A.8.4: Section 8.4 Answers

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1. \(1+u(t-4)(t-1);\quad\frac{1}{s}+e^{-4s}\left(\frac{1}{s^{2}}+\frac{3}{s}\right)\)

2. \(t+u(t-1)(1-t);\quad\frac{1-e^{-s}}{s^{2}}\)

3. \(2t-1-u(t-2)(t-1);\quad\left(\frac{2}{s^{2}}-\frac{1}{s}\right)-e^{-2s}\left(\frac{1}{s^{2}}+\frac{1}{s}\right)\)

4. \(1+u(t-1)(t+1);\quad\frac{1}{s}+e^{-s}\left(\frac{1}{s^{2}}+\frac{2}{s}\right)\)

5. \(t-1+u(t-2)(5-t);\quad\frac{1}{s^{2}}-\frac{1}{s}-e^{-2s}\left(\frac{1}{s^{2}}-\frac{3}{s}\right)\)

6. \(t^{2}(1-u(t-1));\quad\frac{2}{s^{3}}-e^{-s}\left(\frac{2}{s^{3}}+\frac{2}{s^{2}}+\frac{1}{s}\right)\)

7. \(u(t-2)(t^{2}+3t);\quad e^{-2s}\left(\frac{2}{s^{3}}+\frac{7}{s^{2}}+\frac{10}{s}\right)\)

8. \(t^{2}+2+u(t-1)(t-t^{2}-2);\quad\frac{2}{s^{3}}+\frac{2}{s}-e^{-s}\left(\frac{2}{s^{3}}+\frac{1}{s^{2}}+\frac{2}{s}\right)\)

9. \(te^{t}+u(t-1)(e^{t}-te^{t});\quad\frac{1-e^{-(s-1)}}{(s-1)^{2}}\)

10. \(e^{-t}+u(t-1)(e^{-2t}-e^{-t});\quad\frac{1-e^{-(s+1)}}{s+1}+\frac{e^{-(s+2)}}{s+2}\)

11. \(-t+2u(t-2)(t-2)-u(t-3)(t-5);\quad-\frac{1}{s^{2}}+\frac{2e^{-2s}}{s^{2}}+e^{-3s}\left(\frac{2}{s}-\frac{1}{s^{2}}\right)\)

12. \(\left[u(t-1)-u(t-2)\right] t;\quad e^{-s}\left(\frac{1}{s^{2}}+\frac{1}{s}\right)-e^{-2s}\left(\frac{1}{s^{2}}+\frac{2}{s}\right)\)

13. \(t+u(t-1)(t^{2}-t)-u(t-2)t^{2};\quad\frac{1}{s^{2}}+e^{-s}\left(\frac{2}{s^{3}}+\frac{1}{s^{2}}\right)-e^{-2s}\left(\frac{2}{s^{3}}+\frac{4}{s^{2}}+\frac{4}{s}\right)\)

14. \(t+u(t-1)(2-2t)+u(t-2)(4+t);\quad\frac{1}{s^{2}}-2\frac{e^{-s}}{s^{2}}+e^{-2s}\left(\frac{1}{s^{2}}+\frac{6}{s}\right)\)

15. \(\sin t+u(t-\pi /2)\sin t+u(t-\pi )(\cos t-2\sin t);\quad \frac{1+e^{-\frac{\pi }{2}s}s-e^{-\pi s}(s-2)}{s^{2}+1}\)

16. \(2-2u(t-1)t+u(t-3)(5t-2);\quad\frac{2}{s}-e^{-s}\left(\frac{2}{s^{2}}+\frac{2}{s}\right)+e^{-3s}\left(\frac{5}{s^{2}}+\frac{13}{s}\right)\)

17. \(3+u(t-2)(3t-1)+u(t-4)(t-2);\quad\frac{3}{s}+e^{-2s}\left(\frac{3}{s^{2}}+\frac{5}{s}\right)+e^{-4s}\left(\frac{1}{s^{2}}+\frac{2}{s}\right)\)

18. \((t+1)^{2}+u(t-1)(2t+3);\quad\frac{2}{s^{3}}+\frac{2}{s^{2}}+\frac{1}{s}+e^{-s}\left(\frac{2}{s^{2}}+\frac{5}{s}\right)\)

19. \(u(t-2)e^{2(t-2)}=\left\{\begin{array}{cc}{0,}&{0\leq t<2,}\\{e^{2(t-2)},}&{t\geq 2}\end{array} \right.\)

20. \(u(t-1)\left(1-e^{-(t-1)}\right)=\left\{\begin{array}{cc}{0,}&{0\leq t<1,}\\{1-e^{-(t-1)},}&{t\geq 1}\end{array} \right.\)

21. \(u(t-1)\frac{(t-1)^{2}}{2}+u(t-2)(t-2)=\left\{\begin{array}{cc}{0,}&{0\leq t<1,}\\{\frac{(t-1)^{2}}{2},}&{1\leq t<2,}\\{\frac{t^{2}-3}{2},}&{t\geq 2}\end{array} \right.\)

22. \(2+t+u(t-1)(4-t)+u(t-3)(t-2)=\left\{\begin{array}{cc}{2+t,}&{0\leq t<1,}\\{6,}&{1\leq t<3,}\\{t+4,}&{t\geq 3}\end{array} \right.\)

23. \(5-t+u(t-3)(7t-15)+\frac{3}{2}u(t-6)(t-6)^{2}=\left\{\begin{array}{cc}{5-t,}&{0\leq t<3,}\\{6t-10,}&{3\leq t<6,}\\{44-12t+\frac{3}{2}t^{2},}&{t\geq 6}\end{array} \right.\)

24. \(u(t-\pi )e^{-2(t-\pi )}(2\cos t-5\sin t)=\left\{\begin{array}{cc}{0,}&{0\leq t<\pi ,}\\{e^{-2(t-\pi )}(2\cos t-5\sin t)}&{t\geq\pi }\end{array} \right.\)

25. \(1-\cos t+u(t-\pi /2)(3\sin t+\cos t)=\left\{\begin{array}{cc}{1-\cos t,}&{0\leq t<\frac{\pi }{2},}\\{1+3\sin t,}&{t\geq\frac{\pi }{2}}\end{array} \right.\)

26. \(u(t-2)(4e^{-(t-2)}-4e^{2(t-2)}+2e^{(t-2)}=\left\{\begin{array}{cc}{0,}&{0\leq t<2,}\\{4e^{-(t-2)}-4e^{2(t-2)}+2e^{(t-2)},}&{t\geq 2}\end{array} \right.\)

27. \(1+t+u(t-1)(2t+1)+u(t-3)(3t-5)=\left\{\begin{array}{cc}{t+1,}&{0\leq t<1,}\\{3t+2,}&{1\leq t<3,}\\{6t-3,}&{t\geq 3}\end{array} \right.\)

28. \(1-t^{2}+u(t-2)\left(-\frac{t^{2}}{2}+2t+1\right)+u(t-4)(t-4)=\left\{\begin{array}{cc}{1-t^{2},}&{0\leq t<2,}\\{-\frac{3t^{2}}{2}+2t+2,}&{2\leq t<4,}\\{-\frac{3t^{2}}{2}+3t-2,}&{t\geq 4}\end{array} \right.\)

29. \(\frac{e^{-\tau s}}{s}\)

30. For each \(t\) only finitely many terms are nonzero.

33. \(1+\sum_{m=1}^{\infty}u(t-m);\quad\frac{1}{s(1-e^{-s})}\)

34. \(1+2\sum_{m=1}^{\infty}(-1)^{m}u(t-m);\quad\frac{1}{s};\quad\frac{1-e^{-s}}{1+e^{-s}}\)

35. \(1+\sum_{m=1}^{\infty}(2m+1)u(t-m);\quad\frac{e^{-s}(1+e^{-s})}{s(1-e^{-s})^{2}}\)

36. \(\sum_{m=1}^{\infty}(-1)^{m}(2m-1)u(t-m);\quad\frac{1}{s}\frac{(1-e^{s})}{(1+e^{s})^{2}}\)