14.42: Section 8.7 Answers

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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.

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

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