## Problem Set: First-order Linear Equations

Are the following differential equations linear? Explain your reasoning.

1. $\frac{dy}{dx}={x}^{2}y+\sin{x}$

2. $\frac{dy}{dt}=ty$

3. $\frac{dy}{dt}+{y}^{2}=x$

4. $y^{\prime} ={x}^{3}+{e}^{x}$

5. $y^{\prime} =y+{e}^{y}$

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

6. $y^{\prime} ={x}^{3}y+\sin{x}$

7. $y^{\prime} +3y-\text{ln}x=0$

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

9. $\frac{dy}{dt}=4y+ty+\tan{t}$

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

What are the integrating factors for the following differential equations?

11. $y^{\prime} =xy+3$

12. $y^{\prime} +{e}^{x}y=\sin{x}$

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

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

15. $\frac{dy}{dt}+3ty={e}^{t}y$

Solve the following differential equations by using integrating factors.

16. $y^{\prime} =3y+2$

17. $y^{\prime} =2y-{x}^{2}$

18. $xy^{\prime} =3y - 6{x}^{2}$

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

20. $y^{\prime} =3x+xy$

21. $xy^{\prime} =x+y$

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

23. $y^{\prime} =y+{e}^{x}$

24. $xy^{\prime} =3y+{x}^{2}$

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

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

27. [T] $y^{\prime} =3{e}^{\frac{t}{3}}-2y$

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

29. [T] $xy^{\prime} =2\frac{\cos{x}}{x}-3y$

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

31. [T] $\sin\left(x\right)y^{\prime} +\cos\left(x\right)y=2x$

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

33. [T] ${x}^{3}y^{\prime} +2{x}^{2}y=x+1$

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

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

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

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

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

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

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

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

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

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

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

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 $0$.

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 $4?$

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?)

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 $9.8{\text{m/s}}^{2}$ and the proportionality constant is $4?$ 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?

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

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

53. Solve the generic equation $y^{\prime} =x+axy$. How does varying $a$ 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?