8.5E: Exercises for Section 8.5

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

Answer

$Yes$

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

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

Answer

$Yes$

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

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

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

Answer

$y'−x^3y=\sin x$

7) $y'+3y−\ln x=0$

8) $−xy'=(3x+2)y+xe^x$

Answer

$y'+\frac{(3x+2)}{x}y=−e^x$

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

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

Answer

$\dfrac{dy}{dt}−yx(x+1)=0$

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

11) $y'=xy+3$

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

Answer

$e^x$

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

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

Answer

$−\ln(\cosh x)$

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

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

16) $y'=3y+2$

Answer

$y=Ce^{3x}−\frac{2}{3}$

17) $y'=2y−x^2$

18) $xy'=3y−6x^2$

Answer

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

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

20) $y'=3x+xy$

Answer

$y=Ce^{x^2/2}−3$

21) $xy'=x+y$

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

Answer

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

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

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

Answer

$y=Cx^3−x^2$

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] $(x+2)y'=2y−1$

Answer

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

27) [T] $y'=3e^{t/3}−2y$

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

Answer

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

29) [T] $xy'=2\dfrac{\cos x}{x}−3y$

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

Answer

$y=C(x+1)^3−x^2−2x−1$

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

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

Answer

$y=Ce^{\sinh^{−1}x}−2$

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

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

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

Answer

$y=x+4e^x−1$

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

36) $xy'=y−3x^3,\quad y(1)=0$

Answer

$y=−\dfrac{3x}{2}(x^2−1)$

37) $x^2y'=xy−\ln x,\quad y(1)=1$

38) $(1+x^2)y'=y−1,\quad y(0)=0$

Answer

$y=1−e^{\tan^{−1}x}$

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

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

Answer

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

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

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

Answer

$y=2e^{2\sqrt{x}}−2x−2\sqrt{x}−1$

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 $0.$

Answer

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

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 $4$?

Answer

$40.451$ 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?)

Answer

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

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?

Answer

$y=Ce^x−a(x+1)$

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?

Answer

$y=Ce^{x^2/2}−a$

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?

Answer

$y=\dfrac{e^{kt}−e^t}{k−1}$