A.8.6: Section 8.6 Answers

1.

1. \(\frac{1}{2}\int_{0}^{t}\tau\sin 2(t-\tau )d\tau \)
2. \(\int_{0}^{t}e^{-2\tau }\cos 3(t-\tau )d\tau \)
3. \(\frac{1}{2}\int_{0}^{t}\sin 2\tau\cos 3(t-\tau )d\tau\) or \(\frac{1}{3}\int_{0}^{t}\sin 3\tau\cos 2(t-\tau )d\tau \)
4. \(\int_{0}^{t}\cos\tau\sin (t-\tau )d\tau \)
5. \(\int_{0}^{t}e^{a\tau }d\tau \)
6. \(e^{-t}\int_{0}^{t}\sin (t-\tau )d\tau \)
7. \(e^{-2t}\int_{0}^{t}\tau e^{\tau }\sin (t-\tau )d\tau \)
8. \(\frac{e^{-2t}}{2}\int_{0}^{t}\tau ^{2}(t-\tau )e^{3\tau }d\tau \)
9. \(\int_{0}^{t}(t-\tau )e^{\tau }\cos\tau d\tau \)
10. \(\int_{0}^{t}e^{-3\tau }\cos\tau\cos 2(t-\tau )d\tau \)
11. \(\frac{1}{4!5!}\int_{0}^{t}\tau ^{4}(t-\tau )^{5}e^{3\tau }d\tau \)
12. \(\frac{1}{4}\int_{0}^{t}\tau ^{2}e^{\tau }\sin 2(t-\tau )d\tau \)
13. \(\frac{1}{2}\int_{0}^{t}\tau (t-\tau )^{2}e^{2(t-\tau )}d\tau \)
14. \(\frac{1}{5!6!}\int_{0}^{t}(t-\tau )^{5}e^{2(t-\tau )}\tau ^{6}d\tau \)

2.

1. \(\frac{as}{(s^{2}+a^{2})(s^{2}+b^{2})}\)
2. \(\frac{a}{(s-1)(s^{2}+a^{2})}\)
3. \(\frac{as}{(s^{2}-a^{2})^{2}}\)
4. \(\frac{2\omega s(s^{2}-\omega ^{2})}{(s^{2}+\omega ^{2})^{4}}\)
5. \(\frac{(s-1)\omega }{((s-1)^{2}+\omega ^{2})^{2}}\)
6. \(\frac{2}{(s-2)^{3}(s-1)^{2}}\)
7. \(\frac{s+1}{(s+2)^{2}\left[(s+1)^{2}+\omega ^{2}\right]}\)
8. \(\frac{1}{(s-3)((s-1)^{2}-1)}\)
9. \(\frac{2}{(s-2)^{2}(s^{2}+4)}\)
10. \(\frac{6}{s^{4}(s-1)}\)
11. \(\frac{3\cdot 6!}{s^{7}\left[(s+1)^{2}+9\right]}\)
12. \(\frac{12}{s^{7}}\)
13. \(\frac{2\cdot 7!}{s^{8}\left[ (s+1)^{2}+4\right]}\)
14. \(\frac{48}{s^{5}(s^{2}+4)}\)

3.

1. \(y=\frac{2}{\sqrt{5}}\int_{0}^{t}f(t-\tau )e^{-3\tau /2}\sinh\frac{\sqrt{5}\tau }{2}d\tau \)
2. \(y=\frac{1}{2}\int_{0}^{t}f(t-\tau )\sin 2\tau d\tau\)
3. \(y=\int_{0}^{t}\tau e^{-\tau }f(t-\tau )d\tau \)
4. \(y(t)=-\frac{1}{k}\sin kt+\cos kt+\frac{1}{k}\int_{0}^{t} f(t-\tau )\sin k\tau d\tau \)
5. \(y=-2te^{-3t}+\int_{0}^{t}\tau e^{-3\tau }f(t-\tau )d\tau \)
6. \(y=\frac{3}{2}\sinh 2t+\frac{1}{2}\int _{0}^{t} f(t-\tau )\sinh 2\tau d\tau \)
7. \(y=e^{3t}+\int_{0}^{t}(e^{3\tau }-e^{2\tau })f(t-\tau )d\tau \)
8. \(y=\frac{k_{1}}{\omega }\sin\omega t+k_{0}\cos\omega t+\frac{1}{\omega}\int_{0}^{t}f(t-\tau )\sin\omega\tau d\tau \)

4.

1. \(y=\sin t\)
2. \(y=te^{-t}\)
3. \(y=1+2te^{t}\)
4. \(y=t+\frac{t^{2}}{2}\)
5. \(y=4+\frac{5}{2}t^{2}+\frac{1}{24}t^{4}\)
6. \(y=1-t\)

5.

1. \(\frac{7!8!}{16!}t^{16}\)
2. \(\frac{13!7!}{21!}t^{21}\)
3. \(\frac{6!7!}{14!}t^{14}\)
4. \(\frac{1}{2}(e^{-t}+\sin t-\cos t)\)
5. \(\frac{1}{3}(\cos t-\cos 2t)\)