# 11.13: A.12.2- Section 12.2 Answers

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

2. $$u(x,t)=\frac{8}{\pi ^{3}} \sum_{n=1}^{\infty}\frac{1}{(2n-1)^{3}}\cos 3(2n-1)\pi t\sin (2n-1)\pi x$$

3. $$u(x,t)=-\frac{4}{\pi ^{3}} \sum_{n=1}^{\infty}\frac{(1+(-1)^{n}2)}{n^{3}}\cos n\sqrt{7}\pi t\sin n\pi x$$

4. $$u(x,t)=\frac{8}{3\pi ^{4}} \sum_{n=1}^{\infty}\frac{1}{(2n-1)^{4}}\sin 3(2n-1)\pi t\sin (2n-1)\pi x$$

5. $$u(x,t)=-\frac{4}{\sqrt{7}\pi ^{4}} \sum_{n=1}^{\infty}\frac{(1+(-1)^{n}2)}{n^{4}}\sin n\sqrt{7}\pi t\sin n\pi x$$

6. $$u(x,t)=\frac{324}{\pi ^{3}} \sum_{n=1}^{\infty}\frac{(-1)^{n}}{n^{3}}\cos\frac{8n\pi t}{3}\sin\frac{n\pi x}{3}$$

7. $$u(x,t)=\frac{96}{\pi ^{5}} \sum_{n=1}^{\infty}\frac{1}{(2n-1)^{5}}\cos 2(2n-1)\pi t\sin (2n-1)\pi x$$

8. $$u(x,t)=\frac{243}{2\pi ^{4}} \sum_{n=1}^{\infty}\frac{(-1)^{n}}{n^{4}}\sin\frac{8n\pi t}{3}\sin\frac{n\pi x}{3}$$

9. $$u(x,t)=\frac{48}{\pi ^{6}} \sum_{n=1}^{\infty}\frac{1}{(2n-1)^{6}}\sin 2(2n-1)\pi t\sin (2n-1)\pi x$$

10. $$u(x,t)=\frac{\pi }{2}\cos\sqrt{5}t\sin x-\frac{16}{\pi} \sum_{n=1}^{\infty}\frac{n}{(4n^{2}-1)^{2}}\cos 2n\sqrt{5}t\sin 2nx$$

11. $$u(x,t)=-\frac{240}{\pi ^{5}} \sum_{n=1}^{\infty}\frac{1+(-1)^{n}2}{n^{5}}\cos n\pi t\sin n\pi x$$

12. $$u(x,t)=\frac{\pi }{2\sqrt{5}}\sin\sqrt{5}t\sin x-\frac{8}{pi\sqrt{5}} \sum_{n=1}^{\infty}\frac{1}{(4n^{2}-1)^{2}}\sin 2n\sqrt{5}t\sin 2nx$$

13. $$u(x,t)=-\frac{240}{\pi ^{6}} \sum_{n=1}^{\infty}\frac{1+(-1)^{n}2}{n^{6}}\sin n\pi t\sin n\pi x$$

14. $$u(x,t)=-\frac{720}{\pi ^{5}} \sum_{n=1}^{\infty}\frac{(-1)^{n}}{n^{5}}\cos 2n\pi t\sin n\pi x$$

15. $$u(x,t)=-\frac{240}{\pi ^{6}} \sum_{n=1}^{\infty}\frac{(-1)^{n}}{n^{6}}\sin 3n\pi t\sin n\pi x$$

18. $$u(x,t)=-\frac{128}{\pi ^{3}} \sum_{n=1}^{\infty}\frac{(-1)^{n}}{(2n-1)^{3}}\cos\frac{3(2n-1)\pi t}{4}\cos \frac{(2n-1)\pi x}{4}$$

19. $$u(x,t)=-\frac{64}{\pi ^{3}} \sum_{n=1}^{\infty}\frac{1}{(2n-1)^{3}}\left[(-1)^{n}+\frac{3}{(2n-1)\pi}\right]\cos (2n-1)\pi t\cos\frac{(2n-1)\pi x}{2}$$

20. $$u(x,t)=-\frac{512}{3\pi ^{4}} \sum_{n=1}^{\infty}\frac{(-1)^{n}}{(2n-1)^{4}}\sin\frac{3(2n-1)\pi t}{4}\cos\frac{(2n-1)\pi x}{4}$$

21. $$u(x,t)=-\frac{64}{\pi ^{4}} \sum_{n=1}^{\infty}\frac{1}{(2n-1)^{4}}\left[(-1)^{n}+\frac{3}{(2n-1)\pi}\right]\sin (2n-1)\pi t\cos\frac{(2n-1)\pi x}{2}$$

22. $$u(x,t)=\frac{96}{\pi ^{3}} \sum_{n=1}^{\infty}\frac{1}{(2n-1)^{3}}\left[(-1)^{n}3+\frac{4}{(2n-1)\pi}\right]\cos\frac{(2n-1)\sqrt{5}\pi t}{2}\cos\frac{(2n-1)\pi x}{2}$$

23. $$u(x,t)=-96\sum_{n=1}^{\infty}\frac{1}{(2n-1)^{3}}\left[(-1)^{n}+\frac{2}{(2n-1)\pi}\right]\cos\frac{(2n-1)\sqrt{3} t}{2}\cos\frac{(2n-1) x}{2}$$

24. $$u(x,t)=\frac{192}{\pi ^{4}\sqrt{5}}\sum_{n=1}^{\infty}\frac{1}{(2n-1)^{4}}\left[(-1)^{n}3+\frac{4}{(2n-1)\pi}\right]\sin\frac{(2n-1)\sqrt{5}\pi t}{2}\cos\frac{(2n-1)\pi x}{2}$$

25. $$u(x,t)=-\frac{192}{\sqrt{3}}\sum_{n=1}^{\infty}\frac{1}{(2n-1)^{4}}\left[(-1)^{n}+\frac{2}{(2n-1)\pi}\right]\sin\frac{(2n-1)\sqrt{3}t}{2}\sin\frac{(2n-1)x}{2}$$

26. $$u(x,t)=-\frac{384}{\pi ^{4}} \sum_{n=1}^{\infty}\frac{1}{(2n-1)^{4}}\left[1+\frac{(-1)^{n}4}{(2n-1)\pi}\right]\cos\frac{3(2n-1)\pi t}{2}\cos\frac{(2n-1)\pi x}{2}$$

27. $$u(x,t)=\frac{96}{\pi ^{3}} \sum_{n=1}^{\infty}\frac{1}{(2n-1)^{3}}\left[(-1)^{n}5+\frac{8}{(2n-1)\pi}\right]\cos\frac{(2n-1)\sqrt{7}\pi t}{2}\cos\frac{(2n-1)\pi x}{2}$$

28. $$u(x,t)=-\frac{768}{3\pi ^{5}} \sum_{n=1}^{\infty}\frac{1}{(2n-1)^{5}}\left[1+\frac{(-1)^{n}4}{(2n-1)\pi}\right]\sin\frac{3(2n-1)\pi t}{2}\cos\frac{(2n-1)\pi x}{2}$$

29. $$u(x,t)=\frac{192}{\pi ^{4}\sqrt{7}} \sum_{n=1}^{\infty}\frac{1}{(2n-1)^{4}}\left[(-1)^{n}5+\frac{8}{(2n-1)\pi}\right]\sin\frac{(2n-1)\sqrt{7}\pi t}{2}\cos\frac{(2n-1)\pi x}{2}$$

30. $$u(x,t)=-\frac{768}{\pi ^{4}} \sum_{n=1}^{\infty}\frac{1}{(2n-1)^{4}}\left[1+\frac{(-1)^{n}2}{(2n-1)\pi}\right]\cos\frac{(2n-1)\pi t}{2}\cos\frac{(2n-1)\pi x}{2}$$

31. $$u(x,t)=-\frac{1536}{\pi ^{5}} \sum_{n=1}^{\infty}\frac{1}{(2n-1)^{5}}\left[1+\frac{(-1)^{n}2}{(2n-1)\pi}\right]\sin\frac{(2n-1)\pi t}{2}\cos\frac{(2n-1)\pi x}{2}$$

32. $$u(x,t)=\frac{1}{2}\left[C_{Mf}(x+at)+C_{Mf}(x-at)\right]+\frac{1}{2a} \int_{x-at} ^{x+at} C_{Mg}(\tau )d\tau$$

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

36. $$u(x,t)=-\frac{96}{\pi ^{3}} \sum_{n=1}^{\infty}\frac{1}{(2n-1)^{3}}\left[ 1+(-1)^{n}\frac{4}{(2n-1)\pi}\right]\cos\frac{3(2n-1)\pi t}{2}\sin\frac{(2n-1)\pi x}{2}$$

37. $$u(x,t)=\frac{8}{\pi} \sum_{n=1}^{\infty}\frac{1}{(2n-1)^{4}}\sin 4(2n-1)t\sin\frac{(2n-1)x}{2}$$

38. $$u(x,t)=-\frac{64}{\pi ^{4}} \sum_{n=1}^{\infty}\frac{1}{(2n-1)^{4}}\left[1+(-1)^{n}\frac{4}{(2n-1)\pi}\right]\sin\frac{3(2n-1)\pi t}{2}\sin\frac{(2n-1)\pi x}{2}$$

39. $$u(x,t)=\frac{96}{\pi ^{3}} \sum_{n=1}^{\infty}\frac{1}{(2n-1)^{3}}\left[1+(-1)^{n}\frac{2}{(2n-1)\pi}\right]\cos\frac{3(2n-1)\pi t}{2}\sin\frac{(2n-1)\pi x}{2}$$

40. $$u(x,t)=\frac{192}{\pi} \sum_{n=1}^{\infty}\frac{(-1)^{n}}{(2n-1)^{4}}\cos\frac{(2n-1)\sqrt{3}t}{2}\sin\frac{(2n-1)x}{2}$$

41. $$u(x,t)=\frac{64}{\pi ^{4}} \sum_{n=1}^{\infty} \frac{1}{(2n-1)^{4}}\left[1+(-1)^{n}\frac{2}{(2n-1)\pi}\right]\sin\frac{3(2n-1)\pi t}{2}\sin\frac{(2n-1)\pi x}{2}$$

42. $$u(x,t)=\frac{384}{\sqrt{3}\pi} \sum_{n=1}^{\infty}\frac{(-1)^{n}}{(2n-1)^{5}}\sin\frac{(2n-1)\sqrt{3}t}{2}\sin\frac{(2n-1)x}{2}$$

43. $$u(x,t)=\frac{1536}{\pi^{4}} \sum_{n=1}^{\infty}\frac{1}{(2n-1)^{4}}\left[(-1)^{n}+\frac{3}{(2n-1)\pi}\right]\cos\frac{(2n-1)\sqrt{5}\pi t}{2}\sin\frac{(2n-1)\pi x}{2}$$

44. $$u(x,t)=\frac{384}{\pi ^{4}} \sum_{n=1}^{\infty}\frac{1}{(2n-1)^{4}}\left[(-1)^{n}+\frac{4}{(2n-1)\pi}\right]\cos (2n-1)\pi t\sin\frac{(2n-1)\pi x}{2}$$

45. $$u(x,t)=\frac{3072}{\sqrt{5}\pi ^{5}} \sum_{n=1}^{\infty}\frac{1}{(2n-1)^{5}}\left[(-1)^{n}+\frac{3}{(2n-1)\pi}\right]\sin\frac{(2n-1)\sqrt{5}\pi t}{2}\sin\frac{(2n-1)\pi x}{2}$$

46. $$u(x,t)=\frac{384}{\pi ^{5}} \sum_{n=1}^{\infty}\frac{1}{(2n-1)^{5}}\left[(-1)^{n}+\frac{4}{(2n-1)\pi}\right]\sin (2n-1)\pi t\sin \frac{(2n-1)\pi x}{2}$$

47. $$u(x,t)=\frac{1}{2}[S_{Mf}(x+at)+S_{Mf}(x-at)]+\frac{1}{2a} \int_{x-at} ^{x+at} S_{Mg}(\tau )d\tau$$

50. $$u(x,t)=4-\frac{768}{\pi^{4}} \sum_{n=1}^{\infty}\frac{1}{(2n-1)^{4}}\cos\frac{\sqrt{5}(2n-1)\pi t}{2}\cos\frac{(2n-1)\pi x}{2}$$

51. $$u(x,t)=4t-\frac{1536}{\sqrt{5}\pi ^{5}} \sum_{n=1}^{\infty}\frac{1}{(2n-1)^{5}}\sin\frac{\sqrt{5}(2n-1)\pi t}{2}\cos\frac{(2n-1)\pi x}{2}$$

52. $$u(x,t)=-\frac{2\pi ^{4}}{5}-48 \sum_{n=1}^{\infty}\frac{1+(-1)^{n}2}{n^{4}}\cos 2nt\cos nx$$

53. $$u(x,t)=-\frac{7}{5}-\frac{144}{\pi ^{4}} \sum_{n=1}^{\infty}\frac{(-1)^{n}}{n^{4}}\cos n\sqrt{7}\pi t\cos n\pi x$$

54. $$u(x,t)=-\frac{2\pi ^{4}t}{5}-24\sum_{n=1}^{\infty}\frac{1+(-1)^{n}2}{n^{5}}\sin 2nt\cos nx$$

55. $$u(x,t)=-\frac{7t}{5}-\frac{144}{\pi ^{5}\sqrt{7}}\sum_{n=1}^{\infty}\frac{(-1)^{n}}{n^{5}}\sin n\sqrt{7}\pi t\cos n\pi x$$

56. $$u(x,t)=\frac{\pi ^{4}}{30}-3\sum_{n=1}^{\infty}\frac{1}{n^{4}}\cos 8nt\cos 2nx$$

57. $$u(x,t)=\frac{3}{5}-\frac{48}{\pi ^{4}}\sum_{n=1}^{\infty}\frac{2+(-1)^{n}}{n^{4}}\cos n\pi t\cos n\pi x$$

58. $$u(x,t)=\frac{\pi ^{4}t}{30}-\frac{3}{8}\sum_{n=1}^{\infty}\frac{1}{n^{5}}\sin 8nt\cos 2nx$$

59. $$u(x,t)=\frac{3t}{5}-\frac{48}{\pi ^{5}}\sum_{n=1}^{\infty}\frac{2+(-1)^{n}}{n^{5}}\sin n\pi t\cos n\pi x$$

60. $$u(x,t)=\frac{1}{2}\left[ C_{f}(x+at)+C_{f}(x-at)\right] +\frac{1}{2a} \int _{x-at}^{x+at} C_{g}(\tau )d\tau$$

63. c. $$u(x,t)=\frac{f(x+at)+f(x-at)}{2}+\frac{1}{2}\int_{x-at}^{x+at} g(u)du$$

64. $$u(x,t)=x(1+4at$$

65. $$u(x,t)=x^{2}+a^{2}t^{2}+t$$

66. $$u(x,t)=\sin (x+at)$$

67. $$u(x,t)=x^{3}+6tx^{2}+3a^{2}t^{2}x+2a^{2}t^{3}$$

68. $$u(x,t)=x\sin x\cos at+at\cos x\sin at+\frac{\sin x\sin at}{a}$$

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