30: Section 8.4 Answers
- Page ID
- 103648
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)1. \(y=c_1(1+2x-2x^2+{4\over 9}x^3+\cdots)+c_2x^{3/2}(1-{2\over 5}x+{2\over 35}x^2-{4\over 945}x^3+\cdots)\)
2. \(y=c_1(1-2x+{2\over 9}x^2-{4\over 459}x^3+\cdots)+c_2x^{7/8}(1-{2\over 15}x+{2\over 345}x^2-{4\over 32085}x^3+\cdots)\)
3. \(y=c_1(1+{1\over 2}x+{1\over 10}x^2+{1\over 80}x^3+\cdots)+c_2x^{1/3}(1+{1\over 3}x+{1\over 18}x^2+{1\over 162}x^3+\cdots)\)
4. \(y=c_1(1+{1\over 3}x-{1\over 6}x^2-{1\over 6}x^3+\cdots)+c_2x^{5/2}(1+{4\over 7}x+{4\over 21}x^2+{32\over 693}x^3+\cdots)\)
5. \(y=c_1x^{1/3}(1-{1\over 2}x+{1\over 5}x^2-{7\over 120}x^3+\cdots)+c_2x^{2/3}(1-{1\over 2}x+{5\over 28}x^2-{1\over 21}x^3+\cdots)\)
6. \(y=c_1{\sinh x\over x}+c_2{\cosh x\over x}\)
7. \(y=c_1x+c_2(x\ln x-1+{1\over 2}x^2+{1\over 12}x^3+{1\over 72}x^4+\cdots)\)
8. \(y=c_1e^x+c_2[e^x\ln x+e^x(-x+{1\over 4}x^2-{1\over 18}x^3+{1\over 96}x^4+\cdots)]\)
9. \(y=c_1(1-{1\over 14}x^2+{1\over 616}x^4+\cdots)+c_2x^{-3/2}(1-{1\over 2}x^2+{1\over 40}x^4+\cdots)\)
10. \(y=c_1x(1-{1\over 10}x^2+{1\over 360}x^4+\cdots)+c_2x^{1/2}(1-{1\over 6}x^2+{1\over 168}x^4+\cdots)\)
11. \(y=c_1x^{1/3}(1+{3\over 2}x+{9\over 20}x^2+\cdots)+c_2x^{2/3}(1+{3\over 4}x+{9\over 56}x^2+\cdots)\)
12. \(y=c_1x^{-2/3}(1-{3\over 4}x^2+{9\over 128}x^4+\cdots)+c_2x^{2/3}(1-{3\over 20}x^2+{9\over 1280}x^4+\cdots)\)
13. \(y=c_1x^{-1}(1+2x-2x^2+\cdots)+c_2x^{1/2}(1-{2\over 5}x+{2\over 35}x^2+\cdots)\)
14. \(y-c_1x^{-1/2}\cos x+c_2x^{-1/2}\sin x\)
15. \(y=c_1(1+{1\over 4}x^2+{1\over 48}x^4+\cdots)+c_2[-{1\over 2}y_1\ln x+y_1(-{1\over 2x^2}+{7\over 96}x^2-{19\over 2304}x^4+\cdots)]\)
16. \(y=c_1(1-x+{1\over 4}x^2+\cdots)+c_2[y_1\ln x+y_1(2x+{5\over 4}x^2+{23\over 27}x^3+\cdots)]\)
17. \(y=c_1x^{1/2}\left(1-\frac{1}{5}x-\frac{2}{35}x^{2}+\frac{31}{315}x^{3}+\ldots \right)+c_2x^{-1}\left(1+x+\frac{1}{2}x^{2}-\frac{1}{6}x^{3}+\ldots \right)\)
18. \(y= c_1x^{1/3}\left( 1 −\frac{2}{3} x + \frac{8}{9} x^{2} − \frac{40}{81} x^{3} +\ldots\right)+c_2( 1 − x + \frac{6}{5} x^{2} − \frac{4}{5} x^{3} +\ldots )\)
19. \(y= c_1x^{1/3}\left( 1 − \frac{4}{7} x − \frac{7}{45} x^{2} + \frac{970}{2457}x^{3} +\ldots\right)+c_2 x^{−1}\left( 1 − x^{2} + \frac{2}{3} x^{3} +\ldots\right)\)
20. \(y = c_1x^{1/4}\left( 1 − \frac{1}{2} x − \frac{19}{104} x^{2} + \frac{1571}{10608} x^{3} +\ldots\right) +c_2 x^{−1}\left( 1 + 2x − \frac{11}{6} x^{2} − \frac{1}{7} x^{3} + \ldots\right)\)
21. \(y = c_1x^{1/3}\left( 1 − x + \frac{28}{31} x^{2} − \frac{1111}{1333} x^{3} + \ldots\right) +c_2x^{−1/4}\left( 1 − x + \frac{7}{8} x^{2} − \frac{19}{24} x^{3} + \ldots\right)\)
22. \(y= c_1x^{1/5}\left( 1 − \frac{6}{25} x − \frac{1217}{625} x^{2} + \frac{41972}{46875} x^{3} +\ldots\right) +c_2( x − \frac{1}{4} x^{2} − \frac{35}{18} x^{3} + \frac{11}{12} x^{4} +\ldots) \)
23. \(y= c_1x^{3/2}\left( 1 − x + \frac{11}{26} x^{2} − \frac{109}{1326} x^{3} + \ldots\right) +c_2 x^{1/4}\left( 1 + 4x − \frac{131}{24} x^{2} + \frac{39}{14}x^{3} +\ldots\right)\)
24. \(y=c_1 x^{1/3}\left( 1 − \frac{1}{3} x + \frac{2}{15} x^{2} − \frac{5}{63} x^{3} +\ldots\right) +c_2 x^{−1/6}\left( 1 − \frac{1}{12} x^{2} + \frac{1}{18} x^{3} +\ldots\right)\)
25. \(y =c_1( 1 − \frac{1}{14}x^{2} + \frac{1}{105} x^{3} +\ldots)+c_2 x^{−1/3}\left( 1 − \frac{1}{18} x − \frac{71}{405}x^{2} + \frac{719}{34992} x^{3} +\ldots\right)\)
26. \(y= c_1x^{1/5}\left( 1 + \frac{3}{17} x − \frac{7}{153} x^{2} − \frac{547}{5661} x^{3} +\ldots\right)+c_2x^{−1/2}\left( 1 + x + \frac{14}{13} x^{2} − \frac{556}{897} x^{3} +\ldots\right)\)
27. \(y = c_1x^{1/2}\left( 1 − \frac{9}{40} x + \frac{5}{128} x^{2} − \frac{245}{39936} x^{3} +\ldots\right) +c_2 x^{1/4}\left( 1 −\frac{25}{96} x + \frac{675}{14336} x^{2} − \frac{38025}{5046272} x^{3} +\ldots\right)\)
28. \(y=c_1 x^{1/3}\left( 1 + \frac{32}{117} x − \frac{28}{1053} x^{2} + \frac{4480}{540189} x^{3} +\ldots\right) +c_2x^{−3}\left( 1 + \frac{32}{7} x + \frac{48}{7} x^{2}\right)\)
29.\(y= c_1x^{1/2}\left( 1 − \frac{5}{8} x + \frac{55}{96} x^{2} − \frac{935}{1536} x^{3} +\ldots\right) +c_2 x^{−1/2}\left( 1 + \frac{1}{4} x − \frac{5}{32} x^{2} −\frac{55}{384} x^{3} +\ldots\right)\)
30. \(y = c_1x^{1/2} \left( 1 − \frac{3}{4} x + \frac{5}{96} x^{2} + \frac{5}{4224}x ^{3} +\ldots\right)+c_2 x^{−2} ( 1 + 8x + 60x^{2} − 160x^{3 }+\ldots)\)
31. \(y = c_1x^{−1/3}\left( 1 − \frac{10}{63} x + \frac{200}{7371} x^{2} − \frac{17600}{3781323} x^{3} +\ldots\right) +c_2 x^{−1/2}\left( 1 − \frac{3}{20} x + \frac{9}{352} x^{2} − \frac{105}{23936} x^{3} +\ldots\right)\)
32. \(y = c_1x^{1/2}\left( 1 − \frac{6}{13}x^{2} + \frac{36}{325}x^{4} − \frac{216}{12025}x^{6} +\ldots\right)+c_2 x^{1/3}\left( 1 − \frac{1}{2}x^{2} + \frac{1}{8}x^{4} − \frac{1}{48}x^{6} + \ldots\right)\)
33. \(y = c_1x^{1/4}\left( 1 − \frac{13}{64}x^{2} + \frac{273}{8192}x^{4} − \frac{2639}{524288}x^{6} +\ldots\right) +c_2 x^{−1}\left( 1 − \frac{1}{3}x^{2} + \frac{2}{33} x^{4} − \frac{2}{209}x^{6} +\ldots\right)\)
34. \(y = c_1x^{1/3}\left( 1 − \frac{3}{4}x^{2} + \frac{9}{14}x^{4} − \frac{81}{140}x^{6} +\ldots\right)+c_2 x^{−1/3}\left( 1 − \frac{2}{3}x^{2} + \frac{5}{9}x^{4} − \frac{40}{81}x^{6} +\ldots\right)\)
35. \(y = c_1x^{1/2}\left( 1 − \frac{3}{2}x^{2} + \frac{15}{8}x^{4} − \frac{35}{16}x^{6} +\ldots\right) +c_2x^{−1/2}\left( 1 − 2x^{2} + \frac{8}{3}x^{4} − \frac{16}{5}x^{6} +\ldots\right)\)
36. \(y = c_1x^{1/4}\left( 1 − x^{2} + \frac{3}{2}x^{4} − \frac{5}{2}x^{6} +\ldots\right)+c_2 x^{−1/2}\left( 1 − \frac{2}{5} x^{2} + \frac{36}{65}x^{4} − \frac{408}{455}x^{6} +\ldots\right)\)
37. \(y = c_1x\left( 1 − x + \frac{3}{4}x^{2} − \frac{13}{36}x^{3} +\ldots\right) +c_2[ y_{1} \ln x + x^{2}\left( 1 − x + \frac{65}{108} x^{2} +\ldots\right)]\)
38. \(y = c_1x^{−1}\left( 1 − 2x + \frac{9}{2} x^{2} − \frac{20}{3} x^{3} +\ldots\right)+c_2( y_{1} \ln x + 1 − \frac{15}{4} x + \frac{133}{ 18} x^{2} +\ldots )\)
39. \(y= c_1(1 + x − x^{2} + \frac{1}{3} x^{3} + \dots)+c_2[ y_{1} \ln x − x\left( 3 − \frac{1}{2} x − \frac{31}{18} x^{2} +\ldots\right)]\)
40. \(y = c_1x^{1/2}\left( 1 − 2x + \frac{5}{2} x^{2} − 2x^{3} +\ldots\right) +c_2[ y_{1} \ln x + x^{3/2}\left( 1 − \frac{9}{4} x + \frac{17}{6} x^{2} +\ldots\right)]\)
41. \(y = c_1x\left( 1 − 4x + \frac{19}{2} x^{2} − \frac{49}{3} x^{3} +\ldots\right)+c_2[ y_{1} \ln x + x^{2}\left( 3 − \frac{43}{4} x + \frac{208}{9} x^{2} +\ldots\right)]\)
42. \(y = c_1x^{−1/3}\left( 1 − x + \frac{5}{6} x^{2} − \frac{1}{2} x^{3} +\ldots\right) +c_2[y_{1} \ln x + x^{2/3}\left( 1 − \frac{11}{12} x + \frac{25}{36} x^{2} +\ldots\right)]\)
43. \(y=c_1( 1 − 2x + \frac{7}{4} x^{2} − \frac{7}{9} x^{3} +\ldots)+c_2[ y_{1} \ln x + x\left( 3 − \frac{15}{4} x + \frac{239}{108} x^{2} +\ldots\right)]\)
44. \(y =c_1 x^{−2}\left( 1 − 2x + \frac{5}{2} x^{2} − 3x^{3} +\ldots\right)+c_2( y_{1} \ln x + \frac{3}{4} − \frac{13}{6} x +\ldots) \)
45. \(y = c_1x^{−1/2}\left( 1 − x + \frac{1}{4} x^{2} + \frac{1}{18} x^{3} +\ldots\right)+c_2[ y_{1} \ln x + x^{1/2}\left( \frac{3}{2} − \frac{13}{16} x + \frac{1}{54} x^{2} +\ldots\right)]\)
46. \(y =c_1 x^{−1/4}\left( 1 − \frac{1}{4} x − \frac{7}{32} x^{2} + \frac{23}{384}x^{3} +\ldots\right) +c_2[ y_{1} \ln x + x^{3/4}\left( \frac{1}{4} + \frac{5}{64}x − \frac{157}{2304}x^{2} +\ldots\right)]\)
47. \(y= c_1x^{−1/3}\left( 1 − x + \frac{7}{6} x^{2} −\frac{23}{18} x^{3} +\ldots\right)+c_2[ y_{1} \ln x − x^{5/3}\left( \frac{1}{12} − \frac{13}{108} x\ldots\right)]\)
48. \(y= c_1x^{−2}\left( 1 + 3x + \frac{3}{2}^{2} −\frac{1}{2}x^{3} +\ldots\right) +c_2[ y_{1} \ln x − 5x^{−1}\left( 1 + \frac{5}{4} x − \frac{1}{4} x^{2} +\ldots\right)]\)
49. \(y= c_1x^{3} (1 + 20x + 180x^{2} + 1120x^{3} +\ldots )+c_2[ y_{1} \ln x − x^{4}\left( 26 + 324x + 6968 3 x^{2} +\ldots\right)]\)
50. \(y = c_1x\left( 1 − 5x + \frac{85}{4}x^{2} − \frac{3145}{36} x^{3} +\ldots\right)+c_2[y_{1} \ln x + x^{2}\left( 2 − \frac{39}{4} x + \frac{4499}{108}x^{2} +\ldots\right)]\)
51. \(y = c_1(1 − x + \frac{3}{4}x^{2} − \frac{7}{12}x^{3} +\ldots)+c_2[ y_{1} \ln x + x\left( 1 − \frac{3}{4} x + \frac{5}{9} x^{2} +\ldots\right)]\)
52. \(y= c_1x^{−3} (1 + 16x + 36x^{2} + 16x^{3} +\ldots )+c_2[y_{1} \ln x − x^{−2}\left( 40 + 150x + \frac{280}{3}x^{2} +\ldots\right)]\)
53. \(y = c_1x^{−1}\left( 1 −\frac{3}{2}x^{2} + \frac{15}{8}x^{4} −\frac{35}{16}x^{6} +\ldots\right)+c_2[ y_{1} \ln x + x\left(\frac{1}{4} − \frac{13}{32}x^{2} + \frac{101}{192}x^{4} +\ldots\right)]\)
54. \(y = c_1x\left( 1 −\frac{1}{2}x^{2} +\frac{1}{8}x^{4} −\frac{1}{48}x^{6}+\ldots\right)+c_2[ y_{1} \ln x + x^{3}\left(\frac{1}{4} − \frac{3}{32}x^{2}+ \frac{11}{576}x^{4} +\ldots\right)]\)
55. \(y= c_1x^{−2}\left(1 −\frac{3}{4}x^{2} −\frac{9}{64}x^{4} −\frac{25}{256}x^{6} +\ldots\right) +c_2( y_{1} \ln x + \frac{1}{2} − \frac{21}{128}x^{2} − \frac{215}{1536}x^{4} +\ldots) \)
56. \(y = c_1x^{−3}\left( 1 −\frac{17}{8}x^{2} +\frac{85}{256}x^{4} − \frac{85}{18432}x^{6} +\ldots\right)+c_2[ y_{1} \ln x + x^{−1}\left( \frac{25}{8} − \frac{471}{512} x^{2} + \frac{1583}{110592} x^{4} +\ldots\right)]\)
57. \(y= c_1x^{−1}\left( 1 −\frac{3}{4}x^{2} + \frac{45}{64}x^{4} − \frac{175}{256}x^{6} +\ldots\right) +c_2[ y_{1} \ln x − x\left( \frac{1}{4} − \frac{33}{128} x^{2} + \frac{395}{1536}x^{4} +\ldots\right)]\)
58. \(y= c_1\frac{1}{x} +c_2( y_{1} \ln x − 6 + 6x − \frac{8}{3} x^{2})\)
59. \(y=c_1(1-x)+c_2(y_{1}\ln x+4x)\)
60. \(y=c_1(x-4x^{3}+x^{5})+c_2(y_{1}\ln x+6x^{3}-3x^{5})\)