A.8.5: Section 8.5 Answers

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1. \(y=3(1-\cos t)-3u(t-\pi )(1+\cos t)\)

2. \(y=3-2\cos t+2u(t-4)(t-4-\sin (t-4))\)

3. \(y=-\frac{15}{2}+\frac{3}{2}e^{2t}-2t+\frac{u(t-1)}{2}(e^{2(t-1)}-2t+1)\)

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

5. \(y=-7e^{t}+4e^{2t}+u(t-1)\left(\frac{1}{2}-e^{t-1}+\frac{1}{2}e^{2(t-1)}\right)-2u(t-2)\left(\frac{1}{2}-e^{t-2}+\frac{1}{2}e^{2(t-2)}\right)\)

6. \(y=\frac{1}{3}\sin 2t-3\cos 2t+\frac{1}{3}\sin t-2u(t-\pi )\left(\frac{1}{3}\sin t+\frac{1}{6}\sin 2t\right)+u(t-2\pi )\left(\frac{1}{3}\sin t-\frac{1}{6}\sin 2t\right)\)

7. \(y=\frac{1}{4}-\frac{31}{12}e^{4t}+\frac{16}{3}e^{t}+u(t-1)\left(\frac{2}{3}e^{t-1}-\frac{1}{6}e^{4(t-1)}-\frac{1}{2}\right)+u(t-2)\left(\frac{1}{4}+\frac{1}{12}e^{4(t-2)}-\frac{1}{3}e^{t-2}\right)\)

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

9. \(y=\frac{t}{4}-\frac{1}{8}\sin 2t+\frac{1}{8}u\left(t-\frac{\pi }{2}\right)(\pi\cos 2t-\sin 2t+2\pi -2t)\)

10. \(y=t-\sin t-2u(t-\pi )(t+\sin t+\pi\cos t)\)

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

12. \(y=t+\sin t+\cos t-u(t-2\pi )(3t-3\sin t-6\pi\cos t)\)

13. \(y=\frac{1}{2}+\frac{1}{2}e^{-2t}-e^{-t}+u(t-2)\left(2e^{-(t-2)}-e^{-2(t-2)}-1\right)\)

14. \(y=-\frac{1}{3}-\frac{1}{6}e^{3t}+\frac{1}{2}e^{t}+u(t-1)\left(\frac{2}{3}+\frac{1}{3}e^{3(t-1)}-e^{t-1}\right)\)

15. \(y=\frac{1}{4}\left(e^{t}+e^{-t}(11+6t)\right)+u(t-1)(te^{-(t-1)}-1)\)

16. \(y=e^{t}-e^{-t}-2te^{-t}-u(t-1)\left(e^{t}-e^{-(t-2)}-2(t-1)e^{-(t-2)}\right)\)

17. \(y=te^{-t}+e^{-2t}+u(t-1)\left(e^{-t}(2-t)-e^{-(2t-1)}\right)\)

18. \(y=\frac{t^{2}e^{2t}}{2}-te^{2t}-u(t-2)(t-2)^{2}e^{2t}\)

19. \(y=\frac{t^{4}}{12}+1-\frac{1}{12}u(t-1)(t^{4}+2t^{3}-10t+7)+\frac{1}{6}u(t-2)(2t^{3}+3t^{2}-36t+44)\)

20. \(y=\frac{1}{2}e^{-t}(3\cos t+\sin t)+\frac{1}{2}-u(t-2\pi )\left(e^{-(t-2\pi )}\left((\pi -1)\cos t+\frac{2\pi -1}{2}\sin t\right)+1-\frac{t}{2}\right)-\frac{1}{2}u(t-3\pi )(e^{-(t-3\pi )}(3\pi\cos t+(3\pi +1)\sin t)+t)\)

21. \(y=\frac{t^{2}}{2}+\sum_{m=1}^{\infty}u(t-m)\frac{(t-m)^{2}}{2}\)

22.

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