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# 8.3E: Solution of Initial Value Problems (Exercises)

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In Exercises [exer:8.3.1}– [exer:8.3.31} use the Laplace transform to solve the initial value problem.

[exer:8.3.1] $$y''+3y'+2y=e^t, \quad y(0)=1,\quad y'(0)=-6$$

[exer:8.3.2] $$y''-y'-6y=2, \quad y(0)=1,\quad y'(0)=0$$

[exer:8.3.3] $$y''+y'-2y=2e^{3t}, \quad y(0)=-1,\quad y'(0)=4$$

[exer:8.3.4] $$y''-4y=2 e^{3t}, \quad y(0)=1,\quad y'(0)=-1$$

[exer:8.3.5] $$y''+y'-2y=e^{3t}, \quad y(0)=1,\quad y'(0)=-1$$

[exer:8.3.6] $$y''+3y'+2y=6e^t, \quad y(0)=1,\quad y'(0)=-1$$

[exer:8.3.7] $$y''+y=\sin2t, \quad y(0)=0,\quad y'(0)=1$$

[exer:8.3.8] $$y''-3y'+2y=2e^{3t}, \quad y(0)=1,\quad y'(0)=-1$$

[exer:8.3.9] $$y''-3y'+2y=e^{4t}, \quad y(0)=1,\quad y'(0)=-2$$

[exer:8.3.10] $$y''-3y'+2y=e^{3t}, \quad y(0)=-1,\quad y'(0)=-4$$

[exer:8.3.11] $$y''+3y'+2y=2e^t, \quad y(0)=0,\quad y'(0)=-1$$

[exer:8.3.12] $$y''+y'-2y=-4, \quad y(0)=2,\quad y'(0)=3$$

[exer:8.3.13] $$y''+4y=4, \quad y(0)=0,\quad y'(0)=1$$

[exer:8.3.14] $$y''-y'-6y=2, \quad y(0)=1,\quad y'(0)=0$$

[exer:8.3.15] $$y''+3y'+2y=e^t, \quad y(0)=0,\quad y'(0)=1$$

[exer:8.3.16] $$y''-y=1, \quad y(0)=1,\quad y'(0)=0$$

[exer:8.3.17] $$y''+4y=3\sin t, \quad y(0)=1,\quad y'(0)=-1$$

[exer:8.3.18] $$y''+y'=2e^{3t}, \quad y(0)=-1,\quad y'(0)=4$$

[exer:8.3.19] $$y''+y=1, \quad y(0)=2,\quad y'(0)=0$$

[exer:8.3.20] $$y''+y=t, \quad y(0)=0,\quad y'(0)=2$$

[exer:8.3.21] $$y''+y=t-3\sin2t, \quad y(0)=1,\quad y'(0)=-3$$

[exer:8.3.22] $$y''+5y'+6y=2e^{-t}, \quad y(0)=1,\quad y'(0)=3$$

[exer:8.3.23] $$y''+2y'+y=6\sin t-4\cos t, \quad y(0)=-1,\; y'(0)=1$$

[exer:8.3.24] $$y''-2y'-3y=10\cos t, \quad y(0)=2,\quad y'(0)=7$$

[exer:8.3.25] $$y''+y=4\sin t+6\cos t, \quad y(0)=-6,\; y'(0)=2$$

[exer:8.3.26] $$y''+4y=8\sin2t+9\cos t, \quad y(0)=1,\quad y'(0)=0$$

[exer:8.3.27] $$y''-5y'+6y=10e^t\cos t, \quad y(0)=2,\quad y'(0)=1$$

[exer:8.3.28] $$y''+2y'+2y=2t, \quad y(0)=2,\quad y'(0)=-7$$

[exer:8.3.29] $$y''-2y'+2y=5\sin t+10\cos t, \quad y(0)=1,\; y'(0)=2$$

[exer:8.3.30] $$y''+4y'+13y=10e^{-t}-36e^t, \quad y(0)=0,\; y'(0)=-16$$

[exer:8.3.31] $$y''+4y'+5y=e^{-t}(\cos t+3\sin t), \quad y(0)=0,\quad y'(0)=4$$

[exer:8.3.32] $$2y''-3y'-2y=4e^t, \quad y(0)=1,\; y'(0)=-2$$

[exer:8.3.33] $$6y''-y'-y=3e^{2t}, \quad y(0)=0,\; y'(0)=0$$

[exer:8.3.34] $$2y''+2y'+y=2t, \quad y(0)=1,\; y'(0)=-1$$

[exer:8.3.35] $$4y''-4y'+5y=4\sin t-4\cos t, \quad y(0)=0,\; y'(0)=11/17$$

[exer:8.3.36] $$4y''+4y'+y=3\sin t+\cos t, \quad y(0)=2,\; y'(0)=-1$$

[exer:8.3.37] $$9y''+6y'+y=3e^{3t}, \quad y(0)=0,\; y'(0)=-3$$

[exer:8.3.38] 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 $y_1(0)=1,\quad y_1'(0)=0\quad \text{and} \quad y_2(0)=0,\quad y_2'(0)=1.\nonumber$