
# 8.7E: Constant Coefficient Equations with Impulses (Exercises)

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In Exercises [exer:8.7.1}– [exer:8.7.20} solve the initial value problem. Where indicated by , graph the solution.

[exer:8.7.1] $$y''+3y'+2y=6e^{2t}+2\delta(t-1), \quad y(0)=2,\quad y'(0)=-6$$

[exer:8.7.2] $$y''+y'-2y=-10e^{-t}+5\delta(t-1), \quad y(0)=7,\quad y'(0)=-9$$

[exer:8.7.3] $$y''-4y=2e^{-t}+5\delta(t-1), \quad y(0)=-1,\quad y'(0)=2$$

[exer:8.7.4] $$y''+y=\sin3t+2\delta(t-\pi/2), \quad y(0)=1,\quad y'(0)=-1$$

[exer:8.7.5] $$y''+4y=4+\delta(t-3\pi), \quad y(0)=0,\quad y'(0)=1$$

[exer:8.7.6] $$y''-y=8+2\delta(t-2), \quad y(0)=-1,\quad y'(0)=1$$

[exer:8.7.7] $$y''+y'=e^t+3\delta(t-6), \quad y(0)=-1,\quad y'(0)=4$$

[exer:8.7.8] $$y''+4y=8e^{2t}+\delta(t-\pi/2), \quad y(0)=8,\quad y'(0)=0$$

[exer:8.7.9] $$y''+3y'+2y=1+\delta(t-1), \quad y(0)=1,\quad y'(0)=-1$$

[exer:8.7.10] $$y''+2y'+y=e^t+2\delta(t-2), \quad y(0)=-1,\quad y'(0)=2$$

[exer:8.7.11] $$y''+4y=\sin t+\delta(t-\pi/2), \quad y(0)=0,\quad y'(0)=2$$

[exer:8.7.12] $$y''+2y'+2y=\delta(t-\pi)-3\delta(t-2\pi), \quad y(0)=-1,\quad y'(0)=2$$

[exer:8.7.13] $$y''+4y'+13y=\delta(t-\pi/6)+2\delta(t-\pi/3), \quad y(0)=1,\quad y'(0)=2$$

[exer:8.7.14] $$2y''-3y'-2y=1+\delta(t-2), \quad y(0)=-1,\quad y'(0)=2$$

[exer:8.7.15] $$4y''-4y'+5y=4\sin t-4\cos t+\delta(t-\pi/2)-\delta(t-\pi), \quad y(0)=1,\quad y'(0)=1$$

[exer:8.7.16] $$y''+y=\cos2t+2\delta(t-\pi/2)-3\delta(t-\pi), \quad y(0)=0,\quad y'(0)=-1$$

[exer:8.7.17] $$y''-y=4e^{-t}-5\delta(t-1)+3\delta(t-2), \quad y(0)=0,\quad y'(0)=0$$

[exer:8.7.18] $$y''+2y'+y=e^t-\delta(t-1)+2\delta(t-2), \quad y(0)=0,\quad y'(0)=-1$$

[exer:8.7.19] $$y''+y=f(t)+\delta(t-2\pi), \quad y(0)=0,\quad y'(0)=1$$,

$$f(t)=\left\{\begin{array}{cl} \sin2t,&0\le t<\pi,\$4pt]0,&t\ge \pi.\end{array}\right.$$ [exer:8.7.20] $$y''+4y=f(t)+\delta(t-\pi)-3\delta(t-3\pi/2), \quad y(0)=1,\quad y'(0)=-1$$, $$f(t)=\left\{\begin{array}{cl}1,&0\le t<\pi/2,\\[4pt]2,&t\ge \pi/2\end{array}\right.$$ [exer:8.7.21] $$y''+y=\delta(t), \quad y(0)=1,\quad y_-'(0)=-2$$ [exer:8.7.22] $$y''-4y=3\delta(t), \quad y(0)=-1,\quad y_-'(0)=7$$ [exer:8.7.23] $$y''+3y'+2y=-5\delta(t), \quad y(0)=0,\quad y_-'(0)=0$$ [exer:8.7.24] $$y''+4y'+4y=-\delta(t), \quad y(0)=1,\quad y_-'(0)=5$$ [exer:8.7.25] $$4y''+4y'+y=3\delta(t), \quad y(0)=1,\quad y_-'(0)=-6$$ [exer:8.7.26] $$y''+2y'+2y=f_h(t), \quad y(0)=0,\quad y'(0)=0$$ [exer:8.7.27] $$y''+2y'+y=f_h(t), \quad y(0)=0,\quad y'(0)=0$$ [exer:8.7.28] $$y''+3y'+2y=f_h(t), \quad y(0)=0,\quad y'(0)=0$$ [exer:8.7.29] Recall from Section 6.2 that the displacement of an object of mass $$m$$ in a spring–mass system in free damped oscillation is \[my''+cy'+ky=0, \quad y(0)=y_0,\quad y'(0)=v_0,$

and that $$y$$ can be written as

$y=Re^{-ct/2m}\cos(\omega_1t-\phi)$

if the motion is underdamped. Suppose $$y(\tau)=0$$. Find the impulse that would have to be applied to the object at $$t=\tau$$ to put it in equilibrium.

[exer:8.7.30] Solve the initial value problem. Find a formula that does not involve step functions and represents $$y$$ on each subinterval of $$[0,\infty)$$ on which the forcing function is zero.

1. $$y''-y=\sum_{k=1}^\infty\delta(t-k), \quad y(0)=0,\quad y'(0)=1$$
2. $$y''+y=\sum_{k=1}^\infty\delta(t-2k\pi), \quad y(0)=0,\quad y'(0)=1$$
3. $$y''-3y'+2y=\sum_{k=1}^\infty\delta(t-k), \quad y(0)=0,\quad y'(0)=1$$
4. $$y''+y=\sum_{k=1}^\infty\delta(t-k\pi), \quad y(0)=0,\quad y'(0)=0$$
$$f(t)$$ $$F(s)$$
1 $$1\over s$$ $$(s > 0)$$
$$t^n$$ $$n!\over s^{n+1}$$ $$(s > 0)$$
($$n = \mbox{ integer } > 0$$)
$$t^p,\; p > -1$$ $$\Gamma (p+1) \over s^{(p+1)}$$ $$(s>0)$$
$$e^{at}$$ $$1 \over s-a$$ $$(s > a)$$
$$t^ne^{at}$$ $$n! \over (s-a)^{n+1}$$ $$(s > 0)$$
($$n= \text{ integer } > 0$$)
$$\cos \omega t$$ $$s \over s^2+\omega^2}$$ $$(s > 0)$$
$$\sin \omega t$$ $$\omega \over s^2+\omega^2$$ $$(s > 0)$$
$$e^{\lambda t} \cos \omega t$$ $$s - \lambda \over (s-\lambda)^2+\omega^2$$ $$(s > \lambda)$$
$$e^{\lambda t} \sin \omega t$$ $$\omega \over (s-\lambda)^2+\omega^2$$ $$(s > \lambda)$$
$$\cosh bt$$ $$s \over s^2-b^2$$ $$(s > |b|)$$
$$\sinh bt$$ $$b \over s^2-b^2$$ $$(s > |b|)$$
$$t \cos \omega t$$ $$s^2-\omega^2 \over (s^2+\omega^2)^2$$ $$(s>0)$$
 $$t \sin \omega t$$ $$2\omega s \over (s^2+\omega^2)^2$$ $$(s>0)$$ $$\sin \omega t -\omega t\cos \omega t$$ $$2\omega^3\over (s^2+\omega^2)^2$$ $$(s>0)$$ $$\omega t - \sin \omega t$$ $$\omega^3 \over s^2(s^2+\omega^2)^2$$ $$(s>0)$$ $$1 \over t} \sin \omega t }$$ $$\arctan \left({\omega \over s}\right)$$ $$(s>0)$$ $$e^{at}f(t)$$ $$F(s-a)$$ $$t^kf(t)$$ $$(-1)^kF^{(k)}(s)}}$$ $$f(\omega t)$$ $$1\over \omega} F\left({s \over \omega}\right), \quad \omega > 0}$$ $$u(t-\tau)$$ $$e^{-\tau s} \over s$$ $$(s>0)$$ $$u(t-\tau)f(t-\tau)\, (\tau > 0)$$ $$e^{-\tau s}F(s)$$ $$\displaystyle{\int^t_o f(\tau)g(t-\tau)\, d\tau}$$ $$F(s) \cdot G(s)$$ $$\delta(t-a)$$ $$e^{-as}$$ $$(s>0)$$