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8.3E: Solution of Initial Value Problems (Exercises)

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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

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