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# Section 8.6 Answers

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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. $$\frac{1}{3}(\cos t-\cos 2t)$$