8.7E: 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

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

22. $y''-4y=3\delta(t), \quad y(0)=-1,\quad y_-'(0)=7$

23. $y''+3y'+2y=-5\delta(t), \quad y(0)=0,\quad y_-'(0)=0$

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

25. $4y''+4y'+y=3\delta(t), \quad y(0)=1,\quad y_-'(0)=-6$

Q8.7.3

In Exercises 8.7.26-8.7.28, solve the initial value problem $$ay_{h}'' + by_{h}'+cy_{h}=\left\{\begin{array}{ll} {0,}&{0\leq t<t_{0}}\\[4pt]{1/h, }&{t_{0}\leq t< t_{0} +h, }\\[4pt]{0,}&{t\geq t_{0}+h, } \end{array} \right. \quad y_{h}(0)=0, y_{h}'(0)=0\nonumber$$ 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. $y''+y=\sum_{k=1}^\infty\delta(t-k\pi), \quad y(0)=0,\quad y'(0)=0$