Determine the order of the following differential equations.
1. [latex]{y}^{\prime }+y=3{y}^{2}[/latex]
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[latex]1[/latex]
2. [latex]{\left({y}^{\prime }\right)}^{2}={y}^{\prime }+2y[/latex]
3. [latex]y^{\prime\prime\prime}+y^{\prime\prime}{y}^{\prime }=3{x}^{2}[/latex]
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[latex]3[/latex]
4. [latex]{y}^{\prime }=y^{\prime\prime}+3{t}^{2}[/latex]
5. [latex]\frac{dy}{dt}=t[/latex]
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[latex]1[/latex]
6. [latex]\frac{dy}{dx}+\frac{{d}^{2}y}{d{x}^{2}}=3{x}^{4}[/latex]
7. [latex]{\left(\frac{dy}{dt}\right)}^{2}+8\frac{dy}{dt}+3y=4t[/latex]
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[latex]1[/latex]
Verify that the following functions are solutions to the given differential equation.
8. [latex]y=\frac{{x}^{3}}{3}[/latex] solves [latex]{y}^{\prime }={x}^{2}[/latex]
9. [latex]y=2{e}^{\text{-}x}+x - 1[/latex] solves [latex]{y}^{\prime }=x-y[/latex]
10. [latex]y={e}^{3x}-\frac{{e}^{x}}{2}[/latex] solves [latex]{y}^{\prime }=3y+{e}^{x}[/latex]
11. [latex]y=\frac{1}{1-x}[/latex] solves [latex]{y}^{\prime }={y}^{2}[/latex]
12. [latex]y={e}^{\frac{{x}^{2}}{2}}[/latex] solves [latex]{y}^{\prime }=xy[/latex]
13. [latex]y=4+\text{ln}x[/latex] solves [latex]x{y}^{\prime }=1[/latex]
14. [latex]y=3-x+x\text{ln}x[/latex] solves [latex]{y}^{\prime }=\text{ln}x[/latex]
15. [latex]y=2{e}^{x}-x - 1[/latex] solves [latex]{y}^{\prime }=y+x[/latex]
16. [latex]y={e}^{x}+\frac{\sin{x}}{2}-\frac{\cos{x}}{2}[/latex] solves [latex]{y}^{\prime }=\cos{x}+y[/latex]
17. [latex]y=\pi {e}^{\text{-}\cos{x}}[/latex] solves [latex]{y}^{\prime }=y\sin{x}[/latex]
Verify the following general solutions and find the particular solution.
18. Find the particular solution to the differential equation [latex]{y}^{\prime }=4{x}^{2}[/latex] that passes through [latex]\left(-3,-30\right)[/latex], given that [latex]y=C+\frac{4{x}^{3}}{3}[/latex] is a general solution.
19. Find the particular solution to the differential equation [latex]{y}^{\prime }=3{x}^{3}[/latex] that passes through [latex]\left(1,4.75\right)[/latex], given that [latex]y=C+\frac{3{x}^{4}}{4}[/latex] is a general solution.
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[latex]y=4+\frac{3{x}^{4}}{4}[/latex]
20. Find the particular solution to the differential equation [latex]{y}^{\prime }=3{x}^{2}y[/latex] that passes through [latex]\left(0,12\right)[/latex], given that [latex]y=C{e}^{{x}^{3}}[/latex] is a general solution.
21. Find the particular solution to the differential equation [latex]{y}^{\prime }=2xy[/latex] that passes through [latex]\left(0,\frac{1}{2}\right)[/latex], given that [latex]y=C{e}^{{x}^{2}}[/latex] is a general solution.
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[latex]y=\frac{1}{2}{e}^{{x}^{2}}[/latex]
22. Find the particular solution to the differential equation [latex]{y}^{\prime }={\left(2xy\right)}^{2}[/latex] that passes through [latex]\left(1,-\frac{1}{2}\right)[/latex], given that [latex]y=-\frac{3}{C+4{x}^{3}}[/latex] is a general solution.
23. Find the particular solution to the differential equation [latex]{y}^{\prime }{x}^{2}=y[/latex] that passes through [latex]\left(1,\frac{2}{e}\right)[/latex], given that [latex]y=C{e}^{\frac{\text{-}1}{x}}[/latex] is a general solution.
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[latex]y=2{e}^{\frac{\text{-}1}{x}}[/latex]
24. Find the particular solution to the differential equation [latex]8\frac{dx}{dt}=-2\cos\left(2t\right)-\cos\left(4t\right)[/latex] that passes through [latex]\left(\pi ,\pi \right)[/latex], given that [latex]x=C-\frac{1}{8}\sin\left(2t\right)-\frac{1}{32}\sin\left(4t\right)[/latex] is a general solution.
25. Find the particular solution to the differential equation [latex]\frac{du}{dt}=\tan{u}[/latex] that passes through [latex]\left(1,\frac{\pi }{2}\right)[/latex], given that [latex]u={\sin}^{-1}\left({e}^{C+t}\right)[/latex] is a general solution.
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[latex]u={\sin}^{-1}\left({e}^{-1+t}\right)[/latex]
26. Find the particular solution to the differential equation [latex]\frac{dy}{dt}={e}^{\left(t+y\right)}[/latex] that passes through [latex]\left(1,0\right)[/latex], given that [latex]y=\text{-}\text{ln}\left(C-{e}^{t}\right)[/latex] is a general solution.
27. Find the particular solution to the differential equation [latex]{y}^{\prime }\left(1-{x}^{2}\right)=1+y[/latex] that passes through [latex]\left(0,-2\right)[/latex], given that [latex]y=C\frac{\sqrt{x+1}}{\sqrt{1-x}}-1[/latex] is a general solution.
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[latex]y=-\frac{\sqrt{x+1}}{\sqrt{1-x}}-1[/latex]
For the following problems, find the general solution to the differential equation.
28. [latex]{y}^{\prime }=3x+{e}^{x}[/latex]
29. [latex]{y}^{\prime }=\text{ln}x+\tan{x}[/latex]
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[latex]y=C-x+x\text{ln}x-\text{ln}\left(\cos{x}\right)[/latex]
30. [latex]{y}^{\prime }=\sin{x}{e}^{\cos{x}}[/latex]
31. [latex]{y}^{\prime }={4}^{x}[/latex]
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[latex]y=C+\frac{{4}^{x}}{\text{ln}\left(4\right)}[/latex]
32. [latex]{y}^{\prime }={\sin}^{-1}\left(2x\right)[/latex]
33. [latex]{y}^{\prime }=2t\sqrt{{t}^{2}+16}[/latex]
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[latex]y=\frac{2}{3}\sqrt{{t}^{2}+16}\left({t}^{2}+16\right)+C[/latex]
34. [latex]{x}^{\prime }=\text{coth}t+\text{ln}t+3{t}^{2}[/latex]
35. [latex]{x}^{\prime }=t\sqrt{4+t}[/latex]
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[latex]x=\frac{2}{15}\sqrt{4+t}\left(3{t}^{2}+4t - 32\right)+C[/latex]
36. [latex]{y}^{\prime }=y[/latex]
37. [latex]{y}^{\prime }=\frac{y}{x}[/latex]
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[latex]y=Cx[/latex]
Solve the following initial-value problems starting from [latex]y\left(t=0\right)=1[/latex] and [latex]y\left(t=0\right)=-1[/latex]. Draw both solutions on the same graph.
38. [latex]\frac{dy}{dt}=2t[/latex]
39. [latex]\frac{dy}{dt}=\text{-}t[/latex]
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[latex]y=1-\frac{{t}^{2}}{2},y=-\frac{{t}^{2}}{2}-1[/latex]
40. [latex]\frac{dy}{dt}=2y[/latex]
41. [latex]\frac{dy}{dt}=\text{-}y[/latex]
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[latex]y={e}^{\text{-}t},y=\text{-}{e}^{\text{-}t}[/latex]
42. [latex]\frac{dy}{dt}=2[/latex]
Solve the following initial-value problems starting from [latex]{y}_{0}=10[/latex]. At what time does [latex]y[/latex] increase to [latex]100[/latex] or drop to [latex]1?[/latex]
43. [latex]\frac{dy}{dt}=4t[/latex]
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[latex]y=2\left({t}^{2}+5\right),t=3\sqrt{5}[/latex]
44. [latex]\frac{dy}{dt}=4y[/latex]
45. [latex]\frac{dy}{dt}=-2y[/latex]
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[latex]y=10{e}^{-2t},t=-\frac{1}{2}\text{ln}\left(\frac{1}{10}\right)[/latex]
46. [latex]\frac{dy}{dt}={e}^{4t}[/latex]
47. [latex]\frac{dy}{dt}={e}^{-4t}[/latex]
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[latex]y=\frac{1}{4}\left(41-{e}^{-4t}\right)[/latex], 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 [latex]y\left(t=0\right)=-10[/latex] to [latex]y\left(t=0\right)=10[/latex] increasing by [latex]2[/latex]. Is there some critical point where the behavior of the solution begins to change?
48. [T] [latex]{y}^{\prime }=y\left(x\right)[/latex]
49. [T] [latex]x{y}^{\prime }=y[/latex]
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Solution changes from increasing to decreasing at [latex]y\left(0\right)=0[/latex]
50. [T] [latex]{y}^{\prime }={t}^{3}[/latex]
51. [T] [latex]{y}^{\prime }=x+y[/latex] (Hint: [latex]y=C{e}^{x}-x - 1[/latex] is the general solution)
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Solution changes from increasing to decreasing at [latex]y\left(0\right)=0[/latex]
52. [T] [latex]{y}^{\prime }=x\text{ln}x+\sin{x}[/latex]
53. Find the general solution to describe the velocity of a ball of mass [latex]1\text{lb}[/latex] that is thrown upward at a rate [latex]a[/latex] ft/sec.
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[latex]v\left(t\right)=-32t+a[/latex]
54. In the preceding problem, if the initial velocity of the ball thrown into the air is [latex]a=25[/latex] 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 [latex]{m}_{1}[/latex] and [latex]{m}_{2}[/latex] upward into the air with the same initial velocity [latex]a[/latex] ft/s. What is the difference in their velocity after [latex]1[/latex] second?
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[latex]0[/latex] ft/s
56. [T] You throw a ball of mass [latex]1[/latex] kilogram upward with a velocity of [latex]a=25[/latex] m/s on Mars, where the force of gravity is [latex]g=-3.711[/latex] m/s2. Use your calculator to approximate how much longer the ball is in the air on Mars than on Earth, where [latex]g= -9.8 m/s^{2}[/latex].
57. [T] For the previous problem, use your calculator to approximate how much higher the ball went on Mars, where [latex]g= -9.8 m/s^{2}[/latex].
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[latex]4.86[/latex] meters
58. [T] A car on the freeway accelerates according to [latex]a=15\cos\left(\pi t\right)[/latex], where [latex]t[/latex] 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 [latex]51[/latex] mph. After [latex]40[/latex] 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 [latex]t[/latex], assuming an initial distance of [latex]0[/latex]. How long does it take the car to travel [latex]100[/latex] miles? Round your answer to hours and minutes.
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[latex]x=50t-\frac{15}{{\pi }^{2}}\cos\left(\pi t\right)+\frac{3}{{\pi }^{2}},2[/latex] hours [latex]1[/latex] minute
60. [T] For the previous problem, find the total distance traveled in the first hour.
61. Substitute [latex]y=B{e}^{3t}[/latex] into [latex]{y}^{\prime }-y=8{e}^{3t}[/latex] to find a particular solution.
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[latex]y=4{e}^{3t}[/latex]
62. Substitute [latex]y=a\cos\left(2t\right)+b\sin\left(2t\right)[/latex] into [latex]{y}^{\prime }+y=4\sin\left(2t\right)[/latex] to find a particular solution.
63. Substitute [latex]y=a+bt+c{t}^{2}[/latex] into [latex]{y}^{\prime }+y=1+{t}^{2}[/latex] to find a particular solution.
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[latex]y=3 - 2t+{t}^{2}[/latex]
64. Substitute [latex]y=a{e}^{t}\cos{t}+b{e}^{t}\sin{t}[/latex] into [latex]{y}^{\prime }=2{e}^{t}\cos{t}[/latex] to find a particular solution.
65. Solve [latex]{y}^{\prime }={e}^{kt}[/latex] with the initial condition [latex]y\left(0\right)=0[/latex] and solve [latex]{y}^{\prime }=1[/latex] with the same initial condition. As [latex]k[/latex] approaches [latex]0[/latex], what do you notice?
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[latex]y=\frac{1}{k}\left({e}^{kt}-1\right)[/latex] and [latex]y=x[/latex]
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