Problem Set: Basics of Differential Equations

Determine the order of the following differential equations.

1. ${y}^{\prime }+y=3{y}^{2}$

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$1$

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

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

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$3$

${y}^{\prime }=y^{\prime\prime}+3{t}^{2}$

5. $\frac{dy}{dt}=t$

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$1$

$\frac{dy}{dx}+\frac{{d}^{2}y}{d{x}^{2}}=3{x}^{4}$

7. ${\left(\frac{dy}{dt}\right)}^{2}+8\frac{dy}{dt}+3y=4t$

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$1$

Verify that the following functions are solutions to the given differential equation.

$y=\frac{{x}^{3}}{3}$ solves

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

$y=2{e}^{\text{-}x}+x - 1$ solves

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

10.

$y={e}^{3x}-\frac{{e}^{x}}{2}$ solves

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

11.

$y=\frac{1}{1-x}$ solves

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

12.

$y={e}^{\frac{{x}^{2}}{2}}$ solves

${y}^{\prime }=xy$

13.

$y=4+\text{ln}x$ solves

$x{y}^{\prime }=1$

14.

$y=3-x+x\text{ln}x$ solves

${y}^{\prime }=\text{ln}x$

15.

$y=2{e}^{x}-x - 1$ solves

${y}^{\prime }=y+x$

16.

$y={e}^{x}+\frac{\sin{x}}{2}-\frac{\cos{x}}{2}$ solves

${y}^{\prime }=\cos{x}+y$

17.

$y=\pi {e}^{\text{-}\cos{x}}$ solves

${y}^{\prime }=y\sin{x}$

Verify the following general solutions and find the particular solution.

18. Find the particular solution to the differential equation

${y}^{\prime }=4{x}^{2}$ that passes through

$\left(-3,-30\right)$ , given that

$y=C+\frac{4{x}^{3}}{3}$ is a general solution.

19. Find the particular solution to the differential equation ${y}^{\prime }=3{x}^{3}$ that passes through $\left(1,4.75\right)$ , given that $y=C+\frac{3{x}^{4}}{4}$ is a general solution.

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$y=4+\frac{3{x}^{4}}{4}$

20. Find the particular solution to the differential equation

${y}^{\prime }=3{x}^{2}y$ that passes through

$\left(0,12\right)$ , given that

$y=C{e}^{{x}^{3}}$ is a general solution.

21. Find the particular solution to the differential equation ${y}^{\prime }=2xy$ that passes through $\left(0,\frac{1}{2}\right)$ , given that $y=C{e}^{{x}^{2}}$ is a general solution.

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$y=\frac{1}{2}{e}^{{x}^{2}}$

22. Find the particular solution to the differential equation

${y}^{\prime }={\left(2xy\right)}^{2}$ that passes through

$\left(1,-\frac{1}{2}\right)$ , given that

$y=-\frac{3}{C+4{x}^{3}}$ is a general solution.

23. Find the particular solution to the differential equation ${y}^{\prime }{x}^{2}=y$ that passes through $\left(1,\frac{2}{e}\right)$ , given that $y=C{e}^{\frac{\text{-}1}{x}}$ is a general solution.

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$y=2{e}^{\frac{\text{-}1}{x}}$

24. Find the particular solution to the differential equation

$8\frac{dx}{dt}=-2\cos\left(2t\right)-\cos\left(4t\right)$ that passes through

$\left(\pi ,\pi \right)$ , given that

$x=C-\frac{1}{8}\sin\left(2t\right)-\frac{1}{32}\sin\left(4t\right)$ is a general solution.

25. Find the particular solution to the differential equation $\frac{du}{dt}=\tan{u}$ that passes through $\left(1,\frac{\pi }{2}\right)$ , given that $u={\sin}^{-1}\left({e}^{C+t}\right)$ is a general solution.

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$u={\sin}^{-1}\left({e}^{-1+t}\right)$

26. Find the particular solution to the differential equation

$\frac{dy}{dt}={e}^{\left(t+y\right)}$ that passes through

$\left(1,0\right)$ , given that

$y=\text{-}\text{ln}\left(C-{e}^{t}\right)$ is a general solution.

27. Find the particular solution to the differential equation ${y}^{\prime }\left(1-{x}^{2}\right)=1+y$ that passes through $\left(0,-2\right)$ , given that $y=C\frac{\sqrt{x+1}}{\sqrt{1-x}}-1$ is a general solution.

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$y=-\frac{\sqrt{x+1}}{\sqrt{1-x}}-1$

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

28.

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

29. ${y}^{\prime }=\text{ln}x+\tan{x}$

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

30.

${y}^{\prime }=\sin{x}{e}^{\cos{x}}$

31. ${y}^{\prime }={4}^{x}$

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$y=C+\frac{{4}^{x}}{\text{ln}\left(4\right)}$

32.

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

33. ${y}^{\prime }=2t\sqrt{{t}^{2}+16}$

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$y=\frac{2}{3}\sqrt{{t}^{2}+16}\left({t}^{2}+16\right)+C$

34.

${x}^{\prime }=\text{coth}t+\text{ln}t+3{t}^{2}$

35. ${x}^{\prime }=t\sqrt{4+t}$

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$x=\frac{2}{15}\sqrt{4+t}\left(3{t}^{2}+4t - 32\right)+C$

36.

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

37. ${y}^{\prime }=\frac{y}{x}$

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$y=Cx$

Solve the following initial-value problems starting from $y\left(t=0\right)=1$ and $y\left(t=0\right)=-1$ . Draw both solutions on the same graph.

38.

$\frac{dy}{dt}=2t$

39. $\frac{dy}{dt}=\text{-}t$

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$y=1-\frac{{t}^{2}}{2},y=-\frac{{t}^{2}}{2}-1$

40.

$\frac{dy}{dt}=2y$

41. $\frac{dy}{dt}=\text{-}y$

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$y={e}^{\text{-}t},y=\text{-}{e}^{\text{-}t}$

42.

$\frac{dy}{dt}=2$

Solve the following initial-value problems starting from ${y}_{0}=10$ . At what time does $y$ increase to $100$ or drop to $1?$

43. $\frac{dy}{dt}=4t$

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$y=2\left({t}^{2}+5\right),t=3\sqrt{5}$

44.

$\frac{dy}{dt}=4y$

45. $\frac{dy}{dt}=-2y$

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$y=10{e}^{-2t},t=-\frac{1}{2}\text{ln}\left(\frac{1}{10}\right)$

46.

$\frac{dy}{dt}={e}^{4t}$

47. $\frac{dy}{dt}={e}^{-4t}$

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$y=\frac{1}{4}\left(41-{e}^{-4t}\right)$ , never

Recall that a family of solutions includes solutions to a differential equation that differ by a constant. For the following problems, use your calculator to graph a family of solutions to the given differential equation. Use initial conditions from $y\left(t=0\right)=-10$ to $y\left(t=0\right)=10$ increasing by $2$ . Is there some critical point where the behavior of the solution begins to change?

48. [T]

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

49. [T] $x{y}^{\prime }=y$

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Solution changes from increasing to decreasing at

$y\left(0\right)=0$

50. [T]

${y}^{\prime }={t}^{3}$

51. [T] ${y}^{\prime }=x+y$ (Hint: $y=C{e}^{x}-x - 1$ is the general solution)

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Solution changes from increasing to decreasing at

$y\left(0\right)=0$

52. [T]

${y}^{\prime }=x\text{ln}x+\sin{x}$

53. Find the general solution to describe the velocity of a ball of mass $1\text{lb}$ that is thrown upward at a rate $a$ ft/sec.

Show Solution

$v\left(t\right)=-32t+a$

54. In the preceding problem, if the initial velocity of the ball thrown into the air is

$a=25$ ft/s, write the particular solution to the velocity of the ball. Solve to find the time when the ball hits the ground.

55. You throw two objects with differing masses ${m}_{1}$ and ${m}_{2}$ upward into the air with the same initial velocity $a$ ft/s. What is the difference in their velocity after $1$ second?

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$0$ ft/s

56. [T] You throw a ball of mass

$1$ kilogram upward with a velocity of

$a=25$ m/s on Mars, where the force of gravity is

$g=-3.711$ m/s². Use your calculator to approximate how much longer the ball is in the air on Mars than on Earth, where

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

57. [T] For the previous problem, use your calculator to approximate how much higher the ball went on Mars, where $g= -9.8 m/s^{2}$ .

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$4.86$ meters

58. [T] A car on the freeway accelerates according to

$a=15\cos\left(\pi t\right)$ , where

$t$ is measured in hours. Set up and solve the differential equation to determine the velocity of the car if it has an initial speed of

$51$ mph. After

$40$ minutes of driving, what is the driver’s velocity?

59. [T] For the car in the preceding problem, find the expression for the distance the car has traveled in time $t$ , assuming an initial distance of $0$ . How long does it take the car to travel $100$ miles? Round your answer to hours and minutes.

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$x=50t-\frac{15}{{\pi }^{2}}\cos\left(\pi t\right)+\frac{3}{{\pi }^{2}},2$ hours

$1$ minute

60. [T] For the previous problem, find the total distance traveled in the first hour.

61. Substitute $y=B{e}^{3t}$ into ${y}^{\prime }-y=8{e}^{3t}$ to find a particular solution.

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$y=4{e}^{3t}$

62. Substitute

$y=a\cos\left(2t\right)+b\sin\left(2t\right)$ into

${y}^{\prime }+y=4\sin\left(2t\right)$ to find a particular solution.

63. Substitute $y=a+bt+c{t}^{2}$ into ${y}^{\prime }+y=1+{t}^{2}$ to find a particular solution.

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$y=3 - 2t+{t}^{2}$

64. Substitute

$y=a{e}^{t}\cos{t}+b{e}^{t}\sin{t}$ into

${y}^{\prime }=2{e}^{t}\cos{t}$ to find a particular solution.

65. Solve ${y}^{\prime }={e}^{kt}$ with the initial condition $y\left(0\right)=0$ and solve ${y}^{\prime }=1$ with the same initial condition. As $k$ approaches $0$ , what do you notice?

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$y=\frac{1}{k}\left({e}^{kt}-1\right)$ and

$y=x$

Differential Equations: Supplemental Content

Problem Set: Basics of Differential Equations

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