## 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.

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.

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

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

6. $y^{\prime} ={2}^{x}+{\cos}^{-1}x$

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

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

9. $y^{\prime} =3x - 2y$

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

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

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

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

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

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

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 $x>0$

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

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

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?

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 $t$.

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 $1,998,257?$

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