A.7.6: Section 7.6 Answers
( \newcommand{\kernel}{\mathrm{null}\,}\)
1. y1=x(1−x+34x2−1336x3+…);y2=y1lnx+x2(1−x+65108x2+…)
2. y1=x−1(1−2x+92x2−203x3+…);y2=y1lnx+1−154x+13318x2+…
3. y1=1+x−x2+13x3+…;y2=y1lnx−x(3−12x−3118x2+…)
4. y1=x1/2(1−2x+52x2−2x3+…);y2=y1lnx+x3/2(1−94x+176x2+…)
5. y1=x(1−4x+192x2−493x3+…);y2=y1lnx+x2(3−434x+2089x2+…)
6. y1=x−1/3(1−x+56x2−12x3+…);y2=y1lnx+x2/3(1−1112x+2536x2+…)
7. y1=1−2x+74x2−79x3+…;y2=y1lnx+x(3−154x+239108x2+…)
8. y1=x−2(1−2x+52x2−3x3+…);y2=y1lnx+34−136x+…
9. y1=x−1/2(1−x+14x2+118x3+…);y2=y1lnx+x1/2(32−1316x+154x2+…)
10. y1=x−1/4(1−14x−732x2+23384x3+…);y2=y1lnx+x3/4(14+564x−1572304x2+…)
11. y1=x−1/3(1−x+76x2−2318x3+…);y2=y1lnx−x5/3(112−13108x…)
12. y1=x1/2∑∞n=0(−1)n(n!)2xn;y2=y1lnx−2x1/2∑∞n=1(−1)n(n!)2(∑nj=11j)xn
13. y1=x1/6∑∞n=0(23)n∏nj=1(3j+1)n!xn;y2=y1lnx−x1/6∑∞n=1(23)n∏nj=1(3j+1)n!(∑nj=11j(3j+1))xn
14. y1=x2∑∞n=0(−1)n(n+1)2xn;y2=y1lnx−2x2∑∞n=1(−1)nn(n+1)xn
15. y1=x3∑∞n=02n(n+1)xn;y2=y1lnx−x3∑∞n=12nnxn
16. y1=x1/5∑∞n=0(−1)n∏nj=1(5j+1)125n(n!)2xn;y2=y1lnx−x1/5∑∞n=1(−1)n∏nj=1(5j+1)125n(n!)2(∑nj=15j+2j(5j+1))xn
17. y1=x1/2∑∞n=0(−1)n∏nj=1(2j−3)4nn!xn;y2=y1lnx+3x1/2∑∞n=1(−1)n∏nj=1(2j−3)4nn!(∑nj=11j(2j−3))xn
18. y1=x1/3∑∞n=0(−1)n∏nj=1(6j−7)281n(n!)2xn;y2=y1lnx+14x1/3∑∞n=1(−1)n∏nj=1(6j−7)281n(n!)2(∑nj=11j(6j−7))xn
19. y1=x2∑∞n=0(−1)n∏nj=1(2j+5)(n!)2xn;y2=y1lnx−2x2∑∞n=1(−1)n∏nj=1(2j+5)(n!)2(∑nj=1j+5j(2j+5))xn
20. y1=1x∑∞n=0(2)n∏nj=1(2j−1)n!xn;y2=y1lnx+1x∑∞n=1(2)n∏nj=1(2j−1)n!(∑nj=11j(2j−1))xn
21. y1=1x∑∞n=0(−1)n∏nj=1(2j−5)n!xn;y2=y1lnx+5x∑∞n=1(−1)n∏nj=1(2j−5)n!(∑nj=11j(2j−5))xn
22. y1=x2∑∞n=0(−1)n∏nj=1(2j+3)2nn!xn;y2=y1lnx−3x2∑∞n=1(−1)n∏nj=1(2j+3)2nn!(∑nj=11j(2j+3))xn
23. y1=x−2(1+3x+322−12x3+…);y2=y1lnx−5x−1(1+54x−14x2+…)
24. y1=x3(1+20x+180x2+1120x3+…);y2=y1lnx−x4(26+324x+69683x2+…)
25. y1=x(1−5x+854x2−314536x3+…);y2=y1lnx+x2(2−394x+4499108x2+…)
26. y1=1−x+34x2−712x3+…;y2=y1lnx+x(1−34x+59x2+…)
27. y1=x−3(1+16x+36x2+16x3+…);y2=y1lnx−x−2(40+150x+2803x2+…)
28. y1=x∑∞m=0(−1)m2mm!x2m;y2=y1lnx−x2∑∞m=1(−1)m2mm!(∑∞j=11j)x2m
29. y1=x2∑∞m=0(−1)m(m+1)x2m;y2=y1lnx−x22∑∞m=1(−1)mmx2m
30. y1=x1/2∑∞m=0(−1)m4mm!x2m;y2=y1lnx−x1/22∑∞m=1(−1)m4mm!(∑mj=11j)x2m
31. y1=x∑∞m=0(−1)m∏mj=1(2j−1)2mm!x2m;y2=y1lnx+x2∑∞m=1(−1)m∏mj=1(2j−1)2mm!(∑mj=11j(2j−1))x2m
32. y1=x1/2∑∞m=0(−1)m∏mj=1(4j−1)8mm!x2m;y2=y1lnx+x1/22∑∞m=1(−1)m∏mj=1(4j−1)8mm!(∑mj=11j(4j−1))x2m
33. y1=x∑∞m=0(−1)m∏mj=1(2j+1)2mm!x2m;y2=y1lnx−x2∑∞m=1(−1)m∏mj=1(2j+1)2mm!(∑mj=11j(2j+1))x2m
34. y1=x−1/4∑∞m=0(−1)m∏mj=1(8j−13)(32)mm!x2m;y2=y1lnx+132x−1/4∑∞m=1(−1)m∏mj=1(8j−13)(32)mm!(∑mj=11j(8j−13))x2m
35. y1=x1/3∑∞m=0(−1)m∏mj=1(3j−1)9mm!x2m;y2=y1lnx+x1/32∑∞m=1(−1)m∏mj=1(3j−1)9mm!(∑mj=11j(3j−1))x2m
36. y1=x1/2∑∞m=0(−1)m∏mj=1(4j−3)(4j−1)4m(m!)2x2m;y2=y1lnx+x1/2∑∞m=1(−1)m∏mj=1(4j−3)(4j−1)4m(m!)2(∑mj=18j−3j(4j−3)(4j−1))x2m
37. y1=x5/3∑∞m=0(−1)m3mm!x2m;y2=y1lnx−x5/32∑∞m=1(−1)m3mm!(∑mj=11j)x2m
38. y1=1x∑∞m=0(−1)m∏mj=1(4j−7)2mm!x2m;y2=y1lnx+72x∑∞m=1(−1)m∏mj=1(4j−7)2mm!(∑mj=11j(4j−7))x2m
39. y1=x−1(1−32x2+158x4−3516x6+…);y2=y1lnx+x(14−1332x2+101192x4+…)
40. y1=x(1−12x2+18x4−148x6+…);y2=y1lnx+x3(14−332x2+11576x4+…)
41. y1=x−2(1−34x2−964x4−25256x6+…);y2=y1lnx+12−21128x2−2151536x4+…
42. y1=x−3(1−178x2+85256x4−8518432x6+…);y2=y1lnx+x−1(258−471512x2+1583110592x4+…)
43. y1=x−1(1−34x2+4564x4−175256x6+…);y2=y1lnx−x(14−33128x2+3951536x4+…)
44. y1=1x;y2=y1lnx−6+6x−83x2
45. y1=1−x;y2=y1lnx+4x
46. y1=(x−1)2x;y2=y1lnx+3−3x+2∑∞n=21n(n2−1)xn
47. y1=x1/2(x+1)2;y2=y1lnx−x3/2(3+3x+2∑∞n=2(−1)nn(n2−1)xn)
48. y1=x2(1−x)3;y2=y1lnx+x3(4−7x+113x2−6∑∞n=31n(n−2)(n2−1)xn)
49. y1=x−4x3+x5;y2=y1lnx+6x3−3x5
50. y1=x1/3(1−16x2);y2=y1lnx+x7/3(14−112∑∞m=116mm(m+1)(m+1)!x2m)
51. y1=(1+x2)2;y2=y1lnx−32x2−32x4+∑∞m=3(−1)mm(m−1)(m−2)x2m
52. y1=x−1/2(1−12x2+132x4);y2=y1lnx+x3/2(58−9128x2+∑∞m=214m+1(m−1)m(m+1)(m+1)!x2m)
56. y1=∑∞m=0(−1)m4m(m!)2x2m;y2=y1lnx−∑∞m=1(−1)m4m(m!)2(∑mj=11j)x2m
58. x1/21+x;x1/2lnx1+x
59. x1/33+x;x1/3lnx3+x
60. x2−x2;xlnx2−x2
61. x1/31+x2;x1/4lnx1+x2
62. x4+3x;xlnx4+3x
63. x1/21+3x+x2;x1/2lnx1+3x+x2
64. x(1−x)2;xlnx(1−x)2
65. x1/31+x+x2;x1/3lnx1+x+x2