# 8.3.1: Solution of Initial Value Problems (Exercises)


## Q8.3.1

In Exercises 8.3.1-8.3.31 use the Laplace transform to solve the initial value problem.

1. $$y''+3y'+2y=e^t, \quad y(0)=1,\quad y'(0)=-6$$

2. $$y''-y'-6y=2, \quad y(0)=1,\quad y'(0)=0$$

3. $$y''+y'-2y=2e^{3t}, \quad y(0)=-1,\quad y'(0)=4$$

4. $$y''-4y=2 e^{3t}, \quad y(0)=1,\quad y'(0)=-1$$

5. $$y''+y'-2y=e^{3t}, \quad y(0)=1,\quad y'(0)=-1$$

6. $$y''+3y'+2y=6e^t, \quad y(0)=1,\quad y'(0)=-1$$

7. $$y''+y=\sin2t, \quad y(0)=0,\quad y'(0)=1$$

8. $$y''-3y'+2y=2e^{3t}, \quad y(0)=1,\quad y'(0)=-1$$

9. $$y''-3y'+2y=e^{4t}, \quad y(0)=1,\quad y'(0)=-2$$

10. $$y''-3y'+2y=e^{3t}, \quad y(0)=-1,\quad y'(0)=-4$$

11. $$y''+3y'+2y=2e^t, \quad y(0)=0,\quad y'(0)=-1$$

12. $$y''+y'-2y=-4, \quad y(0)=2,\quad y'(0)=3$$

13. $$y''+4y=4, \quad y(0)=0,\quad y'(0)=1$$

14. $$y''-y'-6y=2, \quad y(0)=1,\quad y'(0)=0$$

15. $$y''+3y'+2y=e^t, \quad y(0)=0,\quad y'(0)=1$$

16. $$y''-y=1, \quad y(0)=1,\quad y'(0)=0$$

17. $$y''+4y=3\sin t, \quad y(0)=1,\quad y'(0)=-1$$

18. $$y''+y'=2e^{3t}, \quad y(0)=-1,\quad y'(0)=4$$

19. $$y''+y=1, \quad y(0)=2,\quad y'(0)=0$$

20. $$y''+y=t, \quad y(0)=0,\quad y'(0)=2$$

21. $$y''+y=t-3\sin2t, \quad y(0)=1,\quad y'(0)=-3$$

22. $$y''+5y'+6y=2e^{-t}, \quad y(0)=1,\quad y'(0)=3$$

23. $$y''+2y'+y=6\sin t-4\cos t, \quad y(0)=-1,\; y'(0)=1$$

24. $$y''-2y'-3y=10\cos t, \quad y(0)=2,\quad y'(0)=7$$

25. $$y''+y=4\sin t+6\cos t, \quad y(0)=-6,\; y'(0)=2$$

26. $$y''+4y=8\sin2t+9\cos t, \quad y(0)=1,\quad y'(0)=0$$

27. $$y''-5y'+6y=10e^t\cos t, \quad y(0)=2,\quad y'(0)=1$$

28. $$y''+2y'+2y=2t, \quad y(0)=2,\quad y'(0)=-7$$

29. $$y''-2y'+2y=5\sin t+10\cos t, \quad y(0)=1,\; y'(0)=2$$

30. $$y''+4y'+13y=10e^{-t}-36e^t, \quad y(0)=0,\; y'(0)=-16$$

31. $$y''+4y'+5y=e^{-t}(\cos t+3\sin t), \quad y(0)=0,\quad y'(0)=4$$

## Q8.3.2

32. $$2y''-3y'-2y=4e^t, \quad y(0)=1,\; y'(0)=-2$$

33. $$6y''-y'-y=3e^{2t}, \quad y(0)=0,\; y'(0)=0$$

34. $$2y''+2y'+y=2t, \quad y(0)=1,\; y'(0)=-1$$

35. $$4y''-4y'+5y=4\sin t-4\cos t, \quad y(0)=0,\; y'(0)=11/17$$

36. $$4y''+4y'+y=3\sin t+\cos t, \quad y(0)=2,\; y'(0)=-1$$

37. $$9y''+6y'+y=3e^{3t}, \quad y(0)=0,\; y'(0)=-3$$

38. Suppose $$a,b$$, and $$c$$ are constants and $$a\ne0$$. Let $y_1={\cal L}^{-1}\left(as+b\over as^2+bs+c\right)\quad \text{and} \quad y_2={\cal L}^{-1}\left(a\over as^2+bs+c\right). \nonumber$

Show that $y_1(0)=1,\quad y_1'(0)=0\quad \text{and} \quad y_2(0)=0,\quad y_2'(0)=1.\nonumber$

HINT: Use the Laplace transform to solve the initial value problems

$\begin{array}{lll}{ay''+by'+cy=0,}&{y(0)=1,}&{y'(0)=0}\\{ay''+by'+cy=0,}&{y(0)=0,}&{y'(0)=1} \end{array}\nonumber$

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