Are the following differential equations linear? Explain your reasoning.
1. [latex]\frac{dy}{dx}={x}^{2}y+\sin{x}[/latex]
2. [latex]\frac{dy}{dt}=ty[/latex]
3. [latex]\frac{dy}{dt}+{y}^{2}=x[/latex]
4. [latex]y^{\prime} ={x}^{3}+{e}^{x}[/latex]
5. [latex]y^{\prime} =y+{e}^{y}[/latex]
Write the following first-order differential equations in standard form.
6. [latex]y^{\prime} ={x}^{3}y+\sin{x}[/latex]
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[latex]y^{\prime} -{x}^{3}y=\sin{x}[/latex]
7. [latex]y^{\prime} +3y-\text{ln}x=0[/latex]
8. [latex]\text{-}xy^{\prime} =\left(3x+2\right)y+x{e}^{x}[/latex]
Show Solution
[latex]y^{\prime} +\frac{\left(3x+2\right)}{x}y=\text{-}{e}^{x}[/latex]
9. [latex]\frac{dy}{dt}=4y+ty+\tan{t}[/latex]
10. [latex]\frac{dy}{dt}=yx\left(x+1\right)[/latex]
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[latex]\frac{dy}{dt}-yx\left(x+1\right)=0[/latex]
What are the integrating factors for the following differential equations?
11. [latex]y^{\prime} =xy+3[/latex]
12. [latex]y^{\prime} +{e}^{x}y=\sin{x}[/latex]
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[latex]{e}^{x}[/latex]
13. [latex]y^{\prime} =x\text{ln}\left(x\right)y+3x[/latex]
14. [latex]\frac{dy}{dx}=\text{tanh}\left(x\right)y+1[/latex]
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[latex]\text{-}\text{ln}\left(\text{cosh}x\right)[/latex]
15. [latex]\frac{dy}{dt}+3ty={e}^{t}y[/latex]
Solve the following differential equations by using integrating factors.
16. [latex]y^{\prime} =3y+2[/latex]
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[latex]y=C{e}^{3x}-\frac{2}{3}[/latex]
17. [latex]y^{\prime} =2y-{x}^{2}[/latex]
18. [latex]xy^{\prime} =3y - 6{x}^{2}[/latex]
Show Solution
[latex]y=C{x}^{3}+6{x}^{2}[/latex]
19. [latex]\left(x+2\right)y^{\prime} =3x+y[/latex]
20. [latex]y^{\prime} =3x+xy[/latex]
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[latex]y=C{e}^{\frac{{x}^{2}}{2}}-3[/latex]
21. [latex]xy^{\prime} =x+y[/latex]
22. [latex]\sin\left(x\right)y^{\prime} =y+2x[/latex]
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[latex]y=C\tan\left(\frac{x}{2}\right)-2x+4\tan\left(\frac{x}{2}\right)\text{ln}\left(\sin\left(\frac{x}{2}\right)\right)[/latex]
23. [latex]y^{\prime} =y+{e}^{x}[/latex]
24. [latex]xy^{\prime} =3y+{x}^{2}[/latex]
Show Solution
[latex]y=C{x}^{3}-{x}^{2}[/latex]
25. [latex]y^{\prime} +\text{ln}x=\frac{y}{x}[/latex]
Solve the following differential equations. Use your calculator to draw a family of solutions. Are there certain initial conditions that change the behavior of the solution?
26. [T] [latex]\left(x+2\right)y^{\prime} =2y - 1[/latex]
Show Solution
[latex]y=C{\left(x+2\right)}^{2}+\frac{1}{2}[/latex]
27. [T] [latex]y^{\prime} =3{e}^{\frac{t}{3}}-2y[/latex]
28. [T] [latex]xy^{\prime} +\frac{y}{2}=\sin\left(3t\right)[/latex]
Show Solution
[latex]y=\frac{C}{\sqrt{x}}+2\sin\left(3t\right)[/latex]
29. [T] [latex]xy^{\prime} =2\frac{\cos{x}}{x}-3y[/latex]
30. [T] [latex]\left(x+1\right)y^{\prime} =3y+{x}^{2}+2x+1[/latex]
Show Solution
[latex]y=C{\left(x+1\right)}^{3}-{x}^{2}-2x - 1[/latex]
31. [T] [latex]\sin\left(x\right)y^{\prime} +\cos\left(x\right)y=2x[/latex]
32. [T] [latex]\sqrt{{x}^{2}+1}y^{\prime} =y+2[/latex]
Show Solution
[latex]y=C{e}^{{\text{sinh}}^{-1}x}-2[/latex]
33. [T] [latex]{x}^{3}y^{\prime} +2{x}^{2}y=x+1[/latex]
Solve the following initial-value problems by using integrating factors.
34, [latex]y^{\prime} +y=x,y\left(0\right)=3[/latex]
Show Solution
[latex]y=x+4{e}^{x}-1[/latex]
35. [latex]y^{\prime} =y+2{x}^{2},y\left(0\right)=0[/latex]
36. [latex]xy^{\prime} =y - 3{x}^{3},y\left(1\right)=0[/latex]
Show Solution
[latex]y=-\frac{3x}{2}\left({x}^{2}-1\right)[/latex]
37. [latex]{x}^{2}y^{\prime} =xy-\text{ln}x,y\left(1\right)=1[/latex]
38. [latex]\left(1+{x}^{2}\right)y^{\prime} =y - 1,y\left(0\right)=0[/latex]
Show Solution
[latex]y=1-{e}^{{\tan}^{-1}x}[/latex]
39. [latex]xy^{\prime} =y+2x\text{ln}x,y\left(1\right)=5[/latex]
40. [latex]\left(2+x\right)y^{\prime} =y+2+x,y\left(0\right)=0[/latex]
Show Solution
[latex]y=\left(x+2\right)\text{ln}\left(\frac{x+2}{2}\right)[/latex]
41. [latex]y^{\prime} =xy+2x{e}^{x},y\left(0\right)=2[/latex]
42. [latex]\sqrt{x}y^{\prime} =y+2x,y\left(0\right)=1[/latex]
Show Solution
[latex]y=2{e}^{2\sqrt{x}}-2x - 2\sqrt{x}-1[/latex]
43. [latex]y^{\prime} =2y+x{e}^{x},y\left(0\right)=-1[/latex]
44. A falling object of mass [latex]m[/latex] can reach terminal velocity when the drag force is proportional to its velocity, with proportionality constant [latex]k[/latex]. Set up the differential equation and solve for the velocity given an initial velocity of [latex]0[/latex].
Show Solution
[latex]v\left(t\right)=\frac{gm}{k}\left(1-{e}^{\frac{\text{-}kt}{m}}\right)[/latex]
45. Using your expression from the preceding problem, what is the terminal velocity? (Hint: Examine the limiting behavior; does the velocity approach a value?)
46. [T] Using your equation for terminal velocity, solve for the distance fallen. How long does it take to fall [latex]5000[/latex] meters if the mass is [latex]100[/latex] kilograms, the acceleration due to gravity is [latex]9.8[/latex] m/s2 and the proportionality constant is [latex]4?[/latex]
Show Solution
[latex]40.451[/latex] seconds
47. A more accurate way to describe terminal velocity is that the drag force is proportional to the square of velocity, with a proportionality constant [latex]k[/latex]. Set up the differential equation and solve for the velocity.
48. Using your expression from the preceding problem, what is the terminal velocity? (Hint: Examine the limiting behavior: Does the velocity approach a value?)
Show Solution
[latex]\sqrt{\frac{gm}{k}}[/latex]
49. [T] Using your equation for terminal velocity, solve for the distance fallen. How long does it take to fall [latex]5000[/latex] meters if the mass is [latex]100[/latex] kilograms, the acceleration due to gravity is [latex]9.8{\text{m/s}}^{2}[/latex] and the proportionality constant is [latex]4?[/latex] Does it take more or less time than your initial estimate?
For the following problems, determine how parameter [latex]a[/latex] affects the solution.
50. Solve the generic equation [latex]y^{\prime} =ax+y[/latex]. How does varying [latex]a[/latex] change the behavior?
Show Solution
[latex]y=C{e}^{x}-a\left(x+1\right)[/latex]
51. Solve the generic equation [latex]y^{\prime} =ay+x[/latex]. How does varying [latex]a[/latex] change the behavior?
52. Solve the generic equation [latex]y^{\prime} =ax+xy[/latex]. How does varying [latex]a[/latex] change the behavior?
Show Solution
[latex]y=C{e}^{\frac{{x}^{2}}{2}}-a[/latex]
53. Solve the generic equation [latex]y^{\prime} =x+axy[/latex]. How does varying [latex]a[/latex] change the behavior?
54. Solve [latex]y^{\prime} -y={e}^{kt}[/latex] with the initial condition [latex]y\left(0\right)=0[/latex]. As [latex]k[/latex] approaches [latex]1[/latex], what happens to your formula?
Show Solution
[latex]y=\frac{{e}^{kt}-{e}^{t}}{k - 1}[/latex]
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