11.12: A.12.1- Section 12.1 Answers
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8. u(x,t)=8π3∑∞n=11(2n−1)3e−(2n−1)2π2tsin(2n−1)πx
9. u(x,t)=4π∑∞n=11(2n−1)e−9(2n−1)2π2t/16sin(2n−1)πx4
10. u(x,t)=π2e−3tsinx−16π∑∞n=1n(4n2−1)e−12n2tsin2nx
11. u(x,t)=−32π3∑∞n=1(1(−1)n2)n3e−9n2π2t/4sinnπx2
12. u(x,t)=−324π3∑∞n=1(−1)nn3e−4n2π2t/9sinnπx3
13. u(x,t)=8π2∑∞n=1(−1)n+1(2n−1)2e−(2n−1)2π2tsin(2n−1)πx2
14. u(x,t)=−720π5=−720π5∑∞n=1(−1)nn5e−7n2π2tsinnπx
15. u(x,t)=96π5∑∞n=11(2n−1)5e−5(2n−1)2π2tsin(2n−1)πx
16. u(x,t)=−240π5∑∞n=11+(−1)n2n5e−2n2π2tsinnπx
17. u(x,t)=163+64π2∑∞n=1(−1)nn2e−9π2n2t/16cosnπx4
18. u(x,t)=−83+16π2∑∞n=11n2e−n2π2tcosnπx2
19. u(x,t)=16−1π2∑∞n=11n2e−36n2π2tcos2nπx
20. u(x,t)=4−384π4∑∞n=11(2n−1)4e−3(2n−1)2π2t/4cos(2n−1)πx2
21. u(x,t)=−285−576π4∑∞n=1(−1)nn4e−5n2π2t/2cosnπx√2
22. u(x,t)=−25−48π4∑∞n=11+(−1)n2n4e−3n2π2tcosnπx
23. u(x,t)=35−48π4∑∞n=12+(−1)nn4e−n2π2tcosnπx
24. u(x,t)=π430−3∑∞n=11n4e−4n2tcos2nx
25. u(x,t)=8π∑∞n=1(−1)n(2n+1)(2n−3)e−(2n−1)2π2t/4sin(2n−1)πx2
26. u(x,t)=8∑∞n=11(2n−1)2[(−1)n+4(2n−1)π]e−3(2n−1)2t/4sin(2n−1)x2
27. u(x,t)=128π3∑∞n=11(2n−1)3e−5(2n−1)2t/16sin(2n−1)πx4
28. u(x,t)=−96π3∑∞n=11(2n−1)3[1+(−1)n4(2n−1)π]e−(2n−1)2π2t/4sin(2n−1)πx2
29. u(x,t)=96π3∑∞n=11(2n−1)3[1+(−1)n2(2n−1)π]e−(2n−1)2π2t/4sin(2n−1)πx2
30. u(x,t)=192π4∑∞n=1(−1)n(2n−1)4e−(2n−1)2π2t/4sin(2n−1)πx2
31. u(x,t)=1536π4∑∞n=11(2n−1)4[(−1)n+3(2n−1)π]e−(2n−1)2π2t/4sin(2n−1)πx2
32. u(x,t)=384π4∑∞n=11(2n−1)4[(−1)n+4(2n−1)π]e−(2n−1)2π2t/4sin(2n−1)πx2
33. u(x,t)=−64∑∞n=1e−3(2n−1)2t/4(2n−1)3[(−1)n+3(2n−1)π]cos(2n−1)x2
34. u(x,t)=−16π∑∞n=1(−1)n2n−1e−(2n−1)2tcos(2n−1)x4
35. u(x,t)=−64π∑∞n=1(−1)n2n−1[1−8(2n−1)2π2]e−9(2n−1)2π2t/64cos(2n−1)πx8
36. u(x,t)=8π2∑∞n=11(2n−1)2e−3(2n−1)2π2t/4cos(2n−1)πx2
37. u(x,t)=−96π3∑∞n=11(2n−1)3[(−1)n+2(2n−1)π]e−(2n−1)2π2t/4cos(2n−1)πx2
38. u(x,t)=−32π∑∞n=1(−1)n(2n−1)3e−7(2n−1)2t/4cos(2n−1)x2
39. u(x,t)=96π3∑∞n=11(2n−1)3[(−1)n5+8(2n−1)π]e−(2n−1)2π2t/4cos(2n−1)πx2
40. u(x,t)=96π3∑∞n=11(2n−1)3[(−1)n3+4(2n−1)π]e−(2n−1)2π2t/4cos(2n−1)πx2
41. u(x,t)=−768π4∑∞n=11(2n−1)4[1+(−1)n2(2n−1)π]e−(2n−1)2π2t/4cos(2n−1)πx2
42. u(x,t)=−384π4∑∞n=11(2n−1)4[1+(−1)n4(2n−1)π]e−(2n−1)2π2t/4cos(2n−1)πx2
43. u(x,t)=12−2π∑∞n=1(−1)n2n−1e−(2n−1)2π2a2t/L2cos(2n−1)πxL
44. u(x,t)=2π∑∞n=11n[1−cosnπ2]e−n2π2a2t/L2sinnπxL
45. u(x,t)=4π∑∞n=112n−1sin(2n−1)π4e−(2n−1)2π2a2t/4L2cos(2n−1)πx2L
46. u(x,t)=4π∑∞n=112n−1[1−cos(2n−1)π4]e−(2n−1)2π2a2t/4L2sin(2n−1)πx2L
48. u(x,t)=1−x+x3+4π∑∞n=1e−9π2(2n−1)2t/16(2n−1)sin(2n−1)πx4
49. u(x,t)=1+x+x2−8π3∑∞n=1e−(2n−1)2π2t(2n−1)3sin(2n−1)πx
50. u(x,t)=−1−x+x3+8π2∑∞n=11(2n−1)2e−3(2n−1)2π2t/4cos(2n−1)πx2
51. u(x,t)=x2−x−2−64π∑∞n=1(−1)n2n−1[1−8(2n−1)2π2]e−9(2n−1)2π2t/64cos(2n−1)πx8
52. u(x,t)=sinπx+8π∑∞n=1(−1)n(2n+1)(2n−3)e−(2n−1)2π2t/4sin(2n−1)πx2
53. u(x,t)=x3−x+3+32π3∑∞n=1e−(2n−1)2π2t/4(2n−1)3sin(2n−1)πx2