11.13: A.12.2- Section 12.2 Answers
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( \newcommand{\kernel}{\mathrm{null}\,}\)
1. u(x,t)=43π3∑∞n=1(−1)n+1(2n−1)3sin3(2n−1)πtsin(2n−1)πx
2. u(x,t)=8π3∑∞n=11(2n−1)3cos3(2n−1)πtsin(2n−1)πx
3. u(x,t)=−4π3∑∞n=1(1+(−1)n2)n3cosn√7πtsinnπx
4. u(x,t)=83π4∑∞n=11(2n−1)4sin3(2n−1)πtsin(2n−1)πx
5. u(x,t)=−4√7π4∑∞n=1(1+(−1)n2)n4sinn√7πtsinnπx
6. u(x,t)=324π3∑∞n=1(−1)nn3cos8nπt3sinnπx3
7. u(x,t)=96π5∑∞n=11(2n−1)5cos2(2n−1)πtsin(2n−1)πx
8. u(x,t)=2432π4∑∞n=1(−1)nn4sin8nπt3sinnπx3
9. u(x,t)=48π6∑∞n=11(2n−1)6sin2(2n−1)πtsin(2n−1)πx
10. u(x,t)=π2cos√5tsinx−16π∑∞n=1n(4n2−1)2cos2n√5tsin2nx
11. u(x,t)=−240π5∑∞n=11+(−1)n2n5cosnπtsinnπx
12. u(x,t)=π2√5sin√5tsinx−8pi√5∑∞n=11(4n2−1)2sin2n√5tsin2nx
13. u(x,t)=−240π6∑∞n=11+(−1)n2n6sinnπtsinnπx
14. u(x,t)=−720π5∑∞n=1(−1)nn5cos2nπtsinnπx
15. u(x,t)=−240π6∑∞n=1(−1)nn6sin3nπtsinnπx
18. u(x,t)=−128π3∑∞n=1(−1)n(2n−1)3cos3(2n−1)πt4cos(2n−1)πx4
19. u(x,t)=−64π3∑∞n=11(2n−1)3[(−1)n+3(2n−1)π]cos(2n−1)πtcos(2n−1)πx2
20. u(x,t)=−5123π4∑∞n=1(−1)n(2n−1)4sin3(2n−1)πt4cos(2n−1)πx4
21. u(x,t)=−64π4∑∞n=11(2n−1)4[(−1)n+3(2n−1)π]sin(2n−1)πtcos(2n−1)πx2
22. u(x,t)=96π3∑∞n=11(2n−1)3[(−1)n3+4(2n−1)π]cos(2n−1)√5πt2cos(2n−1)πx2
23. u(x,t)=−96∑∞n=11(2n−1)3[(−1)n+2(2n−1)π]cos(2n−1)√3t2cos(2n−1)x2
24. u(x,t)=192π4√5∑∞n=11(2n−1)4[(−1)n3+4(2n−1)π]sin(2n−1)√5πt2cos(2n−1)πx2
25. u(x,t)=−192√3∑∞n=11(2n−1)4[(−1)n+2(2n−1)π]sin(2n−1)√3t2sin(2n−1)x2
26. u(x,t)=−384π4∑∞n=11(2n−1)4[1+(−1)n4(2n−1)π]cos3(2n−1)πt2cos(2n−1)πx2
27. u(x,t)=96π3∑∞n=11(2n−1)3[(−1)n5+8(2n−1)π]cos(2n−1)√7πt2cos(2n−1)πx2
28. u(x,t)=−7683π5∑∞n=11(2n−1)5[1+(−1)n4(2n−1)π]sin3(2n−1)πt2cos(2n−1)πx2
29. u(x,t)=192π4√7∑∞n=11(2n−1)4[(−1)n5+8(2n−1)π]sin(2n−1)√7πt2cos(2n−1)πx2
30. u(x,t)=−768π4∑∞n=11(2n−1)4[1+(−1)n2(2n−1)π]cos(2n−1)πt2cos(2n−1)πx2
31. u(x,t)=−1536π5∑∞n=11(2n−1)5[1+(−1)n2(2n−1)π]sin(2n−1)πt2cos(2n−1)πx2
32. u(x,t)=12[CMf(x+at)+CMf(x−at)]+12a∫x+atx−atCMg(τ)dτ
35. u(x,t)=32π∑∞n=11(2n−1)3cos4(2n−1)tsin(2n−1)x2
36. u(x,t)=−96π3∑∞n=11(2n−1)3[1+(−1)n4(2n−1)π]cos3(2n−1)πt2sin(2n−1)πx2
37. u(x,t)=8π∑∞n=11(2n−1)4sin4(2n−1)tsin(2n−1)x2
38. u(x,t)=−64π4∑∞n=11(2n−1)4[1+(−1)n4(2n−1)π]sin3(2n−1)πt2sin(2n−1)πx2
39. u(x,t)=96π3∑∞n=11(2n−1)3[1+(−1)n2(2n−1)π]cos3(2n−1)πt2sin(2n−1)πx2
40. u(x,t)=192π∑∞n=1(−1)n(2n−1)4cos(2n−1)√3t2sin(2n−1)x2
41. u(x,t)=64π4∑∞n=11(2n−1)4[1+(−1)n2(2n−1)π]sin3(2n−1)πt2sin(2n−1)πx2
42. u(x,t)=384√3π∑∞n=1(−1)n(2n−1)5sin(2n−1)√3t2sin(2n−1)x2
43. u(x,t)=1536π4∑∞n=11(2n−1)4[(−1)n+3(2n−1)π]cos(2n−1)√5πt2sin(2n−1)πx2
44. u(x,t)=384π4∑∞n=11(2n−1)4[(−1)n+4(2n−1)π]cos(2n−1)πtsin(2n−1)πx2
45. u(x,t)=3072√5π5∑∞n=11(2n−1)5[(−1)n+3(2n−1)π]sin(2n−1)√5πt2sin(2n−1)πx2
46. u(x,t)=384π5∑∞n=11(2n−1)5[(−1)n+4(2n−1)π]sin(2n−1)πtsin(2n−1)πx2
47. u(x,t)=12[SMf(x+at)+SMf(x−at)]+12a∫x+atx−atSMg(τ)dτ
50. u(x,t)=4−768π4∑∞n=11(2n−1)4cos√5(2n−1)πt2cos(2n−1)πx2
51. u(x,t)=4t−1536√5π5∑∞n=11(2n−1)5sin√5(2n−1)πt2cos(2n−1)πx2
52. u(x,t)=−2π45−48∑∞n=11+(−1)n2n4cos2ntcosnx
53. u(x,t)=−75−144π4∑∞n=1(−1)nn4cosn√7πtcosnπx
54. u(x,t)=−2π4t5−24∑∞n=11+(−1)n2n5sin2ntcosnx
55. u(x,t)=−7t5−144π5√7∑∞n=1(−1)nn5sinn√7πtcosnπx
56. u(x,t)=π430−3∑∞n=11n4cos8ntcos2nx
57. u(x,t)=35−48π4∑∞n=12+(−1)nn4cosnπtcosnπx
58. u(x,t)=π4t30−38∑∞n=11n5sin8ntcos2nx
59. u(x,t)=3t5−48π5∑∞n=12+(−1)nn5sinnπtcosnπx
60. u(x,t)=12[Cf(x+at)+Cf(x−at)]+12a∫x+atx−atCg(τ)dτ
63. c. u(x,t)=f(x+at)+f(x−at)2+12∫x+atx−atg(u)du
64. u(x,t)=x(1+4at
65. u(x,t)=x2+a2t2+t
66. u(x,t)=sin(x+at)
67. u(x,t)=x3+6tx2+3a2t2x+2a2t3
68. u(x,t)=xsinxcosat+atcosxsinat+sinxsinata