A.8.2: Section 8.2 Answers

1.

1. \(\frac{t^{3}e^{7t}}{2}\)
2. \(2e^{2t}\cos 3t\)
3. \(\frac{e^{-2t}}{4}\sin 4t\)
4. \(\frac{2}{3}\sin 3t\)
5. \(t\cos t\)
6. \(\frac{e^{2t}}{2}\sinh 2t\)
7. \(\frac{2te^{2t}}{3}\sin 9t\)
8. \(\frac{2e^{3t}}{3}\sinh 3t\)
9. \(e^{2t}t\cos t\)

2.

1. \(t^{2}e^{7t}+\frac{17}{6}t^{3}e^{7t}\)
2. \(e^{2t}\left(\frac{1}{6}t^{3}+\frac{1}{6}t^{4}+\frac{1}{40}t^{5}\right)\)
3. \(e^{-3t}\left(\cos 3t+\frac{2}{3}\sin 3t\right)\)
4. \(2\cos 3t+\frac{1}{3}\sin 3t\)
5. \((1-t)e^{-t}\)
6. \(\cosh 3t+\frac{1}{3}\sinh 3t\)
7. \(\left( 1-t-t^{2}-\frac{1}{6}t^{3}\right)e^{-t}\)
8. \(e^{t}\left(2\cos 2t+\frac{5}{2}\sin 2t\right)\)
9. \(1-\cos t\)
10. \(3\cosh t+4\sinh t\)
11. \(3e^{t}+4\cos 3t+\frac{1}{3}\sin 3t\)
12. \(3te^{-2t}-2\cos 2t-3\sin 2t\)

3.

1. \(\frac{1}{4}e^{2t}-\frac{1}{4}e^{-2t}-e^{-t}\)
2. \(\frac{1}{5}e^{-4t}-\frac{41}{5}e^{t}+5e^{3t}\)
3. \(-\frac{1}{2}e^{2t}-\frac{13}{10}e^{-2t}-\frac{1}{5}e^{3t}\)
4. \(-\frac{2}{5}e^{-4t}-\frac{3}{5}e^{t}\)
5. \(\frac{3}{20}e^{2t}-\frac{37}{12}e^{-2t}+\frac{1}{3}e^{t}+\frac{8}{5}e^{-3t}\)
6. \(\frac{39}{10}e^{t}+\frac{3}{14}e^{3t}+\frac{23}{105}e^{-4t}-\frac{7}{3}e^{2t}\)

4.

1. \(\frac{4}{5}e^{-2t}-\frac{1}{2}e^{-t}-\frac{3}{10}\cos t+\frac{11}{10}\sin t\)
2. \(\frac{2}{5}\sin t+\frac{6}{5}\cos t+\frac{7}{5}e^{-t}\sin t-\frac{6}{5}e^{-t}\cos t\)
3. \(\frac{8}{13}e^{2t}-\frac{8}{13}e^{-t}\cos 2t+\frac{15}{26}e^{-t}\sin 2t\)
4. \(\frac{1}{2}te^{t}+\frac{3}{8}e^{t}+e^{-2t}-\frac{11}{8}e^{-3t}\)
5. \(\frac{2}{3}te^{t}+\frac{1}{9}e^{t}+te^{-2t}-\frac{1}{9}e^{-2t}\)
6. \(-e^{t}+\frac{5}{2}te^{t}+\cos t-\frac{3}{2}\sin t\)

5.

1. \(\frac{3}{5}\cos 2t+\frac{1}{5}\sin 2t-\frac{3}{5}\cos 3t-\frac{2}{15}\sin 3t\)
2. \(-\frac{4}{15}\cos t+\frac{1}{15}\sin t+\frac{4}{15}\cos 4t-\frac{1}{60}\sin 4t\)
3. \(\frac{5}{3}\cos t+\sin t-\frac{5}{3}\cos 2t-\frac{1}{2}\sin 2t\)
4. \(-\frac{1}{3}\cos \frac{t}{2}+\frac{2}{3}\sin\frac{t}{2}+\frac{1}{3}\cos t-\frac{1}{3}\sin t\)
5. \(\frac{1}{15}\cos\frac{t}{4}-\frac{8}{15}\sin\frac{t}{4}-\frac{1}{15}\cos 4t+\frac{1}{30}\sin 4t\)
6. \(\frac{2}{5}\cos\frac{t}{3}-\frac{3}{5}\sin\frac{t}{3}-\frac{2}{5}\cos\frac{t}{2}+\frac{2}{5}\sin\frac{t}{2}\)

6.

1. \(e^{t}(\cos 2t+\sin 2t)-e^{-t}\left( \cos 3t+\frac{4}{3}\sin 3t\right)\)
2. \(e^{3t}\left(-\cos 2t+\frac{3}{2}\sin 2t\right)+e^{-t}\left(\cos 2t+\frac{1}{2}\sin 2t\right)\)
3. \(e^{-2t}\left(\frac{1}{8}\cos t+\frac{1}{4}\sin t\right)-e^{2t}\left(\frac{1}{8}\cos 3t-\frac{1}{12}\sin 3t\right)\)
4. \(e^{2t}\left(\cos t+\frac{1}{2}\sin t\right)-e^{3t}\left(\cos 2t-\frac{1}{4}\sin 2t\right)\)
5. \(e^{t}\left(\frac{1}{5}\cos t+\frac{2}{5}\sin t\right)-e^{-t}\left(\frac{1}{5}\cos 2t+\frac{2}{5}\sin 2t\right)\)
6. \(e^{t/2}\left(-\cos t+\frac{9}{8}\sin t\right)+e^{-t/2}\left(\cos t-\frac{1}{8}\sin t\right)\)

7.

1. \(1-\cos t\)
2. \(\frac{e^{t}}{16}(1-\cos 4t)\)
3. \(\frac{4}{9}e^{2t}+\frac{5}{9}e^{-t}\sin 3t-\frac{4}{9}e^{-t}\cos 3t\)
4. \(3e^{t/2}-\frac{7}{2}e^{t}\sin 2t-3e^{t}\cos 2t\)
5. \(\frac{1}{4}e^{3t}-\frac{1}{4}e^{-t}\cos 2t\)
6. \(\frac{1}{9}e^{2t}-\frac{1}{9}e^{-t}\cos 3t+\frac{5}{9}e^{-t}\sin 3t\)

8.

1. \(-\frac{3}{10}\sin t+\frac{2}{5}\cos t-\frac{3}{4}e^{t}+\frac{7}{20}e^{3t}\)
2. \(-\frac{3}{5}e^{-t}\sin t+\frac{1}{5}e^{-t}\cos t-\frac{1}{2}e^{-t}+\frac{3}{10}e^{t}\)
3. \(-\frac{1}{10}e^{t}\sin t-\frac{7}{10}e^{t}\cos t+\frac{1}{5}e^{-t}+\frac{1}{2}e^{2t}\)
4. \(-\frac{1}{2}e^{t}+\frac{7}{10}e^{-t}-\frac{1}{5}\cos 2t+\frac{3}{5}\sin 2t\)
5. \(\frac{3}{10}+\frac{1}{10}e^{2t}+\frac{1}{10}e^{t}\sin 2t-\frac{2}{5}e^{t}\cos 2t\)
6. \(-\frac{4}{9}e^{2t}\cos 3t+\frac{1}{3}e^{2t}\sin 3t-\frac{5}{9}e^{2t}+e^{t}\)

9. \(\frac{1}{a}e^{\frac{b}{a}t}f\left(\frac{t}{a}\right)\)