A.12.1: Section 12.1 Answers
- Page ID
- 43745
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)8. \(u(x,t)=\frac{8}{\pi ^{3}}\sum_{n=1}^{\infty}\frac{1}{(2n-1)^{3}}e^{-(2n-1)^{2}\pi ^{2}t}\sin (2n-1)\pi x\)
9. \(u(x,t)=\frac{4}{\pi}\sum_{n=1}^{\infty}\frac{1}{(2n-1)}e^{-9(2n-1)^{2}\pi ^{2}t/16}\sin\frac{(2n-1)\pi x}{4}\)
10. \(u(x,t)=\frac{\pi}{2}e^{-3t}\sin x-\frac{16}{\pi} \sum_{n=1}^{\infty}\frac{n}{(4n^{2}-1)}e^{-12n^{2}t}\sin 2nx\)
11. \(u(x,t)=-\frac{32}{\pi ^{3}} \sum_{n=1}^{\infty}\frac{(1(-1)^{n}2)}{n^{3}}e^{-9n^{2}\pi ^{2}t/4}\sin\frac{n\pi x}{2}\)
12. \(u(x,t)=-\frac{324}{\pi ^{3}} \sum_{n=1}^{\infty}\frac{(-1)^{n}}{n^{3}}e^{-4n^{2}\pi ^{2}t/9}\sin\frac{n\pi x}{3}\)
13. \(u(x,t)=\frac{8}{\pi ^{2}} \sum_{n=1}^{\infty}\frac{(-1)^{n+1}}{(2n-1)^{2}}e^{-(2n-1)^{2}\pi ^{2}t}\sin\frac{(2n-1)\pi x}{2}\)
14. \(u(x,t)=-\frac{720}{\pi ^{5}}=-\frac{720}{\pi ^{5}} \sum_{n=1}^{\infty}\frac{(-1)^{n}}{n^{5}}e^{-7n^{2}\pi ^{2}t}\sin n\pi x\)
15. \(u(x,t)=\frac{96}{\pi ^{5}} \sum_{n=1}^{\infty}\frac{1}{(2n-1)^{5}}e^{-5(2n-1)^{2}\pi ^{2}t}\sin (2n-1)\pi x\)
16. \(u(x,t)=-\frac{240}{\pi ^{5}} \sum_{n=1}^{\infty}\frac{1+(-1)^{n}2}{n^{5}}e^{-2n^{2}\pi ^{2}t}\sin n\pi x\)
17. \(u(x,t)=\frac{16}{3}+\frac{64}{\pi ^{2}} \sum_{n=1}^{\infty}\frac{(-1)^{n}}{n^{2}}e^{-9\pi ^{2}n^{2}t/16}\cos\frac{n\pi x}{4}\)
18. \(u(x,t)=-\frac{8}{3}+\frac{16}{\pi ^{2}} \sum_{n=1}^{\infty}\frac{1}{n^{2}}e^{-n^{2}\pi ^{2}t}\cos\frac{n\pi x}{2}\)
19. \(u(x,t)=\frac{1}{6}-\frac{1}{\pi ^{2}} \sum_{n=1}^{\infty}\frac{1}{n^{2}}e^{-36n^{2}\pi ^{2}t}\cos 2n\pi x\)
20. \(u(x,t)=4-\frac{384}{\pi ^{4}} \sum_{n=1}^{\infty}\frac{1}{(2n-1)^{4}}e^{-3(2n-1)^{2}\pi ^{2}t/4}\cos\frac{(2n-1)\pi x}{2}\)
21. \(u(x,t)=-\frac{28}{5}-\frac{576}{\pi ^{4}} \sum_{n=1}^{\infty}\frac{(-1)^{n}}{n^{4}}e^{-5n^{2}\pi ^{2}t/2}\cos\frac{n\pi x}{\sqrt{2}}\)
22. \(u(x,t)=-\frac{2}{5}-\frac{48}{\pi ^{4}} \sum_{n=1}^{\infty}\frac{1+(-1)^{n}2}{n^{4}}e^{-3n^{2}\pi ^{2}t}\cos n\pi x\)
23. \(u(x,t)=\frac{3}{5}-\frac{48}{\pi ^{4}} \sum_{n=1}^{\infty}\frac{2+(-1)^{n}}{n^{4}}e^{-n^{2}\pi ^{2}t}\cos n\pi x\)
24. \(u(x,t)=\frac{\pi ^{4}}{30}-3 \sum_{n=1}^{\infty}\frac{1}{n^{4}}e^{-4n^{2}t}\cos 2nx\)
25. \(u(x,t)=\frac{8}{\pi} \sum_{n=1}^{\infty}\frac{(-1)^{n}}{(2n+1)(2n-3)}e^{-(2n-1)^{2}\pi ^{2}t/4}\sin\frac{ (2n-1)\pi x}{2}\)
26. \(u(x,t)=8 \sum_{n=1}^{\infty}\frac{1}{(2n-1)^{2}}\left[(-1)^{n}+\frac{4}{(2n-1)\pi }\right]e^{-3(2n-1)^{2}t/4}\sin\frac{(2n-1)x}{2}\)
27. \(u(x,t)=\frac{128}{\pi ^{3}} \sum_{n=1}^{\infty}\frac{1}{(2n-1)^{3}}e^{-5(2n-1)^{2}t/16}\sin\frac{(2n-1)\pi x}{4}\)
28. \(u(x,t)=-\frac{96}{\pi ^{3}} \sum_{n=1}^{\infty}\frac{1}{(2n-1)^{3}}\left[1+(-1)^{n}\frac{4}{(2n-1)\pi}\right]e^{-(2n-1)^{2}\pi ^{2}t/4}\sin\frac{(2n-1)\pi x}{2}\)
29. \(u(x,t)=\frac{96}{\pi ^{3}} \sum_{n=1}^{\infty}\frac{1}{(2n-1)^{3}}\left[1+(-1)^{n}\frac{2}{(2n-1)\pi}\right]e^{-(2n-1)^{2}\pi ^{2}t/4}\sin\frac{(2n-1)\pi x}{2}\)
30. \(u(x,t)=\frac{192}{\pi ^{4}}\sum_{n=1}^{\infty}\frac{(-1)^{n}}{(2n-1)^{4}}e^{-(2n-1)^{2}\pi ^{2}t/4}\sin\frac{(2n-1)\pi x}{2}\)
31. \(u(x,t)=\frac{1536}{\pi ^{4}}\sum_{n=1}^{\infty}\frac{1}{(2n-1)^{4}}\left[(-1)^{n}+\frac{3}{(2n-1)\pi}\right]e^{-(2n-1)^{2}\pi ^{2}t/4}\sin\frac{(2n-1)\pi x}{2}\)
32. \(u(x,t)=\frac{384}{\pi ^{4}}\sum_{n=1}^{\infty}\frac{1}{(2n-1)^{4}}\left[(-1)^{n}+\frac{4}{(2n-1)\pi}\right]e^{-(2n-1)^{2}\pi ^{2}t/4}\sin\frac{(2n-1)\pi x}{2}\)
33. \(u(x,t)=-64\sum_{n=1}^{\infty}\frac{e^{-3(2n-1)^{2}t/4}}{(2n-1)^{3}}\left[(-1)^{n}+\frac{3}{(2n-1)\pi}\right]\cos\frac{(2n-1)x}{2}\)
34. \(u(x,t)=-\frac{16}{\pi}\sum_{n=1}^{\infty}\frac{(-1)^{n}}{2n-1}e^{-(2n-1)^{2}t}\cos\frac{(2n-1)x}{4}\)
35. \(u(x,t)=-\frac{64}{\pi}\sum_{n=1}^{\infty}\frac{(-1)^{n}}{2n-1}\left[1-\frac{8}{(2n-1)^{2}\pi ^{2}}\right]e^{-9(2n-1)^{2}\pi ^{2}t/64}\cos\frac{(2n-1)\pi x}{8}\)
36. \(u(x,t)=\frac{8}{\pi ^{2}}\sum_{n=1}^{\infty}\frac{1}{(2n-1)^{2}}e^{-3(2n-1)^{2}\pi ^{2}t/4}\cos\frac{(2n-1)\pi x}{2}\)
37. \(u(x,t)=-\frac{96}{\pi ^{3}} \sum_{n=1}^{\infty}\frac{1}{(2n-1)^{3}}\left[(-1)^{n}+\frac{2}{(2n-1)\pi}\right]e^{-(2n-1)^{2}\pi ^{2}t/4}\cos\frac{(2n-1)\pi x}{2}\)
38. \(u(x,t)=-\frac{32}{\pi} \sum_{n=1}^{\infty}\frac{(-1)^{n}}{(2n-1)^{3}}e^{-7(2n-1)^{2}t/4}\cos\frac{(2n-1)x}{2}\)
39. \(u(x,t)=\frac{96}{\pi ^{3}} \sum_{n=1}^{\infty}\frac{1}{(2n-1)^{3}}\left[(-1)^{n}5+\frac{8}{(2n-1)\pi}\right]e^{-(2n-1)^{2}\pi ^{2}t/4}\cos\frac{(2n-1)\pi x}{2}\)
40. \(u(x,t)=\frac{96}{\pi ^{3}} \sum_{n=1}^{\infty}\frac{1}{(2n-1)^{3}}\left[(-1)^{n}3+\frac{4}{(2n-1)\pi}\right]e^{-(2n-1)^{2}\pi ^{2}t/4}\cos\frac{(2n-1)\pi x}{2}\)
41. \(u(x,t)=-\frac{768}{\pi ^{4}}\sum_{n=1}^{\infty}\frac{1}{(2n-1)^{4}}\left[1+\frac{(-1)^{n}2}{(2n-1)\pi}\right]e^{-(2n-1)^{2}\pi ^{2}t/4}\cos\frac{(2n-1)\pi x}{2}\)
42. \(u(x,t)=-\frac{384}{\pi ^{4}}\sum_{n=1}^{\infty}\frac{1}{(2n-1)^{4}}\left[1+\frac{(-1)^{n}4}{(2n-1)\pi}\right]e^{-(2n-1)^{2}\pi ^{2}t/4}\cos\frac{(2n-1)\pi x}{2}\)
43. \(u(x,t)=\frac{1}{2}-\frac{2}{\pi}\sum_{n=1}^{\infty}\frac{(-1)^{n}}{2n-1}e^{-(2n-1)^{2}\pi ^{2}a^{2}t/L^{2}}\cos\frac{(2n-1)\pi x}{L}\)
44. \(u(x,t)=\frac{2}{\pi}\sum_{n=1}^{\infty}\frac{1}{n}\left[1-\cos\frac{n\pi}{2}\right]e^{-n^{2}\pi ^{2}a^{2}t/L^{2}}\sin\frac{n\pi x}{L}\)
45. \(u(x,t)=\frac{4}{\pi} \sum_{n=1}^{\infty}\frac{1}{2n-1}\sin\frac{(2n-1)\pi}{4}e^{-(2n-1)^{2}\pi ^{2}a^{2}t/4L^{2}}\cos\frac{(2n-1)\pi x}{2L}\)
46. \(u(x,t)=\frac{4}{\pi} \sum_{n=1}^{\infty}\frac{1}{2n-1}\left[1-\cos\frac{(2n-1)\pi}{4}\right]e^{-(2n-1)^{2}\pi ^{2}a^{2}t/4L^{2}}\sin\frac{(2n-1)\pi x}{2L}\)
48. \(u(x,t)=1-x+x^{3}+\frac{4}{\pi} \sum_{n=1}^{\infty}\frac{e^{}-9\pi ^{2}(2n-1)^{2}t/16}{(2n-1)}\sin\frac{(2n-1)\pi x}{4}\)
49. \(u(x,t)=1+x+x^{2}-\frac{8}{\pi ^{3}} \sum_{n=1}^{\infty}\frac{e^{-(2n-1)^{2}\pi ^{2}t}}{(2n-1)^{3}}\sin (2n-1)\pi x\)
50. \(u(x,t)=-1-x+x^{3}+\frac{8}{\pi ^{2}} \sum_{n=1}^{\infty}\frac{1}{(2n-1)^{2}}e^{-3(2n-1)^{2}\pi ^{2}t/4}\cos\frac{(2n-1)\pi x}{2}\)
51. \(u(x,t)=x^{2}-x-2-\frac{64}{\pi} \sum_{n=1}^{\infty}\frac{(-1)^{n}}{2n-1}\left[1-\frac{8}{(2n-1)^{2}\pi ^{2}}\right]e^{-9(2n-1)^{2}\pi ^{2}t/64}\cos\frac{(2n-1)\pi x}{8}\)
52. \(u(x,t)=\sin\pi x+\frac{8}{\pi } \sum_{n=1}^{\infty}\frac{(-1)^{n}}{(2n+1)(2n-3)}e^{-(2n-1)^{2}\pi ^{2}t/4}\sin\frac{(2n-1)\pi x}{2}\)
53. \(u(x,t)=x^{3}-x+3+\frac{32}{\pi ^{3}} \sum_{n=1}^{\infty}\frac{e^{-(2n-1)^{2}\pi ^{2}t/4}}{(2n-1)^{3}}\sin\frac{(2n-1)\pi x}{2}\)