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Mathematics LibreTexts

Section 7.7 Answers

  • Page ID
    29454
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    1. \(y_{1}=2x^{3}\sum_{n=0}^{\infty}\frac{(-4)^{n}}{n!(n+2)!}x^{n};\quad y_{2}=x+4x^{2}-8\left(y_{1}\ln x-4\sum_{n=1}^{\infty}\frac{(-4)^{n}}{n!(n+2)!}\left(\sum_{j=1}^{n}\frac{j+1}{j(j+2)} \right)x^{n} \right)\)

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

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

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

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

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

    7. \(y_{1}=6x^{3/2}\sum_{n=0}^{\infty}\frac{(-1)^{n}}{4^{n}n!(n+3)!}x^{n};\quad y_{2}=x^{-3/2}\left(1+\frac{1}{8}x+\frac{1}{64}x^{2}\right)-\frac{1}{768}\left( y_{1}\ln x-6x^{3/2}\sum_{n=1}^{\infty}\frac{(-1)^{n}}{4^{n}n!(n+3)!}\left(\sum_{j=1}^{n}\frac{2j+3}{j(j+3)}\right)x^{n}\right)\)

    8. \(y_{1}=\frac{120}{x^{2}}\sum_{n=0}^{\infty}\frac{(-1)^{n}}{n!(n+5)!}x^{n};\quad y_{2}=x^{-7}\left(1+\frac{1}{4}x+\frac{1}{24}x^{2}+\frac{1}{144}x^{3}+\frac{1}{576}x^{4}\right)-\frac{1}{2880}\left(y_{1}\ln x-\frac{120}{x^{2}}\sum_{n=1}^{\infty}\frac{(-1)^{n}}{n!(n+f)!}\left(\sum_{j=1}^{n}\frac{2j+5}{j(j+5)}\right)x^{n}\right)\)

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

    10. \(y_{1} = x^{4}\left( 1 −\frac{2}{5} x\right);\quad y_{2} = 1 + 10x + 50x^{2} + 200x^{3} − 300\left(y_{1} \ln x + \frac{27}{25} x^{5} − \frac{1 }{30} x^{ 6}\right)\)

    11. \(y_{1}=x^{3};\quad y_{2}=x^{-3}\left(1-\frac{6}{5}x+\frac{3}{4}x^{2}-\frac{1}{3}x^{3}+\frac{1}{8}x^{4}-\frac{1}{20}x^{5}\right) -\frac{1}{120}\left(y_{1}\ln x+x^{3}\sum_{n=1}^{\infty}\frac{(-1)^{n}6!}{n(n+6)!}x^{n} \right)\)

    12. \(y_{1}=x^{2}\sum_{n=0}^{\infty}\frac{1}{n!}\left(\prod_{j=1}^{n}\frac{2j+3}{j+4} \right)x^{n};\quad y_{2}=x^{-2}\left( 1+x+\frac{1}{4}x^{2}-\frac{1}{12}x^{3}\right) -\frac{1}{16}y_{1}\ln x+\frac{x^{2}}{8}\sum_{n=1}^{\infty}\frac{1}{n!}\left(\prod_{j=1}^{n}\frac{2j+3}{j+4} \right)\left(\sum_{j=1}^{n}\frac{(j^{2}+3j+6)}{j(j+4)(2j+3)} \right)x^{n}\)

    13. \(y_{1}=x^{5}\sum_{n=0}^{\infty}(-1)^{n}(n+1)(n+2)x^{n};\quad y_{2}=1-\frac{x}{2}+\frac{x^{2}}{6}\)

    14. \(y_{1}=\frac{1}{x}\sum_{n=0}^{\infty}\frac{(-1)^{n}}{n!}\left(\prod_{j=1}^{n}\frac{(j+3)(2j-3)}{j+6} \right)x^{n};\quad y_{2}=x^{-7}\left(1+\frac{26}{5}x+\frac{143}{20}x^{2}\right)\)

    15. \(y_{1}=x^{7/2}\sum_{n=0}^{\infty}\frac{(-1)^{n}}{2^{n}(n+4)!}x^{n};\quad y_{2}=x^{-1/2}\left(1-\frac{1}{2}x+\frac{1}{8}x^{2}-\frac{1}{48}x^{3} \right)\)

    16. \(y_{1}=x^{10/3}\sum_{n=0}^{\infty}\frac{(-1)^{n}(n+1)}{9^{n}}\left(\prod_{j=1}^{n}\frac{3j+7}{j+4} \right)x^{n};\quad y_{2}=x^{-2/3}\left(1+\frac{4}{27}x-\frac{1}{243}x^{2}\right)\)

    17. \(y_{1}=x^{3}\sum_{n=0}^{7}(-1)^{n}(n+1)\left(\prod_{j=1}^{n}\frac{j-8}{j+6} \right)x^{n};\quad y_{2}=x^{-3}\left(1+\frac{52}{5}x+\frac{234}{5}x^{2}+\frac{572}{5}x^{3}+143x^{4}\right)\)

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

    19. \(y_{1}=x^{6}\sum_{n=0}^{4}(-1)^{n}2^{n}\left(\prod_{j=1}^{n}\frac{j-5}{j+5} \right)x^{n};\quad y_{2}=x(1+18x+144x^{2}+672x^{3}+2016x^{4})\)

    20. \(y_{1}=x^{6}\left(1+\frac{2}{3}x+\frac{1}{7}x^{2}\right);\quad y_{2}=x\left(1+\frac{21}{4}x+\frac{21}{2}x^{2}+\frac{35}{4}x^{3} \right)\)

    21. \(y_{1}=x^{7/2}\sum_{n=0}^{\infty}(-1)^{n}(n+1)x^{n};\quad y_{2}=x^{-7/2}\left(1-\frac{5}{6}x+\frac{2}{3}x^{2}-\frac{1}{2}x^{3}+\frac{1}{3}x^{4}-\frac{1}{6}x^{5}\right)\)

    22. \(y_{1}=\frac{x^{10}}{6}\sum_{n=0}^{\infty}(-1)^{n}2^{n}(n+1)(n+2)(n+3)x^{n};\quad y_{2}=\left(1-\frac{4}{3}x+\frac{5}{3}x^{2}-\frac{40}{21}x^{3}+\frac{40}{21}x^{4}-\frac{32}{21}x^{5}+\frac{16}{21}x^{6}\right)\)

    23. \(y_{1}=x^{6}\sum_{m=0}^{\infty}\frac{(-1)^{m}\prod_{j=1}^{m}(2j+5)}{2^{m}m!}x^{2m};\quad y_{2}=x^{2}\left(1+\frac{3}{2}x^{2}\right)-\frac{15}{2}y_{1}\ln x+\frac{75}{2}x^{6}\sum_{m=1}^{\infty}\frac{(-1)^{m}\prod_{j=1}^{m}(2j+5)}{2^{m+1}m!}\left(\sum_{j=1}^{m}\frac{1}{j(2j+5)} \right)x^{2m}\)

    24. \(y_{1}=x^{6}\sum_{m=0}^{\infty}\frac{(-1)^{m}}{2^{m}m!}x^{2m}=x^{6}e^{-x^{2}};\quad y_{2}=x^{2}\left(1+\frac{1}{2}x^{2}\right)-\frac{1}{2}y_{1}\ln x+\frac{x^{6}}{4}\sum_{m=1}^{\infty}\frac{(-1)^{m}}{2^{m}m!}\left(\sum_{j=1}^{m}\frac{1}{j} \right)x^{2m}\)

    25. \(y_{1}=6x^{6}\sum_{m=0}^{\infty}\frac{(-1)^{m}}{4^{m}m!(m+3)!}x^{2m};\quad y_{2}=1+\frac{1}{8}x^{2}+\frac{1}{64}x^{4}-\frac{1}{384}\left( y_{1}\ln x-3x^{6}\sum_{m=1}^{\infty}\frac{(-1(^{m}}{4^{m}m!(m+3)!}\left(\sum_{j=1}^{m}\frac{2j+3}{j(j+3)}\right)x^{2m}\right)\)

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

    27. \(y_{1}=2x^{3}\sum_{m=0}^{\infty}\frac{(-1)^{m}}{4^{m}m!(m+2)!}x^{2m};\quad y_{2}=x^{-1}\left(1+\frac{1}{4}x^{2}\right)-\frac{1}{16}\left(y_{1}\ln x-2x^{3}\sum_{m=1}^{\infty}\frac{(-1)^{m}}{4^{m}m!(m+2)!}\left(\sum_{j=1}^{m}\frac{j+1}{j(j+2)}\right)x^{2m}\right)\)

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

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

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

    31. \(y_{1}=6x^{5/2}\sum_{m=0}^{\infty}\frac{(-1)^{m}}{16^{m}m!(m+3)!}x^{2m};\quad y_{2}=x^{-7/2}\left(1+\frac{1}{32}x^{2}+\frac{1}{1024}x^{4}\right) -\frac{1}{24576}\left(y_{1}\ln x-3x^{5/2}\sum_{m=1}^{\infty}\frac{(-1)^{m}}{16^{m}m!(m+3)!}\left(\sum_{j=1}^{m}\frac{2j+3}{j(j+3)}\right) x^{2m}\right)\)

    32. \(y_{1}=2x^{13/3}\sum_{m=0}^{\infty}\frac{\prod_{j=1}^{m}(3j+1)}{9^{m}m!(m+2)!}x^{2m};\quad y_{2}=x^{1/3}\left(1+\frac{2}{9}x^{2}\right)+\frac{2}{81}\left( y_{1}\ln x-x^{13/3}\sum_{m=0}^{\infty}\frac{\prod_{j=1}^{m}(3j+1)}{9^{m}m!(m+2)!}\left(\sum_{j=1}^{m}\frac{3j^{2}+2j+2}{j(j+2)(3j+1)}\right)x^{2m}\right)\)

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

    34. \(y_{1}=x^{2}\left(1-\frac{1}{2}x^{2}\right) ;\quad y_{2}=x^{-2}\left(1+\frac{9}{2}x^{2}\right)-\frac{27}{2}\left(y_{1}\ln x+\frac{7}{12}x^{4}-x^{2}\sum_{m=2}^{\infty}\frac{\left(\frac{3}{2}\right)^{m}}{m(m-1)(m+2)!}x^{2m}\right)\)

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

    36. \(y_{1}=x^{5/2}\sum_{m=0}^{\infty}\frac{(-1)^{m}}{(m+1)(m+2)(m+3)}x^{2m};\quad y_{2}=x^{-7/2}(1+x^{2})^{2}\)

    37.  \(y_{1}=\frac{x^{7}}{5}\sum_{m=0}^{\infty}(-1)^{m}(m+5)x^{2m};\quad y_{2}=x^{-1}(1-2x^{2}+3x^{4}-4x^{6})\)

    38. \(y_{1}=x^{3}\sum_{m=0}^{\infty}(-1)^{m}\frac{m+1}{2^{m}}\left(\prod_{j=1}^{m}\frac{2j+1}{j+5}\right) x^{2m};\quad y_{2}=x^{-7}\left(1+\frac{21}{8}x^{2}+\frac{35}{16}x^{4}+\frac{35}{64}x^{6}\right)\)

    39. \(y_{1}=2x^{4}\sum_{m=0}^{\infty}(-1)^{m}\frac{\prod_{j=1}^{m}(4j+5)}{2^{m}(m+2)!}x^{2m};\quad y_{2}=1-\frac{1}{2}x^{2}\)

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

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