} =y - 2,y\left(0\right)=0$

16. $y^{\prime} =3y-x+6{x}^{2},y\left(0\right)=-1$

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$y\left(x\right)=-2{x}^{2}-2x-\frac{1}{3}-\frac{2}{3}{e}^{3x}$

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.

$y^{\prime} =2y-{y}^{2}$

18. $y^{\prime} =\frac{1}{x}+\text{ln}x-y$ , for $x>0$

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A direction field with arrows pointing up and to the right along a logarithmic curve that approaches negative infinity as x goes to zero and increases as x goes to infinity.

$y\left(x\right)=C{e}^{\text{-}x}+\text{ln}x$

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

19.

$y^{\prime} =-4yx,y\left(0\right)=1$

20. $y^{\prime} ={3}^{x}-2y,y\left(0\right)=0$

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Euler:

$0.6939$ , exact solution:

$y\left(x\right)=\frac{{3}^{x}-{e}^{-2x}}{2+\text{ln}\left(3\right)}$

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

21. A car drives along a freeway, accelerating according to

$a=5\sin\left(\pi t\right)$ , 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 $g=9.8{\text{m/s}}^{2}$ .

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$\frac{40}{49}$ 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?

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$x\left(t\right)=5000+\frac{245}{9}-\frac{49}{3}t-\frac{245}{9}{e}^{\frac{\text{-}5}{3t}},t=307.8$ 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 $t$ .

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$T\left(t\right)=200\left(1-{e}^{\frac{\text{-}}{1000}}\right)$

27. You boil water to make tea. When you pour the water into your teapot, the temperature is

$100^\circ C.$ After

$5$ minutes in your

$15^\circ C$ room, the temperature of the tea is

$85^\circ C.$ 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

$\left(72^\circ C\right)?$

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 $2\text{%}$ 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 $1,998,257?$

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

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

Differential Equations: Supplemental Content

Module 4 Review Problems

Candela Citations