8.3E: Solution of Initial Value Problems (Exercises)

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Jun 10, 2024
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- 8.3: Solution of Initial Value Problems
- 8.4: The Unit Step Function

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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

$\begin{array}{lll}{ay''+by'+cy=0,}&{y(0)=1,}&{y'(0)=0}\\[4pt]{ay''+by'+cy=0,}&{y(0)=0,}&{y'(0)=1} \end{array}\nonumber$

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