14.42: Section 8.7 Answers

1. $y=\frac{1}{2}e^{2t}-4e^{-t}+\frac{11}{2}e^{-2t}+2u(t-1)(e^{-(t-1)}-e^{-2(t-1)})$

2. $y=2e^{-2t}+5e^{-t}+\frac{5}{3}u(t-1)(e^{(t-1)}-e^{-2(t-1)})$

3. $y=\frac{1}{6}e^{2t}-\frac{2}{3}e^{-t}-\frac{1}{2}e^{-2t}+\frac{5}{2}u(t-1)\sinh 2(t-1)$

4. $y=\frac{1}{8}\left(8\cos t-5\sin t-\sin 3t\right)-2u(t-\pi /2)\cos t$

5. $y=1-\cos 2t+\frac{1}{2}\sin 2t+\frac{1}{2}u(t-3\pi )\sin 2t$

6. $y=4e^{t}+3e^{-t}-8+2u(t-2)\sinh (t-2)$

7. $y=\frac{1}{2}e^{t}-\frac{7}{2}e^{-t}+2+3u (t-6)(1-e^{-(t-6)})$

8. $y=e^{2t}+7\cos 2t-\sin 2t-\frac{1}{2}u(t-\pi /2)\sin 2t$

9. $y=\frac{1}{2}(1+e^{-2t})+u(t-1)(e^{-(t-1)}-e^{-2(t-1)})$

10. $y=\frac{1}{4}e^{t}+\frac{1}{4}e^{-t}(2t-5)+2u(t-2)(t-2)e^{-(t-2)}$

11. $y=\frac{1}{6}(2\sin t+5\sin 2t)-\frac{1}{2}u(t-\pi /2)\sin 2t$

12. $y=e^{-t}(\sin t-\cos t)-e^{-(t-\pi )}\sin t-3u(t-2\pi )e^{-(t-2\pi )}\sin t$

13. $y=e^{-2t}\left(\cos 3t+\frac{4}{3}\sin 3t\right)-\frac{1}{3}u(t-\pi /6)e^{-2(t-\pi /6)}\cos 3t-\frac{2}{3}u(t-\pi /3)e^{-2(t-\pi /3)}\sin 3t$

14. $y=\frac{7}{10}e^{2t}-\frac{6}{5}e^{-t/2}-\frac{1}{2}+\frac{1}{5}u(t-2)(e^{2(t-2)}-e^{-(t-2)/2})$

15. $y=\frac{1}{17}(12\cos t+20\sin t)+\frac{1}{34}e^{t/2}(10\cos t-11\sin t)-u(t-\pi /2)e^{(2t-\pi )/4}\cos t+u(t-\pi )e^{(t-\pi )/2}\sin t$

16. $y=\frac{1}{3}(\cos t-\cos 2t-3\sin t)-2u(t-\pi /2)\cos t+3u(t-\pi )\sin t$

17. $y=e^{t}-e^{-t}(1+2t)-5u(t-1)\sinh (t-1)+3u(t-2)\sinh (t-2)$

18. $y=\frac{1}{4}(e^{t}-e^{-t}(1+6t))-u(t-1)e^{-(t-1)}+2u(t-2)e^{-(t-2)})$

19. $y=\frac{5}{3}\sin t-\frac{1}{3}\sin 2t+\frac{1}{3}u(t-\pi )(\sin 2t+2\sin t)+u(t-2\pi )\sin t$

20. $y=\frac{3}{4}\cos 2t-\frac{1}{2}\sin 2t+\frac{1}{4}+\frac{1}{4}u(t-\pi /2)(1+cos2t)+\frac{1}{2}u(t-\pi )\sin 2t+\frac{3}{2}u(t-3\pi /2)\sin 2t$

21. $y=\cos t-\sin t$

22. $y=\frac{1}{4}(8e^{3t}-12e^{-2t})$

23. $y=5(e^{-2t}-e^{-t})$

24. $y=e^{-2t}(1+6t)$

25. $y=\frac{1}{4}e^{-t/2}(4-19t)$

29. $y=(-1)^{k}m\omega _{1}Re^{-c\tau /2m}\delta (t-\tau )$ if $\omega _{1}\tau -\phi =(2k+1)\pi /2(k=$ integer $)$

30.

1. $y=\frac{(e^{m+1}-1)(e^{t-m}-e^{-t})}{2(e-1)},\quad m\leq t<m+1,\quad (m=0,1,\ldots )$
2. $y=(m+1)\sin t,\quad 2m\pi\leq t<2(m+1)\pi ,\quad (m=0,1,\ldots )$
3. $y=e^{2(t-m)}\frac{e^{2m+2}-1}{2^{2}-1}-e^{(t-m)}\frac{e^{m+1}-1}{e-1},\quad m\leq t<m+1,\quad (m=0,1,\ldots )$
4. $y=\left\{ \begin{array}{cc}{0,}&{2m\pi\leq t<(2m+1)\pi ,}\\[4pt]{-\sin t,}&{(2m+1)\pi\leq t<(2m+2)\pi ,}\end{array}\right.\quad (m=0,1,\ldots )$