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# 8.7.1: Constant Coefficient Equations with Impulses (Exercises)


## Q8.7.1

In Exercises 8.7.1-8.7.20 solve the initial value problem. Graph the solution for Exercises 8.7.2, 8.7.4, 8.7.9, and 8.7.19.

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

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

3. $$y''-4y=2e^{-t}+5\delta(t-1), \quad y(0)=-1,\quad y'(0)=2$$

4. $$y''+y=\sin3t+2\delta(t-\pi/2), \quad y(0)=1,\quad y'(0)=-1$$

5. $$y''+4y=4+\delta(t-3\pi), \quad y(0)=0,\quad y'(0)=1$$

6. $$y''-y=8+2\delta(t-2), \quad y(0)=-1,\quad y'(0)=1$$

7. $$y''+y'=e^t+3\delta(t-6), \quad y(0)=-1,\quad y'(0)=4$$

8. $$y''+4y=8e^{2t}+\delta(t-\pi/2), \quad y(0)=8,\quad y'(0)=0$$

9. $$y''+3y'+2y=1+\delta(t-1), \quad y(0)=1,\quad y'(0)=-1$$

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

11. $$y''+4y=\sin t+\delta(t-\pi/2), \quad y(0)=0,\quad y'(0)=2$$

12. $$y''+2y'+2y=\delta(t-\pi)-3\delta(t-2\pi), \quad y(0)=-1,\quad y'(0)=2$$

13. $$y''+4y'+13y=\delta(t-\pi/6)+2\delta(t-\pi/3), \quad y(0)=1,\quad y'(0)=2$$

14. $$2y''-3y'-2y=1+\delta(t-2), \quad y(0)=-1,\quad y'(0)=2$$

15. $$4y''-4y'+5y=4\sin t-4\cos t+\delta(t-\pi/2)-\delta(t-\pi), \quad y(0)=1,\quad y'(0)=1$$

16. $$y''+y=\cos2t+2\delta(t-\pi/2)-3\delta(t-\pi), \quad y(0)=0,\quad y'(0)=-1$$

17. $$y''-y=4e^{-t}-5\delta(t-1)+3\delta(t-2), \quad y(0)=0,\quad y'(0)=0$$

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

19. $$y''+y=f(t)+\delta(t-2\pi), \quad y(0)=0,\quad y'(0)=1$$,

$$f(t)=\left\{\begin{array}{cl} \sin2t,&0\le t<\pi,\$4pt]0,&t\ge \pi.\end{array}\right.$$ 20. $$y''+4y=f(t)+\delta(t-\pi)-3\delta(t-3\pi/2), \quad y(0)=1,\quad y'(0)=-1$$, $$f(t)=\left\{\begin{array}{cl}1,&0\le t<\pi/2,\\[4pt]2,&t\ge \pi/2\end{array}\right.$$ ## Q8.7.2 1. $$y''+y=\delta(t), \quad y(0)=1,\quad y_-'(0)=-2$$ 2. $$y''-4y=3\delta(t), \quad y(0)=-1,\quad y_-'(0)=7$$ 3. $$y''+3y'+2y=-5\delta(t), \quad y(0)=0,\quad y_-'(0)=0$$ 4. $$y''+4y'+4y=-\delta(t), \quad y(0)=1,\quad y_-'(0)=5$$ 5. $$4y''+4y'+y=3\delta(t), \quad y(0)=1,\quad y_-'(0)=-6$$ ## Q8.7.3 In Exercises 8.7.26-8.7.28, solve the initial value problem \[ay_{h}'' + by_{h}'+cy_{h}=\left\{\begin{array}{ll} {0,}&{0\leq t<t_{0}}\\{1/h, }&{t_{0}\leq t< t_{0} +h, }\\{0,}&{t\geq t_{0}+h, } \end{array} \right. \quad y_{h}(0)=0, y_{h}'(0)=0\nonumber$ where $$t_{0}>0$$ and $$h>0$$. Then find $w=\cal{L}^{-1}\left(\frac{1}{as^{2}+bs+c} \right) \nonumber$ and verify Theorem 8.7.1 by graphing $$w$$ and $$y_{h}$$ on the same axes, for small positive values of $$h$$.

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

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

28. $$y''+3y'+2y=f_h(t), \quad y(0)=0,\quad y'(0)=0$$

## Q8.7.4

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

$my''+cy'+ky=0, \quad y(0)=y_0,\quad y'(0)=v_0,\nonumber$

and that $$y$$ can be written as

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

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.

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. $$y''+y=\sum_{k=1}^\infty\delta(t-k\pi), \quad y(0)=0,\quad y'(0)=0$$