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# Section 7.5 Answers

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1. $$y_{1}=x^{1/2}\left(1-\frac{1}{5}x-\frac{2}{35}x^{2}+\frac{31}{315}x^{3}+\ldots \right) ;\quad y_{2}=x^{-1}\left(1+x+\frac{1}{2}x^{2}-\frac{1}{6}x^{3}+\ldots \right)$$

2. $$y_{1} = x^{1/3}\left( 1 −\frac{2}{3} x + \frac{8}{9} x^{2} − \frac{40}{81} x^{3} +\ldots\right) ;\quad y_{2} = 1 − x + \frac{6}{5} x^{2} − \frac{4}{5} x^{3} +\ldots$$

3. $$y_{1} = x^{1/3}\left( 1 − \frac{4}{7} x − \frac{7}{45} x^{2} + \frac{970}{2457}x^{3} +\ldots\right) ;\quad y_{2} = x^{−1}\left( 1 − x^{2} + \frac{2}{3} x^{3} +\ldots\right)$$

4. $$y_{1} = x^{1/4}\left( 1 − \frac{1}{2} x − \frac{19}{104} x^{2} + \frac{1571}{10608} x^{3} +\ldots\right) ;\quad y_{2} = x^{−1}\left( 1 + 2x − \frac{11}{6} x^{2} − \frac{1}{7} x^{3} + \ldots\right)$$

5. $$y_{1} = x^{1/3}\left( 1 − x + \frac{28}{31} x^{2} − \frac{1111}{1333} x^{3} + \ldots\right) ;\quad y_{2} = x^{−1/4}\left( 1 − x + \frac{7}{8} x^{2} − \frac{19}{24} x^{3} + \ldots\right)$$

6. $$y_{1} = x^{1/5}\left( 1 − \frac{6}{25} x − \frac{1217}{625} x^{2} + \frac{41972}{46875} x^{3} +\ldots\right) ;\quad y_{2} = x − \frac{1}{4} x^{2} − \frac{35}{18} x^{3} + \frac{11}{12} x^{4} +\ldots$$

7. $$y_{1} = x^{3/2}\left( 1 − x + \frac{11}{26} x^{2} − \frac{109}{1326} x^{3} + \ldots\right) ;\quad y_{2} = x^{1/4}\left( 1 + 4x − \frac{131}{24} x^{2} + \frac{39}{14}x^{3} +\ldots\right)$$

8. $$y_{1} = x^{1/3}\left( 1 − \frac{1}{3} x + \frac{2}{15} x^{2} − \frac{5}{63} x^{3} +\ldots\right) ;\quad y_{2} = x^{−1/6}\left( 1 − \frac{1}{12} x^{2} + \frac{1}{18} x^{3} +\ldots\right)$$

9. $$y_{1} = 1 − \frac{1}{14}x^{2} + \frac{1}{105} x^{3} +\ldots ;\quad y_{2} = x^{−1/3}\left( 1 − \frac{1}{18} x − \frac{71}{405}x^{2} + \frac{719}{34992} x^{3} +\ldots\right)$$

10. $$y_{1} = x^{1/5}\left( 1 + \frac{3}{17} x − \frac{7}{153} x^{2} − \frac{547}{5661} x^{3} +\ldots\right) ;\quad y_{2} = x^{−1/2}\left( 1 + x + \frac{14}{13} x^{2} − \frac{556}{897} x^{3} +\ldots\right)$$

14. $$y_{1}=x^{1/2}\sum_{n=0}^{\infty}\frac{(-2)^{n}}{\prod_{j=1}^{n}(2j+3)}x^{n};\quad y_{2}=x^{-1}\sum_{n=0}^{\infty}\frac{(-1)^{n}}{n!}x^{n}$$

15. $$y_{1}=x^{1/3}\sum_{n=0}^{\infty}\frac{(-1)^{n}\prod_{j=1}^{n}(3j+1)}{9^{n}n!}x^{n};\quad x^{-1}$$

16. $$y_{1}=x^{1/2}\sum_{n=0}^{\infty}\frac{(-1)^{n}}{\prod_{j=1}^{n}(3j+4)}x^{n};\quad y_{2}=\frac{1}{x^{2}}\sum_{n=0}^{\infty}\frac{2^{n}}{n!\prod_{j=1}^{n}(2j-5)}x^{n}$$

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

18. $$y_{1}=x\sum_{n=0}^{\infty}\frac{2^{n}}{n!\prod_{j=1}^{n}(2j+1)}x^{n};\quad y_{2}=x^{1/2}\sum_{n=0}^{\infty}\frac{2^{n}}{n!\prod_{j=1}^{n}(2j-1)}x^{n}$$

19. $$y_{1}=x^{1/3}\sum_{n=0}^{\infty}\frac{1}{n!\prod_{j=1}^{n}(3j+2)}x^{n};\quad y_{2}=x^{-1/3}\sum_{n=0}^{\infty}\frac{1}{n!\prod_{j=1}^{n}(3j-2)}x^{n}$$

20. $$y_{1}=x\left(1+\frac{2}{7}x+\frac{1}{70}x^{2}\right) ;\quad y_{2}=x^{-1/3}\sum_{n=0}^{\infty}\frac{(-1)^{n}}{3^{n}n!}\left(\prod_{j=1}^{n}\frac{3j-13}{3j-4} \right)x^{n}$$

21. $$y_{1}=x^{1/2}\sum_{n=0}^{\infty}(-1)^{n}\left(\prod_{j=1}^{n}\frac{2j+1}{6j+1} \right) x^{n};\quad y_{2}=x^{1/3}\sum_{n=0}^{\infty}\frac{(-1)^{n}}{9^{n}n!}\left(\prod_{j=1}^{n}(3j+1) \right)x^{n}$$

22. $$y_{1}=x\sum_{n=0}^{\infty}\frac{(-1)^{n}(n+2)!}{2\prod_{j=1}^{n}(4j+3)}x^{n};\quad y_{2}=x^{1/4}\sum_{n=0}^{\infty}\frac{(-1)^{n}}{16^{n}n!}\prod_{j=1}^{n}(4j+5)x^{n}$$

23. $$y_{1}=x^{-1/2}\sum_{n=0}^{\infty}\frac{(1)^{n}}{n!\prod_{j=1}^{n}(2j+1)}x^{n};\quad y_{2}=x^{-1}\sum_{n=0}^{\infty}\frac{(1)^{n}}{n!\prod_{j=1}^{n}(2j-1)}x^{n}$$

24. $$y_{1}=x^{1/3}\sum_{n=0}^{\infty}\frac{(-1)^{n}}{n!}\left(\frac{2}{9}\right)^{n}\left(\prod_{j=1}^{n}(6j+5) \right)x^{n};\quad y_{2}=x^{-1}\sum_{n=0}^{\infty}(-1)^{n}2^{n}\left(\prod_{j=1}^{n}\frac{2j-1}{3j-4} \right)x^{n}$$

25. $$y_{1}=4x^{1/3}\sum_{n=0}^{\infty}\frac{1}{6^{n}n!(3n+4)}x^{n};\quad x^{-1}$$

28. $$y_{1} = x^{1/2}\left( 1 − \frac{9}{40} x + \frac{5}{128} x^{2} − \frac{245}{39936} x^{3} +\ldots\right) ;\quad y_{2} = x^{1/4}\left( 1 −\frac{25}{96} x + \frac{675}{14336} x^{2} − \frac{38025}{5046272} x^{3} +\ldots\right)$$

29. $$y_{1} = x^{1/3}\left( 1 + \frac{32}{117} x − \frac{28}{1053} x^{2} + \frac{4480}{540189} x^{3} +\ldots\right) ;\quad y_{2} = x^{−3}\left( 1 + \frac{32}{7} x + \frac{48}{7} x^{2}\right)$$

30.$$y_{1} = x^{1/2}\left( 1 − \frac{5}{8} x + \frac{55}{96} x^{2} − \frac{935}{1536} x^{3} +\ldots\right) ;\quad y_{2} = x^{−1/2}\left( 1 + \frac{1}{4} x − \frac{5}{32} x^{2} −\frac{55}{384} x^{3} +\ldots\right)$$

31. $$y_{1} = x^{1/2} \left( 1 − \frac{3}{4} x + \frac{5}{96} x^{2} + \frac{5}{4224}x ^{3} +\ldots\right) ;\quad y_{2} = x^{−2} ( 1 + 8x + 60x^{2} − 160x^{3 }+\ldots)$$

32. $$y_{1} = x^{−1/3}\left( 1 − \frac{10}{63} x + \frac{200}{7371} x^{2} − \frac{17600}{3781323} x^{3} +\ldots\right) ;\quad y_{2} = x^{−1/2}\left( 1 − \frac{3}{20} x + \frac{9}{352} x^{2} − \frac{105}{23936} x^{3} +\ldots\right)$$

33. $$y_{1}=x^{1/2}\sum_{m=0}^{\infty}\frac{(-1)^{m}}{8^{m}m!}\left(\prod_{j=1}^{m}\frac{4j-3}{8j+1} \right)x^{2m};\quad y_{2}=x^{1/4}\sum_{m=0}^{\infty}\frac{(-1)^{m}}{16^{m}m!}\left(\prod_{j=1}^{m}\frac{8j-7}{8j-1} \right)x^{2m}$$

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

35. $$y_{1}=x^{4}\sum_{m=0}^{\infty}(-1)^{m}(m+1)x^{2m};\quad y_{2}=-x\sum_{m=0}^{\infty}(-1)^{m}(2m-1)x^{2m}$$

36. $$y_{1}=x^{1/3}\sum_{m=0}^{\infty}\frac{(-1)^{m}}{18^{m}m!}\left(\prod_{j=1}^{m}(6j-17) \right)x^{2m};\quad y_{2}=1+\frac{4}{5}x^{2}+\frac{8}{55}x^{4}$$

37. $$y_{1}=x^{1/4}\sum_{m=0}^{\infty}\left(\prod_{j=1}^{m}\frac{8j+1}{8j+5} \right)x^{2m};\quad y_{2}=x^{-1}\sum_{m=0}^{\infty}\frac{\prod_{j=1}^{m}(2j-1)}{2^{m}m!}x^{2m}$$

38. $$y_{1}=x^{1/2}\sum_{m=0}^{\infty}\frac{1}{8^{m}m!}\left(\prod_{j=1}^{m}(4j-1) \right)x^{2m};\quad y_{2}=x^{1/3}\sum_{m=0}^{\infty}2^{m}\left(\prod_{j=1}^{m}\frac{3j-1}{12j-1} \right)x^{2m}$$

39. $$y_{1}=x^{7/2}\sum_{m=0}^{\infty}(-1)^{m}\frac{\prod_{j=1}^{m}(4j+5)}{8^{m}m!}x^{2m};\quad y_{2}=x^{1/2}\sum_{m=0}^{\infty}\frac{(-1)^{m}}{4^{m}}\left(\prod_{j=1}^{m}\frac{4j-1}{2j-3} \right)x^{2m}$$

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

41. $$y_{1}=x^{1/2}\sum_{m=0}^{\infty}\frac{(-1)^{m}}{m!}\left(\prod_{j=1}^{m}(2j+1) \right)x^{2m};\quad y_{2}=\frac{1}{x^{2}}\sum_{m=0}^{\infty}(-2)^{m}\left(\prod_{j=1}^{m}\frac{4j-3}{4j-5} \right)x^{2m}$$

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

43. $$y_{1}=\sum_{m=0}^{\infty}(-1)^{m}\frac{2^{m}(m+1)!}{\prod_{j=1}^{m}(2j+3)}x^{2m};\quad y_{2}=\frac{1}{x^{3}}\sum_{m=0}^{\infty}(-1)^{m}\frac{\prod_{j=1}^{m}(2j-1)}{2^{m}m!}x^{2m}$$

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

45. $$y_{1}=x\sum_{m=0}^{\infty}(-2)^{m}\left(\prod_{j=1}^{m}\frac{2j+1}{4j+5} \right)x^{2m};\quad y_{2}=x^{-3/2}\sum_{m=0}^{\infty}\frac{(-1)^{m}}{4^{m}m!}\left(\prod_{j=1}^{m}(4j-3) \right)x^{2m}$$

46. $$y_{1}=x^{1/3}\sum_{m=0}^{\infty}\frac{(-1)^{m}}{2^{m}\prod_{j=1}^{m}(3j+1)}x^{2m};\quad y_{2}=x^{-1/3}\sum_{m=0}^{\infty}\frac{(-1)^{m}}{6^{m}m!}x^{2m}$$

47. $$y_{1} = x^{1/2}\left( 1 − \frac{6}{13}x^{2} + \frac{36}{325}x^{4} − \frac{216}{12025}x^{6} +\ldots\right);\quad y_{2} = x^{1/3}\left( 1 − \frac{1}{2}x^{2} + \frac{1}{8}x^{4} − \frac{1}{48}x^{6} + \ldots\right)$$

48. $$y_{1} = x^{1/4}\left( 1 − \frac{13}{64}x^{2} + \frac{273}{8192}x^{4} − \frac{2639}{524288}x^{6} +\ldots\right) ;\quad y_{2} = x^{−1}\left( 1 − \frac{1}{3}x^{2} + \frac{2}{33} x^{4} − \frac{2}{209}x^{6} +\ldots\right)$$

49. $$y_{1} = x^{1/3}\left( 1 − \frac{3}{4}x^{2} + \frac{9}{14}x^{4} − \frac{81}{140}x^{6} +\ldots\right) ;\quad y_{2} = x^{−1/3}\left( 1 − \frac{2}{3}x^{2} + \frac{5}{9}x^{4} − \frac{40}{81}x^{6} +\ldots\right)$$

50. $$y_{1} = x^{1/2}\left( 1 − \frac{3}{2}x^{2} + \frac{15}{8}x^{4} − \frac{35}{16}x^{6} +\ldots\right) ;\quad y_{2} = x^{−1/2}\left( 1 − 2x^{2} + \frac{8}{3}x^{4} − \frac{16}{5}x^{6} +\ldots\right)$$

51. $$y_{1} = x^{1/4}\left( 1 − x^{2} + \frac{3}{2}x^{4} − \frac{5}{2}x^{6} +\ldots\right);\quad y_{2} = x^{−1/2}\left( 1 − \frac{2}{5} x^{2} + \frac{36}{65}x^{4} − \frac{408}{455}x^{6} +\ldots\right)$$

53. (a) $$y_{1}=x^{v}\sum_{m=0}^{\infty}\frac{(-1)^{m}}{4^{m}m!\prod_{j=1}^{m}(j+v)}x^{2m};\quad y_{2}=x^{-v}\sum_{m=0}^{\infty}\frac{(-1)^{m}}{4^{m}m!\prod_{j=1}^{m}(j-v)}x^{2m}\quad y_{1}=\frac{\sin x}{\sqrt{x}};\quad y_{2}=\frac{\cos x}{\sqrt{x}}$$

61. $$y_{1}=\frac{x^{1/2}}{1+x};\quad y_{2}=\frac{x}{1+x}$$

62. $$y_{1}=\frac{x^{1/3}}{1+2x^{2}};\quad y_{2}=\frac{x^{1/2}}{1+2x^{2}}$$

63. $$y_{1}=\frac{x^{1/4}}{1-3x};\quad y_{2}=\frac{x^{2}}{1-3x}$$

64. $$y_{1}=\frac{x^{1/3}}{5+x};\quad y_{2}=\frac{x^{-1/3}}{5+x}$$

65. $$y_{1}=\frac{x^{1/4}}{2-x^{2}};\quad y_{2}=\frac{x^{-1/2}}{2-x^{2}}$$

66. $$y_{1}=\frac{x^{1/2}}{1+3x+x^{2}};\quad y_{2}=\frac{x^{3/2}}{1+3x+x^{2}}$$

67. $$y_{1}=\frac{x}{(1+x)^{2}};\quad y_{2}=\frac{x^{1/3}}{(1+x)^{2}}$$

68. $$y_{1}=\frac{x}{3+2x+x^{2}};\quad y_{2}=\frac{x^{1/4}}{3+2x+x^{2}}$$