A.7.6: Section 7.6 Answers

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1. \(y_{1} = x\left( 1 − x + \frac{3}{4}x^{2} − \frac{13}{36}x^{3} +\ldots\right) ;\quad y_{2} = y_{1} \ln x + x^{2}\left( 1 − x + \frac{65}{108} x^{2} +\ldots\right)\)

2. \(y_{1} = x^{−1}\left( 1 − 2x + \frac{9}{2} x^{2} − \frac{20}{3} x^{3} +\ldots\right) ;\quad y_{2} = y_{1} \ln x + 1 − \frac{15}{4} x + \frac{133}{ 18} x^{2} +\ldots \)

3. \(y_{1} = 1 + x − x^{2} + \frac{1}{3} x^{3} + \dots ;\quad y_{2} = y_{1} \ln x − x\left( 3 − \frac{1}{2} x − \frac{31}{18} x^{2} +\ldots\right)\)

4. \(y_{1} = x^{1/2}\left( 1 − 2x + \frac{5}{2} x^{2} − 2x^{3} +\ldots\right) ;\quad y_{2} = y_{1} \ln x + x^{3/2}\left( 1 − \frac{9}{4} x + \frac{17}{6} x^{2} +\ldots\right)\)

5. \(y_{1} = x\left( 1 − 4x + \frac{19}{2} x^{2} − \frac{49}{3} x^{3} +\ldots\right) ;\quad y_{2} = y_{1} \ln x + x^{2}\left( 3 − \frac{43}{4} x + \frac{208}{9} x^{2} +\ldots\right)\)

6. \(y_{1} = x^{−1/3}\left( 1 − x + \frac{5}{6} x^{2} − \frac{1}{2} x^{3} +\ldots\right) ;\quad y_{2} = y_{1} \ln x + x^{2/3}\left( 1 − \frac{11}{12} x + \frac{25}{36} x^{2} +\ldots\right)\)

7. \(y_{1} = 1 − 2x + \frac{7}{4} x^{2} − \frac{7}{9} x^{3} +\ldots ;\quad y_{2} = y_{1} \ln x + x\left( 3 − \frac{15}{4} x + \frac{239}{108} x^{2} +\ldots\right)\)

8. \(y_{1} = x^{−2}\left( 1 − 2x + \frac{5}{2} x^{2} − 3x^{3} +\ldots\right) ;\quad y_{2} = y_{1} \ln x + \frac{3}{4} − \frac{13}{6} x +\ldots \)

9. \(y_{1} = x^{−1/2}\left( 1 − x + \frac{1}{4} x^{2} + \frac{1}{18} x^{3} +\ldots\right) ;\quad y_{2} = y_{1} \ln x + x^{1/2}\left( \frac{3}{2} − \frac{13}{16} x + \frac{1}{54} x^{2} +\ldots\right)\)

10. \(y_{1} = x^{−1/4}\left( 1 − \frac{1}{4} x − \frac{7}{32} x^{2} + \frac{23}{384}x^{3} +\ldots\right) ;\quad y_{2} = y_{1} \ln x + x^{3/4}\left( \frac{1}{4} + \frac{5}{64}x − \frac{157}{2304}x^{2} +\ldots\right)\)

11. \(y_{1} = x^{−1/3}\left( 1 − x + \frac{7}{6} x^{2} −\frac{23}{18} x^{3} +\ldots\right) ;\quad y_{2} = y_{1} \ln x − x^{5/3}\left( \frac{1}{12} − \frac{13}{108} x\ldots\right)\)

12. \(y_{1}=x^{1/2}\sum_{n=0}^{\infty}\frac{(-1)^{n}}{(n!)^{2}}x^{n};\quad y_{2}=y_{1}\ln x-2x^{1/2}\sum_{n=1}^{\infty}\frac{(-1)^{n}}{(n!)^{2}}\left(\sum_{j=1}^{n}\frac{1}{j} \right)x^{n}\)

13. \(y_{1}=x^{1/6}\sum_{n=0}^{\infty}\left(\frac{2}{3}\right)^{n}\frac{\prod_{j=1}^{n}(3j+1)}{n!}x^{n};\quad y_{2}=y_{1}\ln x-x^{1/6}\sum_{n=1}^{\infty}\left(\frac{2}{3}\right)^{n}\frac{\prod_{j=1}^{n}(3j+1)}{n!}\left(\sum_{j=1}^{n}\frac{1}{j(3j+1)} \right)x^{n}\)

14. \(y_{1}=x^{2}\sum_{n=0}^{\infty}(-1)^{n}(n+1)^{2}x^{n};\quad y_{2}=y_{1}\ln x-2x^{2}\sum_{n=1}^{\infty}(-1)^{n}n(n+1)x^{n}\)

15. \(y_{1}=x^{3}\sum_{n=0}^{\infty}2^{n}(n+1)x^{n};\quad y_{2}=y_{1}\ln x-x^{3}\sum_{n=1}^{\infty}2^{n}nx^{n}\)

16. \(y_{1}=x^{1/5}\sum_{n=0}^{\infty}\frac{(-1)^{n}\prod_{j=1}^{n}(5j+1)}{125^{n}(n!)^{2}}x^{n};\quad y_{2}=y_{1}\ln x-x^{1/5}\sum_{n=1}^{\infty}\frac{(-1)^{n}\prod_{j=1}^{n}(5j+1)}{125^{n}(n!)^{2}}\left(\sum_{j=1}^{n}\frac{5j+2}{j(5j+1)} \right)x^{n}\)

17. \(y_{1}=x^{1/2}\sum_{n=0}^{\infty}\frac{(-1)^{n}\prod_{j=1}^{n}(2j-3)}{4^{n}n!}x^{n};\quad y_{2}=y_{1}\ln x+3x^{1/2}\sum_{n=1}^{\infty}\frac{(-1)^{n}\prod_{j=1}^{n}(2j-3)}{4^{n}n!}\left(\sum_{j=1}^{n}\frac{1}{j(2j-3)} \right)x^{n}\)

18. \(y_{1}=x^{1/3}\sum_{n=0}^{\infty}\frac{(-1)^{n}\prod_{j=1}^{n}(6j-7)^{2}}{81^{n}(n!)^{2}}x^{n};\quad y_{2}=y_{1}\ln x+14x^{1/3}\sum_{n=1}^{\infty}\frac{(-1)^{n}\prod_{j=1}^{n}(6j-7)^{2}}{81^{n}(n!)^{2}}\left(\sum_{j=1}^{n}\frac{1}{j(6j-7)} \right)x^{n}\)

19. \(y_{1}=x^{2}\sum_{n=0}^{\infty}\frac{(-1)^{n}\prod_{j=1}^{n}(2j+5)}{(n!)^{2}}x^{n};\quad y_{2}=y_{1}\ln x-2x^{2}\sum_{n=1}^{\infty}\frac{(-1)^{n}\prod_{j=1}^{n}(2j+5)}{(n!)^{2}}\left(\sum_{j=1}^{n}\frac{j+5}{j(2j+5)} \right)x^{n}\)

20. \(y_{1}=\frac{1}{x}\sum_{n=0}^{\infty}\frac{(2)^{n}\prod_{j=1}^{n}(2j-1)}{n!}x^{n};\quad y_{2}=y_{1}\ln x+\frac{1}{x}\sum_{n=1}^{\infty}\frac{(2)^{n}\prod_{j=1}^{n}(2j-1)}{n!}\left(\sum_{j=1}^{n}\frac{1}{j(2j-1)} \right)x^{n}\)

21. \(y_{1}=\frac{1}{x}\sum_{n=0}^{\infty}\frac{(-1)^{n}\prod_{j=1}^{n}(2j-5)}{n!}x^{n};\quad y_{2}=y_{1}\ln x+\frac{5}{x}\sum_{n=1}^{\infty}\frac{(-1)^{n}\prod_{j=1}^{n}(2j-5)}{n!}\left(\sum_{j=1}^{n}\frac{1}{j(2j-5)} \right)x^{n}\)

22. \(y_{1}=x^{2}\sum_{n=0}^{\infty}\frac{(-1)^{n}\prod_{j=1}^{n}(2j+3)}{2^{n}n!}x^{n};\quad y_{2}=y_{1}\ln x-3x^{2}\sum_{n=1}^{\infty}\frac{(-1)^{n}\prod_{j=1}^{n}(2j+3)}{2^{n}n!}\left(\sum_{j=1}^{n}\frac{1}{j(2j+3)} \right)x^{n}\)

23. \(y_{1} = x^{−2}\left( 1 + 3x + \frac{3}{2}^{2} −\frac{1}{2}x^{3} +\ldots\right) ;\quad y_{2} = y_{1} \ln x − 5x^{−1}\left( 1 + \frac{5}{4} x − \frac{1}{4} x^{2} +\ldots\right)\)

24. \(y_{1} = x^{3} (1 + 20x + 180x^{2} + 1120x^{3} +\ldots );\quad y_{2} = y_{1} \ln x − x^{4}\left( 26 + 324x + 6968 3 x^{2} +\ldots\right)\)

25. \(y_{1} = x\left( 1 − 5x + \frac{85}{4}x^{2} − \frac{3145}{36} x^{3} +\ldots\right) ;\quad y_{2} = y_{1} \ln x + x^{2}\left( 2 − \frac{39}{4} x + \frac{4499}{108}x^{2} +\ldots\right)\)

26. \(y_{1} = 1 − x + \frac{3}{4}x^{2} − \frac{7}{12}x^{3} +\ldots ;\quad y_{2} = y_{1} \ln x + x\left( 1 − \frac{3}{4} x + \frac{5}{9} x^{2} +\ldots\right)\)

27. \(y_{1} = x^{−3} (1 + 16x + 36x^{2} + 16x^{3} +\ldots );\quad y_{2} = y_{1} \ln x − x^{−2}\left( 40 + 150x + \frac{280}{3}x^{2} +\ldots\right)\)

28. \(y_{1}=x\sum_{m=0}^{\infty}\frac{(-1)^{m}}{2^{m}m!}x^{2m};\quad y_{2}=y_{1}\ln x-\frac{x}{2}\sum_{m=1}^{\infty}\frac{(-1)^{m}}{2^{m}m!}\left(\sum_{j=1}^{\infty}\frac{1}{j} \right)x^{2m}\)

29. \(y_{1}=x^{2}\sum_{m=0}^{\infty}(-1)^{m}(m+1)x^{2m};\quad y_{2}=y_{1}\ln x-\frac{x^{2}}{2}\sum_{m=1}^{\infty}(-1)^{m}mx^{2m}\)

30. \(y_{1}=x^{1/2}\sum_{m=0}^{\infty}\frac{(-1)^{m}}{4^{m}m!}x^{2m};\quad y_{2}=y_{1}\ln x-\frac{x^{1/2}}{2}\sum_{m=1}^{\infty}\frac{(-1)^{m}}{4^{m}m!}\left(\sum_{j=1}^{m}\frac{1}{j} \right)x^{2m}\)

31. \(y_{1}=x\sum_{m=0}^{\infty}\frac{(-1)^{m}\prod_{j=1}^{m}(2j-1)}{2^{m}m!}x^{2m};\quad y_{2}=y_{1}\ln x+\frac{x}{2}\sum_{m=1}^{\infty}\frac{(-1)^{m}\prod_{j=1}^{m}(2j-1)}{2^{m}m!}\left(\sum_{j=1}^{m}\frac{1}{j(2j-1)} \right)x^{2m}\)

32. \(y_{1}=x^{1/2}\sum_{m=0}^{\infty}\frac{(-1)^{m}\prod_{j=1}^{m}(4j-1)}{8^{m}m!}x^{2m};\quad y_{2}=y_{1}\ln x+\frac{x^{1/2}}{2}\sum_{m=1}^{\infty}\frac{(-1)^{m}\prod_{j=1}^{m}(4j-1)}{8^{m}m!}\left(\sum_{j=1}^{m}\frac{1}{j(4j-1)} \right)x^{2m}\)

33. \(y_{1}=x\sum_{m=0}^{\infty}\frac{(-1)^{m}\prod_{j=1}^{m}(2j+1)}{2^{m}m!}x^{2m};\quad y_{2}=y_{1}\ln x-\frac{x}{2}\sum_{m=1}^{\infty}\frac{(-1)^{m}\prod_{j=1}^{m}(2j+1)}{2^{m}m!}\left(\sum_{j=1}^{m}\frac{1}{j(2j+1)} \right)x^{2m}\)

34. \(y_{1}=x^{-1/4}\sum_{m=0}^{\infty}\frac{(-1)^{m}\prod_{j=1}^{m}(8j-13)}{(32)^{m}m!}x^{2m};\quad y_{2}=y_{1}\ln x+\frac{13}{2}x^{-1/4}\sum_{m=1}^{\infty}\frac{(-1)^{m}\prod_{j=1}^{m}(8j-13)}{(32)^{m}m!}\left(\sum_{j=1}^{m}\frac{1}{j(8j-13)} \right)x^{2m}\)

35. \(y_{1}=x^{1/3}\sum_{m=0}^{\infty}\frac{(-1)^{m}\prod_{j=1}^{m}(3j-1)}{9^{m}m!}x^{2m};\quad y_{2}=y_{1}\ln x+\frac{x^{1/3}}{2}\sum_{m=1}^{\infty}\frac{(-1)^{m}\prod_{j=1}^{m}(3j-1)}{9^{m}m!}\left(\sum_{j=1}^{m}\frac{1}{j(3j-1)} \right)x^{2m}\)

36. \(y_{1}=x^{1/2}\sum_{m=0}^{\infty}\frac{(-1)^{m}\prod_{j=1}^{m}(4j-3)(4j-1)}{4^{m}(m!)^{2}}x^{2m};\quad y_{2}=y_{1}\ln x+x^{1/2}\sum_{m=1}^{\infty}\frac{(-1)^{m}\prod_{j=1}^{m}(4j-3)(4j-1)}{4^{m}(m!)^{2}}\left(\sum_{j=1}^{m}\frac{8j-3}{j(4j-3)(4j-1)} \right)x^{2m}\)

37. \(y_{1}=x^{5/3}\sum_{m=0}^{\infty}\frac{(-1)^{m}}{3^{m}m!}x^{2m};\quad y_{2}=y_{1}\ln x-\frac{x^{5/3}}{2}\sum_{m=1}^{\infty}\frac{(-1)^{m}}{3^{m}m!}\left(\sum_{j=1}^{m}\frac{1}{j} \right)x^{2m}\)

38. \(y_{1}=\frac{1}{x}\sum_{m=0}^{\infty}\frac{(-1)^{m}\prod_{j=1}^{m}(4j-7)}{2^{m}m!}x^{2m};\quad y_{2}=y_{1}\ln x+\frac{7}{2x}\sum_{m=1}^{\infty}\frac{(-1)^{m}\prod_{j=1}^{m}(4j-7)}{2^{m}m!}\left(\sum_{j=1}^{m}\frac{1}{j(4j-7)} \right)x^{2m}\)

39. \(y_{1} = x^{−1}\left( 1 −\frac{3}{2}x^{2} + \frac{15}{8}x^{4} −\frac{35}{16}x^{6} +\ldots\right) ;\quad y_{2} = y_{1} \ln x + x\left(\frac{1}{4} − \frac{13}{32}x^{2} + \frac{101}{192}x^{4} +\ldots\right)\)

40. \(y_{1} = x\left( 1 −\frac{1}{2}x^{2} +\frac{1}{8}x^{4} −\frac{1}{48}x^{6}+\ldots\right);\quad y_{2} = y_{1} \ln x + x^{3}\left(\frac{1}{4} − \frac{3}{32}x^{2}+ \frac{11}{576}x^{4} +\ldots\right)\)

41. \(y_{1} = x^{−2}\left(1 −\frac{3}{4}x^{2} −\frac{9}{64}x^{4} −\frac{25}{256}x^{6} +\ldots\right) ;\quad y_{2} = y_{1} \ln x + \frac{1}{2} − \frac{21}{128}x^{2} − \frac{215}{1536}x^{4} +\ldots \)

42. \(y_{1} = x^{−3}\left( 1 −\frac{17}{8}x^{2} +\frac{85}{256}x^{4} − \frac{85}{18432}x^{6} +\ldots\right) ;\quad y_{2} = y_{1} \ln x + x^{−1}\left( \frac{25}{8} − \frac{471}{512} x^{2} + \frac{1583}{110592} x^{4} +\ldots\right)\)

43. \(y_{1} = x^{−1}\left( 1 −\frac{3}{4}x^{2} + \frac{45}{64}x^{4} − \frac{175}{256}x^{6} +\ldots\right) ;\quad y_{2} = y_{1} \ln x − x\left( \frac{1}{4} − \frac{33}{128} x^{2} + \frac{395}{1536}x^{4} +\ldots\right)\)

44. \(y_{1} = \frac{1}{x} ;\quad y_{2} = y_{1} \ln x − 6 + 6x − \frac{8}{3} x^{2}\)

45. \(y_{1}=1-x;\quad y_{2}=y_{1}\ln x+4x\)

46. \(y_{1}=\frac{(x-1)^{2}}{x};\quad y_{2}=y_{1}\ln x+3-3x+2\sum_{n=2}^{\infty}\frac{1}{n(n^{2}-1)}x^{n}\)

47. \(y_{1}=x^{1/2}(x+1)^{2};\quad y_{2}=y_{1}\ln x-x^{3/2}\left(3+3x+2\sum_{n=2}^{\infty}\frac{(-1)^{n}}{n(n^{2}-1)}x^{n} \right)\)

48. \(y_{1}=x^{2}(1-x)^{3};\quad y_{2}=y_{1}\ln x+x^{3}\left(4-7x+\frac{11}{3}x^{2}-6\sum_{n=3}^{\infty}\frac{1}{n(n-2)(n^{2}-1)}x^{n} \right)\)

49. \(y_{1}=x-4x^{3}+x^{5};\quad y_{2}=y_{1}\ln x+6x^{3}-3x^{5}\)

50. \(y_{1}=x^{1/3}\left(1-\frac{1}{6}x^{2}\right) ;\quad y_{2}=y_{1}\ln x+x^{7/3}\left(\frac{1}{4}-\frac{1}{12}\sum_{m=1}^{\infty}\frac{1}{6^{m}m(m+1)(m+1)!}x^{2m} \right)\)

51. \(y_{1}=(1+x^{2})^{2};\quad y_{2}=y_{1}\ln x-\frac{3}{2}x^{2}-\frac{3}{2}x^{4}+\sum_{m=3}^{\infty}\frac{(-1)^{m}}{m(m-1)(m-2)}x^{2m}\)

52. \(y_{1}=x^{-1/2}\left(1-\frac{1}{2}x^{2}+\frac{1}{32}x^{4}\right);\quad y_{2}=y_{1}\ln x+x^{3/2}\left(\frac{5}{8}-\frac{9}{128}x^{2}+\sum_{m=2}^{\infty}\frac{1}{4^{m+1}(m-1)m(m+1)(m+1)!}x^{2m} \right)\)

56. \(y_{1}=\sum_{m=0}^{\infty}\frac{(-1)^{m}}{4^{m}(m!)^{2}}x^{2m};\quad y_{2}=y_{1}\ln x-\sum_{m=1}^{\infty}\frac{(-1)^{m}}{4^{m}(m!)^{2}}\left(\sum_{j=1}^{m}\frac{1}{j} \right)x^{2m}\)

58. \(\frac{x^{1/2}}{1+x};\quad \frac{x^{1/2}\ln x}{1+x}\)

59. \(\frac{x^{1/3}}{3+x};\quad \frac{x^{1/3}\ln x}{3+x}\)

60. \(\frac{x}{2-x^{2}};\quad \frac{x\ln x}{2-x^{2}}\)

61. \(\frac{x^{1/3}}{1+x^{2}};\quad \frac{x^{1/4}\ln x}{1+x^{2}}\)

62. \(\frac{x}{4+3x};\quad \frac{x\ln x}{4+3x}\)

63. \(\frac{x^{1/2}}{1+3x+x^{2}};\quad \frac{x^{1/2}\ln x}{1+3x+x^{2}}\)

64. \(\frac{x}{(1-x)^{2}};\quad \frac{x\ln x}{(1-x)^{2}}\)

65. \(\frac{x^{1/3}}{1+x+x^{2}};\quad \frac{x^{1/3}\ln x}{1+x+x^{2}}\)