Problem Set: First-order Linear Equations

Are the following differential equations linear? Explain your reasoning.

\frac{d y}{d x} = x^{2} y + \sin x

$\frac{dy}{dx}={x}^{2}y+\sin{x}$

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

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\frac{d y}{d t} + y^{2} = x

$\frac{dy}{dt}+{y}^{2}=x$

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

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y^{'} = y + e^{y}

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

Write the following first-order differential equations in standard form.

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

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y^{'} - x^{3} y = \sin x

$y^{\prime} -{x}^{3}y=\sin{x}$

y^{'} + 3 y - ln x = 0

$y^{\prime} +3y-\text{ln}x=0$

8. $\text{-}xy^{\prime} =\left(3x+2\right)y+x{e}^{x}$

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y^{'} + \frac{(3 x + 2)}{x} y = - e^{x}

$y^{\prime} +\frac{\left(3x+2\right)}{x}y=\text{-}{e}^{x}$

\frac{d y}{d t} = 4 y + t y + \tan t

$\frac{dy}{dt}=4y+ty+\tan{t}$

10. $\frac{dy}{dt}=yx\left(x+1\right)$

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\frac{d y}{d t} - y x (x + 1) = 0

$\frac{dy}{dt}-yx\left(x+1\right)=0$

What are the integrating factors for the following differential equations?

11.

y^{'} = x y + 3

$y^{\prime} =xy+3$

12. $y^{\prime} +{e}^{x}y=\sin{x}$

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e^{x}

${e}^{x}$

13.

y^{'} = x ln (x) y + 3 x

$y^{\prime} =x\text{ln}\left(x\right)y+3x$

14. $\frac{dy}{dx}=\text{tanh}\left(x\right)y+1$

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- ln (cosh x)

$\text{-}\text{ln}\left(\text{cosh}x\right)$

15.

\frac{d y}{d t} + 3 t y = e^{t} y

$\frac{dy}{dt}+3ty={e}^{t}y$

Solve the following differential equations by using integrating factors.

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

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

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

17.

y^{'} = 2 y - x^{2}

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

18. $xy^{\prime} =3y - 6{x}^{2}$

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y = C x^{3} + 6 x^{2}

$y=C{x}^{3}+6{x}^{2}$

19.

(x + 2) y^{'} = 3 x + y

$\left(x+2\right)y^{\prime} =3x+y$

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

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

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

21.

x y^{'} = x + y

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

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

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y = C \tan (\frac{x}{2}) - 2 x + 4 \tan (\frac{x}{2}) ln (\sin (\frac{x}{2}))

$y=C\tan\left(\frac{x}{2}\right)-2x+4\tan\left(\frac{x}{2}\right)\text{ln}\left(\sin\left(\frac{x}{2}\right)\right)$

23.

y^{'} = y + e^{x}

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

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

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y = C x^{3} - x^{2}

$y=C{x}^{3}-{x}^{2}$

25.

y^{'} + ln x = \frac{y}{x}

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

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

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y = C {(x + 2)}^{2} + \frac{1}{2}

$y=C{\left(x+2\right)}^{2}+\frac{1}{2}$

27. [T]

y^{'} = 3 e^{\frac{t}{3}} - 2 y

$y^{\prime} =3{e}^{\frac{t}{3}}-2y$

28. [T] $xy^{\prime} +\frac{y}{2}=\sin\left(3t\right)$

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y = \frac{C}{\sqrt{x}} + 2 \sin (3 t)

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

29. [T]

x y^{'} = 2 \frac{\cos x}{x} - 3 y

$xy^{\prime} =2\frac{\cos{x}}{x}-3y$

30. [T] $\left(x+1\right)y^{\prime} =3y+{x}^{2}+2x+1$

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y = C {(x + 1)}^{3} - x^{2} - 2 x - 1

$y=C{\left(x+1\right)}^{3}-{x}^{2}-2x - 1$

31. [T]

\sin (x) y^{'} + \cos (x) y = 2 x

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

32. [T] $\sqrt{{x}^{2}+1}y^{\prime} =y+2$

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y = C e^{{sinh}^{- 1} x} - 2

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

33. [T]

x^{3} y^{'} + 2 x^{2} y = x + 1

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

Solve the following initial-value problems by using integrating factors.

34, $y^{\prime} +y=x,y\left(0\right)=3$

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y = x + 4 e^{x} - 1

$y=x+4{e}^{x}-1$

35.

y^{'} = y + 2 x^{2}, y (0) = 0

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

36. $xy^{\prime} =y - 3{x}^{3},y\left(1\right)=0$

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y = - \frac{3 x}{2} (x^{2} - 1)

$y=-\frac{3x}{2}\left({x}^{2}-1\right)$

37.

x^{2} y^{'} = x y - ln x, y (1) = 1

${x}^{2}y^{\prime} =xy-\text{ln}x,y\left(1\right)=1$

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

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y = 1 - e^{\tan^{- 1} x}

$y=1-{e}^{{\tan}^{-1}x}$

39.

x y^{'} = y + 2 x ln x, y (1) = 5

$xy^{\prime} =y+2x\text{ln}x,y\left(1\right)=5$

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

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y = (x + 2) ln (\frac{x + 2}{2})

$y=\left(x+2\right)\text{ln}\left(\frac{x+2}{2}\right)$

41.

y^{'} = x y + 2 x e^{x}, y (0) = 2

$y^{\prime} =xy+2x{e}^{x},y\left(0\right)=2$

42. $\sqrt{x}y^{\prime} =y+2x,y\left(0\right)=1$

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y = 2 e^{2 \sqrt{x}} - 2 x - 2 \sqrt{x} - 1

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

43.

y^{'} = 2 y + x e^{x}, y (0) = - 1

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

44. A falling object of mass $m$ can reach terminal velocity when the drag force is proportional to its velocity, with proportionality constant $k$ . Set up the differential equation and solve for the velocity given an initial velocity of $0$ .

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v (t) = \frac{g m}{k} (1 - e^{\frac{- k t}{m}})

$v\left(t\right)=\frac{gm}{k}\left(1-{e}^{\frac{\text{-}kt}{m}}\right)$

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 $5000$ meters if the mass is $100$ kilograms, the acceleration due to gravity is $9.8$ m/s² and the proportionality constant is $4?$

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40.451

$40.451$ 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

k

$k$ . 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?)

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\sqrt{\frac{g m}{k}}

$\sqrt{\frac{gm}{k}}$

49. [T] Using your equation for terminal velocity, solve for the distance fallen. How long does it take to fall

5000

$5000$ meters if the mass is

100

$100$ kilograms, the acceleration due to gravity is

9.8 {m/s}^{2}

$9.8{\text{m/s}}^{2}$ and the proportionality constant is

4 ?

$4?$ Does it take more or less time than your initial estimate?

For the following problems, determine how parameter $a$ affects the solution.

50. Solve the generic equation $y^{\prime} =ax+y$ . How does varying $a$ change the behavior?

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y = C e^{x} - a (x + 1)

$y=C{e}^{x}-a\left(x+1\right)$

51. Solve the generic equation

y^{'} = a y + x

$y^{\prime} =ay+x$ . How does varying

a

$a$ change the behavior?

52. Solve the generic equation $y^{\prime} =ax+xy$ . How does varying $a$ change the behavior?

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y = C e^{\frac{x^{2}}{2}} - a

$y=C{e}^{\frac{{x}^{2}}{2}}-a$

53. Solve the generic equation

y^{'} = x + a x y

$y^{\prime} =x+axy$ . How does varying

a

$a$ change the behavior?

54. Solve $y^{\prime} -y={e}^{kt}$ with the initial condition $y\left(0\right)=0$ . As $k$ approaches $1$ , what happens to your formula?

Show Solution

y = \frac{e^{k t} - e^{t}}{k - 1}

$y=\frac{{e}^{kt}-{e}^{t}}{k - 1}$

Differential Equations: Supplemental Content

Problem Set: First-order Linear Equations

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