A.12.1: Section 12.1 Answers

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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. \(u(x,t)=x^{3}-x+3+\frac{32}{\pi ^{3}} \sum_{n=1}^{\infty}\frac{e^{-(2n-1)^{2}\pi ^{2}t/4}}{(2n-1)^{3}}\sin\frac{(2n-1)\pi x}{2}\)