True or False? Justify your answer with a proof or a counterexample.
1. The differential equation [latex]y^{\prime} =3{x}^{2}y-\cos\left(x\right)y^{\prime}\prime[/latex] is linear.
2. The differential equation [latex]y^{\prime} =x-y[/latex] 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. [latex]{y}^{\prime }={x}^{2}+3{e}^{x}-2x[/latex]
6. [latex]y^{\prime} ={2}^{x}+{\cos}^{-1}x[/latex]
Show Solution
[latex]y\left(x\right)=\frac{{2}^{x}}{\text{ln}\left(2\right)}+x{\cos}^{-1}x-\sqrt{1-{x}^{2}}+C[/latex]
7. [latex]y^{\prime} =y\left({x}^{2}+1\right)[/latex]
8. [latex]y^{\prime} ={e}^{\text{-}y}\sin{x}[/latex]
Show Solution
[latex]y\left(x\right)=\text{ln}\left(C-\cos{x}\right)[/latex]
9. [latex]y^{\prime} =3x - 2y[/latex]
10. [latex]y^{\prime} =y\text{ln}y[/latex]
Show Solution
[latex]y\left(x\right)={e}^{{e}^{C+x}}[/latex]
For the following problems, find the solution to the initial value problem.
11. [latex]y^{\prime} =8x-\text{ln}x - 3{x}^{4},y\left(1\right)=5[/latex]
12. [latex]y^{\prime} =3x-\cos{x}+2,y\left(0\right)=4[/latex]
Show Solution
[latex]y\left(x\right)=4+\frac{3}{2}{x}^{2}+2x-\sin{x}[/latex]
13. [latex]xy^{\prime} =y\left(x - 2\right),y\left(1\right)=3[/latex]
14. [latex]y^{\prime} =3{y}^{2}\left(x+\cos{x}\right),y\left(0\right)=-2[/latex]
Show Solution
[latex]y\left(x\right)=-\frac{2}{1+3\left({x}^{2}+2\sin{x}\right)}[/latex]
15. [latex]\left(x - 1\right)y^{\prime} =y - 2,y\left(0\right)=0[/latex]
16. [latex]y^{\prime} =3y-x+6{x}^{2},y\left(0\right)=-1[/latex]
Show Solution
[latex]y\left(x\right)=-2{x}^{2}-2x-\frac{1}{3}-\frac{2}{3}{e}^{3x}[/latex]
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. [latex]y^{\prime} =2y-{y}^{2}[/latex]
18. [latex]y^{\prime} =\frac{1}{x}+\text{ln}x-y[/latex], for [latex]x>0[/latex]
Show Solution
[latex]y\left(x\right)=C{e}^{\text{-}x}+\text{ln}x[/latex]
For the following problems, use Euler’s Method with [latex]n=5[/latex] steps over the interval [latex]t=\left[0,1\right][/latex]. Then solve the initial-value problem exactly. How close is your Euler’s Method estimate?
19. [latex]y^{\prime} =-4yx,y\left(0\right)=1[/latex]
20. [latex]y^{\prime} ={3}^{x}-2y,y\left(0\right)=0[/latex]
Show Solution
Euler: [latex]0.6939[/latex], exact solution: [latex]y\left(x\right)=\frac{{3}^{x}-{e}^{-2x}}{2+\text{ln}\left(3\right)}[/latex]
For the following problems, set up and solve the differential equations.
21. A car drives along a freeway, accelerating according to [latex]a=5\sin\left(\pi t\right)[/latex], where [latex]t[/latex] represents time in minutes. Find the velocity at any time [latex]t[/latex], assuming the car starts with an initial speed of [latex]60[/latex] mph.
22. You throw a ball of mass [latex]2[/latex] kilograms into the air with an upward velocity of [latex]8[/latex] m/s. Find exactly the time the ball will remain in the air, assuming that gravity is given by [latex]g=9.8{\text{m/s}}^{2}[/latex].
Show Solution
[latex]\frac{40}{49}[/latex] second
23. You drop a ball with a mass of [latex]5[/latex] kilograms out an airplane window at a height of [latex]5000[/latex] m. How long does it take for the ball to reach the ground?
24. You drop the same ball of mass [latex]5[/latex] 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 [latex]3[/latex] 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?
Show Solution
[latex]x\left(t\right)=5000+\frac{245}{9}-\frac{49}{3}t-\frac{245}{9}{e}^{\frac{\text{-}5}{3t}},t=307.8[/latex] seconds
25. A drug is administered to a patient every [latex]24[/latex] hours and is cleared at a rate proportional to the amount of drug left in the body, with proportionality constant [latex]0.2[/latex]. If the patient needs a baseline level of [latex]5[/latex] mg to be in the bloodstream at all times, how large should the dose be?
26. A [latex]1000[/latex] -liter tank contains pure water and a solution of [latex]0.2[/latex] kg salt/L is pumped into the tank at a rate of [latex]1[/latex] L/min and is drained at the same rate. Solve for total amount of salt in the tank at time [latex]t[/latex].
Show Solution
[latex]T\left(t\right)=200\left(1-{e}^{\frac{\text{-}}{1000}}\right)[/latex]
27. You boil water to make tea. When you pour the water into your teapot, the temperature is [latex]100^\circ C.[/latex] After [latex]5[/latex] minutes in your [latex]15^\circ C[/latex] room, the temperature of the tea is [latex]85^\circ C.[/latex] Solve the equation to determine the temperatures of the tea at time [latex]t[/latex]. How long must you wait until the tea is at a drinkable temperature [latex]\left(72^\circ C\right)?[/latex]
28. The human population (in thousands) of Nevada in [latex]1950[/latex] was roughly [latex]160[/latex]. If the carrying capacity is estimated at [latex]10[/latex] million individuals, and assuming a growth rate of [latex]2\text{%}[/latex] per year, develop a logistic growth model and solve for the population in Nevada at any time (use [latex]1950[/latex] as time = 0). What population does your model predict for [latex]2000?[/latex] How close is your prediction to the true value of [latex]1,998,257?[/latex]
Show Solution
[latex]P\left(t\right)=\frac{1600000{e}^{0.02t}}{9840+160{e}^{0.02t}}[/latex]
29. Repeat the previous problem but use Gompertz growth model. Which is more accurate?
