5.6E: 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

\(y=\dfrac{e^{kt}−e^t}{k−1}\)