# 8.5E: Exercises for Section 8.5

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In exercises 1 - 5, state if each of the following differential equations is linear? Explain your reasoning.

1) $$\dfrac{dy}{dx}=x^2y+\sin x$$

2) $$\dfrac{dy}{dt}=ty$$

$$Yes$$

3) $$\dfrac{dy}{dt}+y^2=x$$

4) $$y'=x^3+e^x$$

$$Yes$$

5) $$y'=y+e^y$$

In exercises 6 - 10, write the following first-order differential equations in standard form.

6) $$y'=x^3y+\sin x$$

$$y'−x^3y=\sin x$$

7) $$y'+3y−\ln x=0$$

8) $$−xy'=(3x+2)y+xe^x$$

$$y'+\frac{(3x+2)}{x}y=−e^x$$

9) $$\dfrac{dy}{dt}=4y+ty+\tan t$$

10) $$\dfrac{dy}{dt}=yx(x+1)$$

$$\dfrac{dy}{dt}−yx(x+1)=0$$

In exercises 11 - 15, state the integrating factors for each of the following differential equations.

11) $$y'=xy+3$$

12) $$y'+e^xy=\sin x$$

$$e^x$$

13) $$y'=x\ln(x)y+3x$$

14) $$\dfrac{dy}{dx}=\tanh(x)y+1$$

$$−\ln(\cosh x)$$

15) $$\dfrac{dy}{dt}+3ty=e^ty$$

In exercises 16 - 25, solve each differential equation by using integrating factors.

16) $$y'=3y+2$$

$$y=Ce^{3x}−\frac{2}{3}$$

17) $$y'=2y−x^2$$

18) $$xy'=3y−6x^2$$

$$y=Cx^3+6x^2$$

19) $$(x+2)y'=3x+y$$

20) $$y'=3x+xy$$

$$y=Ce^{x^2/2}−3$$

21) $$xy'=x+y$$

22) $$\sin(x)y'=y+2x$$

$$y=C\tan\left(\dfrac{x}{2}\right)−2x+4\tan\left(\dfrac{x}{2})\ln\left(\sin(\dfrac{x}{2}\right)\right)$$

23) $$y'=y+e^x$$

24) $$xy'=3y+x^2$$

$$y=Cx^3−x^2$$

25) $$y'+\ln x=\dfrac{y}{x}$$

In exercises 26 - 33, solve the given differential equation. Use your calculator to draw a family of solutions. Are there certain initial conditions that change the behavior of the solution?

26) [T] $$(x+2)y'=2y−1$$

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

27) [T] $$y'=3e^{t/3}−2y$$

28) [T] $$xy'+\dfrac{y}{2}=\sin(3t)$$

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

29) [T] $$xy'=2\dfrac{\cos x}{x}−3y$$

30) [T] $$(x+1)y'=3y+x^2+2x+1$$

$$y=C(x+1)^3−x^2−2x−1$$

31) [T] $$\sin(x)y'+\cos(x)y=2x$$

32) [T] $$\sqrt{x^2+1}y'=y+2$$

$$y=Ce^{\sinh^{−1}x}−2$$

33) [T] $$x^3y'+2x^2y=x+1$$

In exercises 34 - 43, solve each initial-value problem by using integrating factors.

34) $$y'+y=x,\quad y(0)=3$$

$$y=x+4e^x−1$$

35) $$y'=y+2x^2,\quad y(0)=0$$

36) $$xy'=y−3x^3,\quad y(1)=0$$

$$y=−\dfrac{3x}{2}(x^2−1)$$

37) $$x^2y'=xy−\ln x,\quad y(1)=1$$

38) $$(1+x^2)y'=y−1,\quad y(0)=0$$

$$y=1−e^{\tan^{−1}x}$$

39) $$xy'=y+2x\ln x,\quad y(1)=5$$

40) $$(2+x)y'=y+2+x,\quad y(0)=0$$

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

41) $$y'=xy+2xe^x,\quad y(0)=2$$

42) $$\sqrt{x}y'=y+2x,\quad y(0)=1$$

$$y=2e^{2\sqrt{x}}−2x−2\sqrt{x}−1$$

43) $$y'=2y+xe^x,\quad y(0)=−1$$

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 $$0.$$

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

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 $$4$$?

$$40.451$$ 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 $$k$$. 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?)

$$\sqrt{\dfrac{gm}{k}}$$

49) [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 $$4$$? Does it take more or less time than your initial estimate?

In exercises 50 - 54, determine how parameter $$a$$ affects the solution.

50) Solve the generic equation $$y'=ax+y$$. How does varying $$a$$ change the behavior?

$$y=Ce^x−a(x+1)$$

51) Solve the generic equation $$y'=ax+y.$$ How does varying $$a$$ change the behavior?

52) Solve the generic equation $$y'=ax+xy$$. How does varying $$a$$ change the behavior?

$$y=Ce^{x^2/2}−a$$

53) Solve the generic equation $$y'=x+axy.$$ How does varying $$a$$ change the behavior?

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

$$y=\dfrac{e^{kt}−e^t}{k−1}$$