Chapter 7 Series Solutions of Linear Second Order Equations

(a) \(R=2\); \(I=(-1,3)\); (b) \(R=1/2\); \(I=(3/2,5/2)\) (c) \(R=0\); (d) \(R=16\);

     \(I=(-14,18)\) (e) \(R=\infty\); \(I=(-\infty,\infty)\) (f) \(R=4/3\); \(I=(-25/3,-17/3)\)

(a) \(R=1\); \(I=(0,2)\) (b) \(R=\sqrt2\); \(I=(-2-\sqrt2,-2+\sqrt2)\); (c) \(R=\infty\);

     \(I=(-\infty,\infty)\) (d) \(R=0\) (e) \(R=\sqrt3\); \(I=(-\sqrt3,\sqrt3)\) (f) \(R=1\) \(I=(0,2)\)

(a) \(R=3\); \(I=(0,6)\) (b) \(R=1\); \(I=(-1,1)\) (c) \(R=1/\sqrt3\)

     \(I=(3-1/\sqrt3,3+1/\sqrt3)\) (d) \(R=\infty\); \(I=(-\infty,\infty)\) (e) \(R=0\) (f) \(R=2\);

     \(I=(-1,3)\)

\(b_n=2(n+2)(n+1)a_{n+2}+(n+1)na_{n+1}+(n+3)a_n\)

\(b_0=2a_2-2a_0\) \(b_n=(n+2)(n+1)a_{n+2}+[3n(n-1)-2]a_n+3(n-1)a_{n-1},\; n\ge1\)

\(b_n=(n+2)(n+1)a_{n+2}+2(n+1)a_{n+1}+(2n^2-5n+4)a_n\)

\(b_n=(n+2)(n+1)a_{n+2}+2(n+1)a_{n+1}+(n^2-2n+3)a_n\)

\(b_n=(n+2)(n+1)a_{n+2}+(3n^2-5n+4)a_n\)

\(b_0=-2a_2+2a_1+a_0\),

     \(b_n =-(n+2)(n+1)a_{n+2}+(n+1)(n+2)a_{n+1}+(2n+1)a_n +a_{n-1}\),  \(n\ge2\)

\(b_0=8a_2+4a_1-6a_0\),

     \(b_n=4(n+2)(n+1)a_{n+2}+4(n+1)^2a_{n+1}+(n^2+n-6)a_n-3a_{n-1}\),  \(n\ge1\)

\(b_0=(r+1)(r+2)a_0\), 

     \(b_n=(n+r+1)(n+r+2)a_n-(n+r-2)^2a_{n-1}\),  \(n\ge1\).

\(b_0=(r-2)(r+2)a_0\), 

     \(b_n=(n+r-2)(n+r+2)a_n+(n+r+2)(n+r-3)a_{n-1}\),  \(n\ge14\)

\(b_0=(r-1)^2a_0\),  \(b_1=r^2a_1+(r+2)(r+3)a_0\),

     \(b_n=(n+r-1)^2a_n+(n+r+1)(n+r+2)a_{n-1}+(n+r-1)a_{n-2}\),  \(n\ge2\)

\(b_0=r(r+1)a_0\),  \(b_1=(r+1)(r+2)a_1+3(r+1)(r+2)a_0\),

     \(b_n=(n+r)(n+r+1)a_n+3(n+r)(n+r+1)a_{n-1}+(n+r)a_{n-2}\),  \(n\ge2\)

\(b_0=(r+2)(r+1)a_0\)   \(b_1=(r+3)(r+2)a_1\),

     \(b_n=(n+r+2)(n+r+1)a_n+2(n+r-1)(n+r-3)a_{n-2}\),  \(n\ge2\)

\(b_0=2(r+1)(r+3)a_0\),  \(b_1=2(r+2)(r+4)a_1\),

     \(b_n=2(n+r+1)(n+r+3)a_n+(n+r-3)(n+r)a_{n-2}\),  \(n\ge2\)

\(y=\dst{a_0\sum^\infty_{m=0}(-1)^m(2m+1)x^{2m}+ a_1\sum^\infty_{m=0}(-1)^m(m+1)x^{2m+1}}\)

\(y=\dst{a_0\sum_{m=0}^\infty(-1)^{m+1}{x^{2m}\over2m-1}+a_1x}\)

\(y=\dst{a_0(1-10x^2+5x^4)+a_1\left(x-2x^3+{1 \over5}x^5\right)}\)

\(y=\dst{a_0\sum_{m=0}^\infty(m+1)(2m+1)x^{2m}+{a_1\over3}\sum_{m=0}^\infty (m+1)(2m+3)x^{2m+1}}\)

\(y=\dst{a_0\sum_{m=0}^\infty (-1)^m\left[\prod_{j=0}^{m-1}{4j+1\over2j+1}\right]x^{2m} +a_1\sum_{m=0}^\infty (-1)^m\left[\prod_{j=0}^{m-1}(4j+3)\right]{x^{2m+1}\over2^mm!}}\)

\(y=\dst{a_0\sum_{m=0}^\infty (-1)^m\left[\prod_{j=0}^{m-1}{(4j+1)^2\over2j+1}\right]{x^{2m}\over8^mm!} +a_1\sum_{m=0}^\infty (-1)^m\left[\prod_{j=0}^{m-1}{(4j+3)^2\over2j+3}\right] {x^{2m+1}\over8^mm!}}\)

\(y=\dst{a_0\sum_{m=0}^\infty{2^mm!\over\prod_{j=0}^{m-1}(2j+1)}x^{2m} +a_1\sum_{m=0}^\infty{\prod_{j=0}^{m-1}(2j+3)\over2^mm!}x^{2m+1}}\)

\(y=\dst{a_0\left(1-14x^2+{35\over3}x^4\right)+a_1\left(x-3x^3+{3\over5}x^5 +{1\over35}x^7\right)}\)

(a) \(y=\dst{a_0\sum_{m=0}^\infty(-1)^m {x^{2m}\over\prod_{j=0}^{m-1}(2j+1)} +a_1\sum_{m=0}^\infty(-1)^m{x^{2m+1}\over2^mm!}}\)

(a) \(y=\dst{a_0\sum_{m=0}^\infty (-1)^m\left[\prod_{j=0}^{m-1}{4j+3\over2j+1}\right]{x^{2m}\over2^mm!} +a_1\sum_{m=0}^\infty (-1)^m\left[\prod_{j=0}^{m-1}{4j+5\over2j+3}\right]{x^{2m+1}\over 2^mm!}}\)

\(y=\dst{2-x-x^2+{1\over3}x^3+{5\over12}x^4-{1\over6}x^5-{17\over72}x^6 +{13\over126}x^7+\cdots}\)

\(y=\dst{1-x+3x^2-{5\over2}x^3+5x^4-{21\over8}x^5+3x^6 -{11\over16}x^7+\cdots}\)

\(y=\dst{2-x-2x^2+{1\over3}x^3+3x^4-{5\over6}x^5-{49\over5}x^6 +{45\over14}x^7+\cdots}\)

\(y=\dst{a_0\sum_{m=0}^\infty{(x-3)^{2m}\over(2m)!}+a_1\sum_{m=0}^\infty {(x-3)^{2m+1}\over(2m+1)!}}\)

\(y=\dst{a_0\sum_{m=0}^\infty{(x-3)^{2m}\over2^mm!}+a_1\sum_{m=0}^\infty {(x-3)^{2m+1}\over\prod_{j=0}^{m-1}(2j+3)}}\)

\(y=\dst{a_0\sum_{m=0}^\infty \left[\prod_{j=0}^{m-1}(2j+3)\right]{ (x-1)^{2m}\over m!}+a_1\sum_{m=0}^\infty {4^m(m+1)!\over\prod_{j=0}^{m-1}(2j+3)}(x-1)^{2m+1}}\)

\(y=\dst{a_0\left(1-6(x-2)^2+{4\over3}(x-2)^4+{8\over135}(x-2)^6\right) +a_1\left((x-2)-{10\over9}(x-2)^3\right)}\)

\(y=\dst{a_0\sum_{m=0}^\infty(-1)^m\left[\prod_{j=0}^{m-1}(2j+1)\right] {3^m\over4^mm!}(x+1)^{2m}+a_1\sum_{m=0}^\infty(-1)^m{3^mm!\over \prod_{j=0}^{m-1}(2j+3)}(x+1)^{2m+1}}\)

\(y=\dst{-1+2x+{3\over8}x^2-{1\over3}x^3-{3\over128}x^4-{1\over1024}x^6+\cdots}\)

\(y=\dst{-2+3(x-3)+3(x-3)^2-2(x-3)^3-{5\over4}(x-3)^4+{3\over5}(x-3)^5 +{7\over24}(x-3)^6-{4\over35}(x-3)^7+\cdots}\)

\(y=\dst{-1+(x-1)+3(x-1)^2-{5\over2}(x-1)^3-{27\over4}(x-1)^4+{21\over4}(x-1)^5 +{27\over2}(x-1)^6-{81\over8}(x-1)^7+\cdots}\)

\(y=\dst{4-6(x-3)-2(x-3)^2+(x-3)^3+{3\over2}(x-3)^4-{5\over4}(x-3)^5- {49\over20}(x-3)^6+{135\over56}(x-3)^7+\cdots}\)

\(y=\dst{3-4(x-4)+15(x-4)^2-4(x-4)^3+{15\over4}(x-4)^4-{1\over5}(x-4)^5}\)

\(y=\dst{3-3(x+1)-30(x+1)^2+{20\over3}(x+1)^3+20(x+1)^4-{4\over3}(x+1)^5-{8\over 9}(x+1)^6}\)

(a)\(y=\dst{a_0\sum_{m=0}^\infty(-1)^m x^{2m}+a_1\sum_{m=0}^\infty(-1)^mx^{2m+1}}\) (b)\(y=\dst{a_0+a_1x\over1+x^2}\)

\(y=\dst{a_0\sum_{m=0}^\infty {x^{3m}\over3^mm!\prod_{j=0}^{m-1}(3j+2)}+a_1\sum_{m=0}^\infty {x^{3m+1}\over3^mm!\prod_{j=0}^{m-1}(3j+4)}}\)

\(y=\dst{a_0\sum_{m=0}^\infty \left(2\over3\right)^m\left[\prod_{j=0}^{m-1}(3j+2)\right]{x^{3m}\over m!} +a_1\sum_{m=0}^\infty{6^mm!\over\prod_{j=0}^{m-1}(3j+4)}x^{3m+1}}\)

\(y=\dst{a_0\sum_{m=0}^\infty (-1)^m{3^mm!\over\prod_{j=0}^{m-1}(3j+2)}x^{3m} +a_1\sum_{m=0}^\infty (-1)^m\left[\prod_{j=0}^{m-1}(3j+4)\right]{x^{3m+1}\over 3^mm!}}\)

\(y=\dst{a_0(1-4x^3+4x^6)+a_1\sum_{m=0}^\infty 2^m\left[\prod_{j=0}^{m-1}{3j-5\over3j+4}\right]x^{3m+1}}\)

\(y=\dst{a_0\left(1+{21\over2}x^3+{42\over5}x^6+{7\over20}x^9\right) +a_1\left(x+4x^4+{10\over7}x^7\right)}\)

\(y=\dst{a_0\sum_{m=0}^\infty (-2)^m\left[\prod_{j=0}^{m-1}{5j+1\over5j+4}\right]x^{5m}+ a_1\sum_{m=0}^\infty \left(-{2\over5}\right)^m\left[\prod_{j=0}^{m-1}(5j+2)\right]{x^{5m+1}\over m!}}\)

\(y=\dst{a_0\sum_{m=0}^\infty (-1)^m{x^{4m}\over4^mm!\prod_{j=0}^{m-1}(4j+3)}+a_1\sum_{m=0}^\infty (-1)^m{x^{4m+1}\over4^mm!\prod_{j=0}^{m-1}(4j+5)}}\)

\(y=\dst{a_0\sum_{m=0}^\infty(-1)^m{x^{7m}\over\prod_{j=0}^{m-1}(7j+6)}+ a_1\sum_{m=0}^\infty(-1)^m{x^{7m+1}\over7^mm!}}\)

\(y=\dst{a_0\left(1-{9\over7}x^8\right)+a_1\left(x-{7\over9}x^9\right)}\)

\(y=\dst{a_0\sum_{m=0}^\infty x^{6m}+a_1\sum_{m=0}^\infty x^{6m+1}}\)

\(y=\dst{a_0\sum_{m=0}^\infty(-1)^m{x^{6m}\over\prod_{j=0}^{m-1}(6j+5)}+ a_1\sum_{m=0}^\infty(-1)^m{x^{6m+1}\over6^mm!}}\)

\(y=\dst{2-3x-2x^2+{7\over2}x^3-{55\over12}x^4+{59\over8}x^5-{83\over6}x^6 +{9547\over336}x^7+\cdots}\)

\(y=\dst{-1+2x-4x^3+4x^4+4x^5-12x^6+4x^7+\cdots}\)

\(y=\dst{1+x^2-{2\over3}x^3+{11\over6}x^4-{9\over5}x^5+ {329\over90}x^6-{1301\over315}x^7+\cdots}\)

\(y=\dst{x-x^2-{7\over2}x^3+{15\over2}x^4+{45\over8}x^5 -{261\over8}x^6+{207\over16}x^7+\cdots}\)

\(y=\dst{4+3x-{15\over4}x^2+{1\over4}x^3+{11\over16}x^4-{5\over16}x^5 +{1\over20}x^6+{1\over120}x^7+\cdots}\)

\(y=\dst{7+3x-{16\over3}x^2+{13\over3}x^3-{23\over9}x^4+{10\over9}x^5 -{7\over27}x^6-{1\over9}x^7+\cdots}\)

\(y=\dst{2+5x-{7\over4}x^2-{3\over16}x^3+{37\over192}x^4 -{7\over192}x^5-{1\over1920}x^6+{19\over11520}x^7+\cdots}\)

\(y=\dst{1-(x-1)+{4\over3}(x-1)^3-{4\over3}(x-1)^4-{4\over5}(x-1)^5 +{136\over45}(x-1)^6-{104\over63}(x-1)^7+\cdots}\)

\(y=\dst{1-(x+1)+4(x+1)^2-{13\over3}(x+1)^3+{77\over6}(x+1)^4 -{278\over15}(x+1)^5+{1942\over45}(x+1)^6-{23332\over315}(x+1)^7+\cdots}\)

\(y=\dst{2-(x-1)-{1\over2}(x-1)^2+{5\over3}(x-1)^3-{19\over12}(x-1)^4 +{7\over30}(x-1)^5+{59\over45}(x-1)^6-{1091\over630}(x-1)^7+\cdots}\)

\(y=\dst{-2+3(x+1)-{1\over2}(x+1)^2-{2\over3}(x+1)^3+{5\over8}(x+1)^4 -{11\over30}(x+1)^5+{29\over144}(x+1)^6-{101\over840}(x+1)^7+\cdots}\)

\(y=\dst{1-2(x-1)-3(x-1)^2+8(x-1)^3-4(x-1)^4-{42\over5}(x-1)^5 +19(x-1)^6-{604\over35}(x-1)^7+\cdots}\)

\(y=\dst{2-7x-4x^2-{17\over6}x^3-{3\over4}x^4-{9\over40}x^5+\cdots}\)

\(y=\dst{1-2(x-1)+{1\over2}(x-1)^2-{1\over6}(x-1)^3+{5\over36}(x-1)^4 -{73\over1080}(x-1)^5+\cdots}\)

\(y=\dst{2-(x+2)-{7\over2}(x+2)^2+{4\over3}(x+2)^3-{1\over24}(x+2)^4 +{1\over60}(x+2)^5+\cdots}\)

\(y=\dst{2-2(x+3)-(x+3)^2+(x+3)^3-{11\over12}(x+3)^4+ {67\over60}(x+3)^5+\cdots}\)

\(y=\dst{-1+2x+{1\over3}x^3-{5\over12}x^4+{2\over5}x^5+\cdots}\)

\(y=\dst{2-3(x+1)+{7\over2}(x+1)^2-5(x+1)^3+{197\over24}(x+1)^4 -{287\over20}(x+1)^5+\cdots}\)

\(y=\dst{-2+3(x+2)-{9\over2}(x+2)^2+{11\over6}(x+2)^3+{5\over24}(x+2)^4 +{7\over20}(x+2)^5+\cdots}\)

\(y=\dst{2-4(x-2)-{1\over2}(x-2)^2+{2\over9}(x-2)^3+{49\over432}(x-2)^4 +{23\over1080}(x-2)^5+\cdots}\)

\(y=\dst{1+2(x+4)-{1\over6}(x+4)^2-{10\over27}(x+4)^3+{19\over648}(x+4)^4 +{13\over324}(x+4)^5+\cdots}\)

\(y=\dst{-1+2(x+1)-{1\over4}(x+1)^2+{1\over2}(x+1)^3-{65\over96}(x+1)^4 +{67\over80}(x+1)^5+\cdots}\)

(a) \(y=\dst{{c_1\over1+x}+{c_2\over1+2x}}\) (b) \(y=\dst{{c_1\over1-2x}+{c_2\over1-3x}}\) (c) \(y=\dst{{c_1\over1-2x}+{c_2x\over(1-2x)^2}}\)

     (d) \(y=\dst{{c_1\over2+x}+{c_2x\over(2+x)^2}}\) (e) \(y=\dst{{c_1\over2+x}+{c_2\over2+3x}}\)

\(y=\dst{1-2x-{3\over2}x^2+{5\over3}x^3+{17\over24}x^4-{11\over20}x^5+\cdots}\)

\(y=\dst{1-2x-{5\over2}x^2+{2\over3}x^3-{3\over8}x^4+{1\over3}x^5+\cdots}\)

\(y=\dst{6-2x+9x^2+{2\over3}x^3-{23\over4}x^4-{3\over10}x^5+\cdots}\)

\(y=\dst{2-5x+2x^2-{10\over3}x^3+{3\over2}x^4-{25\over12}x^5+\cdots}\)

\(y=\dst{3+6x-3x^2+x^3-2x^4-{17\over20}x^5+\cdots}\)

\(y=\dst{3-2x-3x^2+{3\over2}x^3+{3\over2}x^4-{49\over80}x^5+\cdots}\)

\(y=\dst{-2+3x+{4\over3}x^2-x^3-{19\over54}x^4+{13\over60}x^5+\cdots}\)

\(\dst{y_1=\sum^\infty_{m=0} {(-1)^mx^{2m}\over m!}=e^{-x^2}, \quad y_2=\sum^\infty_{m=0} {(-1)^mx^{2m+1}\over m!}=xe^{-x^2}}\)

\(y=\dst{-2+3x+x^2-{1\over6}x^3-{3\over4}x^4+{31\over120}x^5+\cdots}\)

\(y=\dst{2+3x-{7\over2}x^2-{5\over6}x^3+{41\over24}x^4+{41\over120}x^5+\cdots}\)

\(y=\dst{-3+5x-5x^2+{23\over6}x^3-{23\over12}x^4+{11\over30}x^5+\cdots}\)

\(y=\dst{-2+3(x-1)+{3\over2}(x-1)^2-{17\over12}(x-1)^3-{1\over12}(x-1)^4+ {1\over8}(x-1)^5+\cdots}\)

\(y=\dst{2-3(x+2)+{1\over2}(x+2)^2-{1\over3}(x+2)^3+{31\over24}(x+2)^4- {53\over120}(x+2)^5+\cdots}\)

\(y=\dst{1-2x+{3\over2}x^2-{11\over6}x^3+{15\over8}x^4-{71\over60}x^5+\cdots}\)

\(y=\dst{2-(x+2)-{7\over2}(x+2)^2-{43\over6}(x+2)^3-{203\over24}(x+2)^4-{167 \over30}(x+2)^5+\cdots}\)

\(y=\dst{2-x-x^2+{7\over6}x^3-x^4+{89\over120}x^5+\cdots}\)

\(y=\dst{1+{3\over2}(x-1)^2+{1\over6}(x-1)^3-{1\over8}(x-1)^5+\cdots}\)

\(y=\dst{1-2(x-3)+{1\over2}(x-3)^2-{1\over6}(x-3)^3+{1\over4}(x-3)^4 -{1\over6}(x-3)^5+\cdots}\)

\(y=c_1x^{-4}+c_2x^{-2}\)

\(y=c_1x+c_2x^7\)

\(y=x(c_1+c_2 \ln x)\)

\(y=x^{-2}(c_1+c_2 \ln x)\)

\(y=c_1 \cos (\ln x)+c_2 \sin (\ln x)\)

\(y=x^2[c_1 \cos (3 \ln x)+c_2 \sin (3 \ln x)]\)

\(y=\dst{c_1x+{c_2\over x^3}}\)

\(y=c_1x^{2/3}+c_2 x^{3/4}\)

\(y=x^{-1/2} (c_1+c_2 \ln x)\)

\(y=c_1x+c_2x^{1/3}\)

\(y=c_1x^2+c_2 x^{1/2}\)

\(y=\dst{ {1\over x}\left[c_1\cos(2 \ln x)+c_2\sin(2 \ln x\right]}\)

\(y=x^{-1/3} (c_1+c_2 \ln x)\)

\(y=x\left[c_1\cos(3 \ln x)+c_2\sin(3 \ln x)\right]\)

\(y=\dst{c_1x^3+{c_2\over x^2}}\)

\(y=\dst{{c_1\over x}+c_2 x^{1/2}}\)

\(y=x^2(c_1+c_2 \ln x)\)

\(y=\dst{{1\over x^2} \left[c_1\cos\left({1\over\sqrt{2}} \ln x\right)+c_2\sin \left({1\over\sqrt{2}} \ln x\right) \right]}\)

\(y_1=\dst{x^{1/2}\left(1-{1\over5}x-{2\over35}x^2 +{31\over315}x^3+\cdots\right)}\)   \(y_2=\dst{x^{-1}\left(1+x+{1\over2}x^2-{1\over6}x^3+\cdots\right)}\);  

\(y_1=\dst{x^{1/3}\left(1-{2\over3}x+{8\over9}x^2- {40\over81}x^3+\cdots\right)}\);   \(y_2=\dst{1-x+{6\over5}x^2-{4\over5}x^3+\cdots}\)

\(y_1=\dst{x^{1/3}\left(1-{4\over7}x-{7\over45}x^2+ {970\over2457}x^3+\cdots\right)}\);   \(y_2=\dst{x^{-1}\left(1-x^2+{2\over3}x^3+\cdots\right)}\)

\(y_1=\dst{x^{1/4}\left(1-{1\over2}x-{19\over104}x^2+ {1571\over10608}x^3+\cdots\right)}\);   \(y_2=\dst{x^{-1}\left(1+2x-{11\over6}x^2-{1\over7}x^3+\cdots\right)}\)

\(y_1=\dst{x^{1/3}\left(1-x+{28\over31}x^2-{1111\over1333}x^3 +\cdots\right)}\);   \(y_2=\dst{x^{-1/4}\left(1-x+{7\over8}x^2-{19\over24}x^3+ \cdots\right)}\);  

\(y_1=\dst{x^{1/5}\left(1-{6\over25}x-{1217\over625}x^2+ {41972\over46875}x^3 +\cdots\right)}\);   \(y_2=\dst{x-{1\over4}x^2-{35\over18}x^3+{11\over12}x^4+\cdots}\)

\(y_1=\dst{x^{3/2}\left(1-x+{11\over26}x^2-{109\over1326}x^3+ \cdots\right)}\);   \(y_2=\dst{x^{1/4}\left(1+4x-{131\over24}x^2+{39\over14}x^3+ \cdots\right)}\)

\(y_1=\dst{x^{1/3}\left(1-{1\over3}x+{2\over15}x^2- {5\over63}x^3+\cdots\right)}\);   \(y_2=\dst{x^{-1/6}\left(1-{1\over12}x^2+{1\over18}x^3+\cdots\right)}\)

\(y_1=\dst{1-{1\over14}x^2+{1\over105}x^3+\cdots}\);   \(y_2=\dst{x^{-1/3}\left(1-{1\over18}x-{71\over405}x^2+ {719\over34992}x^3+\cdots\right)}\)

\(y_1=\dst{x^{1/5}\left(1+{3\over17}x-{7\over153}x^2- {547\over5661}x^3+\cdots\right)}\);   \(y_2=\dst{x^{-1/2}\left(1+x+{14\over13}x^2-{556\over897}x^3+\cdots\right)}\)

\(y_1=\dst{x^{1/2}\sum_{n=0}^\infty{(-2)^n\over\prod_{j=1}^n(2j+3)}x^n}\); \(y_2=\dst{x^{-1}\sum_{n=0}^\infty{(-1)^n\over n!}x^n}\)

\(y_1=\dst{x^{1/3}\sum_{n=0}^\infty{(-1)^n\prod_{j=1}^n(3j+1)\over9^nn!}x^n}\); \(x^{-1}\)

\(y_1=\dst{x^{1/2}\sum_{n=0}^\infty {(-1)^n\over2^nn!}x^n}\); \(y_2=\dst{{1\over x^2}\sum_{n=0}^\infty{(-1)^n\over\prod_{j=1}^n(2j-5)} x^n}\)

\(y_1=\dst{x\sum_{n=0}^\infty{(-1)^n\over\prod_{j=1}^n(3j+4)}x^n}\); \(y_2=\dst{x^{-1/3}\sum_{n=0}^\infty{(-1)^n\over3^nn!}x^n}\)

\(y_1=\dst{x\sum_{n=0}^\infty{2^n\over n!\prod_{j=1}^n(2j+1)}x^n}\); \(y_2=\dst{x^{1/2}\sum_{n=0}^\infty{2^n\over n!\prod_{j=1}^n(2j-1)}x^n}\)

\(y_1=\dst{x^{1/3}\sum_{n=0}^\infty{1\over n!\prod_{j=1}^n(3j+2)} x^n}\); \(y_2=\dst{x^{-1/3}\sum_{n=0}^\infty{1\over n!\prod_{j=1}^n(3j-2)} x^n}\)

\(y_1=\dst{x\left(1+{2\over7}x+{1\over70}x^2\right)}\); \(y_2=\dst{x^{-1/3}\sum_{n=0}^\infty{(-1)^n\over3^nn!}\left(\prod_{j=1}^n {3j-13\over3j-4}\right) x^n}\)

\(y_1=\dst{x^{1/2}\sum_{n=0}^\infty(-1)^n\left(\prod_{j=1}^n{2j+1 \over6j+1}\right); x^n}\) \(y_2=\dst{x^{1/3}\sum_{n=0}^\infty{(-1)^n\over9^nn!} \left(\prod_{j=1}^n(3j+1)\right)x^n}\)

\(y_1=\dst{x\sum_{n=0}^\infty{(-1)^n(n+2)!\over2\prod_{j=1}^n(4j+3)}; x^n}\) \(y_2=\dst{x^{1/4}\sum_{n=0}^\infty{(-1)^n\over16^nn!}\prod_{j=1}^n(4j+5) x^n}\)

\(y_1=\dst{x^{-1/2}\sum_{n=0}^\infty{(-1)^n\over n!\prod_{j=1}^n(2j+1)} x^n}\); \(y_2=\dst{x^{-1}\sum_{n=0}^\infty{(-1)^n\over n!\prod_{j=1}^n(2j-1)} x^n}\)

\(y_1=\dst{x^{1/3}\sum_{n=0}^\infty{(-1)^n\over n!}\left(2\over9\right)^n\left(\prod_{j=1}^n(6j+5)\right) x^n}\); \(y_2=\dst{x^{-1}\sum_{n=0}^\infty(-1)^n2^n\left(\prod_{j=1}^n {2j-1\over3j-4}\right) x^n}\)

\(y_1=4\dst{x^{1/3}\sum_{n=0}^\infty{1\over6^nn!(3n+4)} x^n}\); \(x^{-1}\)

\(y_1=\dst{x^{1/2}\left(1-{9\over40}x+{5\over128}x^2-{245\over39936}x^3 +\cdots\right)}\);   \(y_2=\dst{x^{1/4}\left(1-{25\over96}x+{675\over14336}x^2- {38025\over5046272}x^3 +\cdots\right)}\)

\(y_1=\dst{x^{1/3}\left(1+{32\over117}x-{28\over1053}x^2+ {4480\over540189}x^3+\cdots\right)}\);   \(y_2=\dst{x^{-3}\left(1+{32\over7}x+{48\over7}x^2\right)}\)

\(y_1=\dst{x^{1/2}\left(1-{5\over8}x+{55\over96}x^2 -{935\over1536}x^3+\cdots\right)}\);   \(y_2=\dst{x^{-1/2}\left(1+{1\over4}x-{5\over32}x^2 -{55\over384}x^3+\cdots\right)}\).

\(y_1=\dst{x^{1/2}\left(1-{3\over4}x+{5\over96}x^2+{5\over4224}x^3 +\cdots\right)}\);   \(y_2=\dst{x^{-2}\left(1+8x+60x^2-160x^3+\cdots\right)}\)

\(y_1=\dst{x^{-1/3}\left(1-{10\over63}x+{200\over7371}x^2- {17600\over3781323}x^3; +\cdots\right)}\);   \(y_2=\dst{x^{-1/2}\left(1-{3\over20}x+{9\over352}x^2 -{105\over23936}x^3 +\cdots\right)}\)

\(y_1=\dst{x^{1/2}\sum_{m=0}^\infty{(-1)^m\over8^mm!}\left(\prod_{j=1}^m{ 4j-3\over8j+1}\right)x^{2m}}\); \(y_2=\dst{x^{1/4}\sum_{m=0}^\infty{(-1)^m\over16^mm!}\left(\prod_{j=1}^m{ 8j-7\over8j-1}\right)x^{2m}}\)

\(y_1=\dst{x^{1/2}\sum_{m=0}^\infty\left(\prod_{j=1}^m{8j-3\over8j+ 1}\right) x^{2m}}\); \(y_2=\dst{x^{1/4}\sum_{m=0}^\infty{1\over2^mm!}\left(\prod_{j=1}^m(2j-1) \right)x^{2m}}\)

\(y_1=\dst{x^4\sum_{m=0}^\infty(-1)^m(m+1)x^{2m}}\); \(y_2=-\dst{x\sum_{m=0}^\infty(-1)^m(2m-1)x^{2m}}\)

\(y_1=\dst{x^{1/3}\sum_{m=0}^\infty{(-1)^m\over18^mm!}\left(\prod_{j=1}^m (6j-17)\right) x^{2m}}\); \(y_2=\dst{1+{4\over5}x^2+{8\over55}x^4}\)

\(y_1=\dst{x^{1/4}\sum_{m=0}^\infty\left(\prod_{j=1}^m{8j+1\over8j+5} \right)x^{2m}}\); \(y_2=\dst{x^{-1}\sum_{m=0}^\infty{\prod_{j=1}^m(2j-1)\over2^mm!} x^{2m}}\)

\(y_1=\dst{x^{1/2}\sum_{m=0}^\infty{1\over8^mm!}\left(\prod_{j=1}^m(4j-1) \right)x^{2m}}\); \(y_2=\dst{x^{1/3}\sum_{m=0}^\infty2^m\left(\prod_{j=1}^m{3j-1\over12j-1} \right) x^{2m}}\)

\(y_1=\dst{x^{7/2}\sum_{m=0}^\infty(-1)^m{\prod_{j=1}^m(4j+5)\over8^mm!}x^{2m}}\); \(y_2=\dst{x^{1/2}\sum_{m=0}^\infty{(-1)^m\over4^m}\left(\prod_{j=1}^m {4j-1\over2j-3} \right)x^{2m}}\)

\(y_1=\dst{x^{1/2}\sum_{m=0}^\infty{(-1)^m\over4^m}\left(\prod_{j=1}^m {4j-1\over2j+1}\right)x^{2m}}\); \(y_2=\dst{x^{-1/2}\sum_{m=0}^\infty{(-1)^m\over8^mm!}\left(\prod_{j=1}^m (4j-3)\right)x^{2m}}\)

\(y_1=\dst{x^{1/2}\sum_{m=0}^\infty{(-1)^m\over m!}\left(\prod_{j=1}^m(2j+1)\right) x^{2m}}\); \(y_2=\dst{{1\over x^2}\sum_{m=0}^\infty(-2)^m\left(\prod_{j=1}^m{4j-3\over 4j-5}\right) x^{2m}}\)

\(y_1=\dst{x^{1/3}\sum_{m=0}^\infty(-1)^m\left(\prod_{j=1}^m {3j-4\over3j+2}\right) x^{2m}}\); \(y_2=\dst{x^{-1}(1+x^2)}\)

\(y_1=\dst{\sum_{m=0}^\infty(-1)^m{2^m(m+1)!\over\prod_{j=1}^m(2j+3)}x^{2m}}\);   \(y_2=\dst{{1\over x^3 }\sum_{m=0}^\infty(-1)^m{\prod_{j=1}^m(2j-1)\over2^mm!}x^{2m}}\)

\(y_1=\dst{x^{1/2}\sum_{m=0}^\infty{(-1)^m\over8^mm!}\left(\prod_{j=1}^m {(4j-3)^2\over4j+3}\right)x^{2m}}\); \(y_2=\dst{x^{-1}\sum_{m=0}^\infty{(-1)^m\over2^mm!}\left(\prod_{j=1}^m {(2j-3)^2\over4j-3}\right) x^{2m}}\)

\(y_1=\dst{x\sum_{m=0}^\infty(-2)^m\left(\prod_{j=1}^m{2j+1\over 4j+5}\right) x^{2m}}\); \(y_2=\dst{x^{-3/2}\sum_{m=0}^\infty{(-1)^m\over4^mm!}\left(\prod_{j=1}^m(4j-3) \right)x^{2m}}\)

\(y_1=\dst{x^{1/3}\sum_{m=0}^\infty{(-1)^m\over2^m\prod_{j=1}^m(3j+1)} x^{2m}}\); \(y_2=\dst{x^{-1/3}\sum_{m=0}^\infty{(-1)^m\over6^mm!} x^{2m}}\)

\(y_1=\dst{x^{1/2}\left(1-{6\over13}x^2+{36\over325}x^4- {216\over12025}x^6 +\cdots\right)}\);   \(y_2=\dst{x^{1/3}\left(1-{1\over2}x^2+{1\over8}x^4-{1\over48}x^6 +\cdots\right)}\)

\(y_1=\dst{x^{1/4}\left(1-{13\over64}x^2+{273\over8192}x^4- {2639\over524288}x^6 +\cdots\right)}\);   \(y_2=\dst{x^{-1}\left(1-{1\over3}x^2+{2\over33}x^4-{2\over209}x^6 +\cdots\right)}\)

\(y_1=\dst{x^{1/3}\left(1-{3\over4}x^2+{9\over14}x^4-{81\over140}x^6 +\cdots\right)}\);   \(y_2=\dst{x^{-1/3}\left(1-{2\over3}x^2+{5\over9}x^4- {40\over81}x^6+\cdots\right)}\)

\(y_1=\dst{x^{1/2}\left(1-{3\over2}x^2+{15\over8}x^4-{35\over16}x^6 +\cdots\right)}\);   \(y_2=\dst{x^{-1/2}\left(1-2x^2+{8\over3}x^4-{16\over5}x^6 +\cdots\right)}\)

\(y_1=\dst{x^{1/4}\left(1-x^2+{3\over2}x^4-{5\over2}x^6 +\cdots\right)}\);   \(y_2=\dst{x^{-1/2}\left(1-{2\over5}x^2+{36\over65}x^4- {408\over455}x^6+\cdots\right)}\)

(a) \(y_1=\dst{x^\nu\sum_{m=0}^\infty{(-1)^m\over4^mm!\prod_{j=1}^m(j+\nu)}x^{2m}}\); \(y_2=\dst{x^{-\nu}\sum_{m=0}^\infty{(-1)^m\over4^mm!\prod_{j=1}^m(j-\nu)}x^{2m}}\)

     \(y_1=\dst{\sin x\over\sqrt x}\); \(y_2=\dst{\cos x\over\sqrt x}\)

\(y_1=\dst{x^{1/2}\over1+x}\); \(y_2=\dst{x\over1+x}\)

\(y_1=\dst{x^{1/3}\over1+2x^2}\); \(y_2=\dst{x^{1/2}\over1+2x^2}\)

\(y_1=\dst{x^{1/4}\over1-3x}\); \(y_2=\dst{x^2\over1-3x}\)

\(y_1=\dst{x^{1/3}\over5+x}\); \(y_2=\dst{x^{-1/3}\over5+x}\)

\(y_1=\dst{x^{1/4}\over2-x^2}\); \(y_2=\dst{x^{-1/2}\over2-x^2}\)

\(y_1=\dst{x^{1/2}\over1+3x+x^2}\); \(y_2=\dst{x^{3/2}\over1+3x+x^2}\)

\(y_1=\dst{x\over(1+x)^2}\); \(y_2=\dst{x^{1/3}\over(1+x)^2}\)

\(y_1=\dst{x\over3+2x+x^2}\); \(y_2=\dst{x^{1/4}\over3+2x+x^2}\)

\(\dst{y_1=x\left(1-x+{3\over4}x^2-{13\over36}x^3+\cdots\right)}\);  \(\dst{y_2=y_1 \ln x + x^2\left(1-x+{65\over108}x^2+ \cdots\right)}\)

\(\dst{y_1=x^{-1}\left(1-2x+{9\over2}x^2-{20\over3}x^3+\cdots\right)}\);  \(\dst{y_2= y_1\ln x+1-{15\over4}x+{133\over18}x^2+\cdots}\)

\(\dst{y_1=1+x-x^2+{1\over3}x^3+\cdots}\);    \(\dst{y_2= y_1\ln x-x\left(3-{1\over2}x-{31\over18}x^2+\cdots\right)}\)

\(\dst{y_1=x^{1/2}\left(1-2x+{5\over2}x^2-2x^3+\cdots\right)}\);  \(\dst{y_2= y_1\ln x+x^{3/2}\left(1-{9\over4}x+{17\over6}x^2+ \cdots\right)}\)

\(\dst{y_1=x\left(1-4x+{19\over2}x^2-{49\over3}x^3+\cdots\right)}\);  \(\dst{y_2= y_1\ln x+x^2\left(3-{43\over4}x+{208\over9}x^2+\cdots \right)}\)

\(\dst{y_1=x^{-1/3}\left(1-x+{5\over6}x^2-{1\over2}x^3+\cdots\right)}\);  \(\dst{y_2= y_1\ln x+x^{2/3}\left(1-{11\over12}x+{25\over36}x^2+ \cdots\right)}\)

\(\dst{y_1=1-2x+{7\over4}x^2-{7\over9}x^3+\cdots}\);    \(\dst{y_2= y_1\ln x+x\left(3-{15\over4}x+{239\over108}x^2+\cdots \right)}\)

\(\dst{y_1=x^{-2}\left(1-2x+{5\over2}x^2-3x^3+\cdots\right)}\);  \(\dst{y_2=y_1\ln x+{3\over4}-{13\over6}x+\cdots}\)

\(\dst{y_1=x^{-1/2}\left(1-x+{1\over4}x^2+{1\over18}x^3+\cdots\right)}\);  \(\dst{y_2=y_1\ln x+x^{1/2}\left({3\over2}-{13\over16}x +{1\over54}x^2+\cdots\right)}\)

\(\dst{y_1=x^{-1/4}\left(1-{1\over4}x-{7\over32}x^2+{23\over384}x^3 +\cdots\right)}\);  \(\dst{y_2= y_1\ln x+x^{3/4}\left({1\over4}+{5\over64}x-{157\over2304}x^2 +\cdots\right)}\)

\(\dst{y_1=x^{-1/3}\left(1-x+{7\over6}x^2-{23\over18}x^3+\cdots\right)}\);  \(\dst{y_2= y_1\ln x-x^{5/3}\left({1\over12}-{13\over108}x\cdots\right)}\)

\(\dst{y_1=x^{1/2}\sum_{n=0}^\infty {(-1)^n\over(n!)^2}x^n}\);  \(\dst{y_2=y_1\ln x-2x^{1/2}\sum_{n=1}^\infty {(-1)^n\over(n!)^2}\left(\sum_{j=1}^n{1\over j}\right)x^n}\); 

\(\dst{y_1=x^{1/6}\sum_{n=0}^\infty\left(2\over3\right)^n {\prod_{j=1}^n(3j+1)\over n!}x^n}\); 

       \(\dst{y_2=y_1\ln x-x^{1/6}\sum_{n=1}^\infty \left(2\over3\right)^n {\prod_{j=1}^n(3j+1)\over n!}\left(\sum_{j=1}^n{1\over j(3j+1)}\right)x^n}\)

\(\dst{y_1=x^2\sum_{n=0}^\infty} (-1)^n(n+1)^2x^n\);  \(\dst{y_2=y_1\ln x-2x^2\sum_{n=1}^\infty(-1)^nn(n+1)x^n}\)

\(\dst{y_1=x^3\sum_{n=0}^\infty 2^n(n+1)x^n}\);  \(\dst{y_2= y_1\ln x-x^3\sum_{n=1}^\infty2^nnx^n}\)

\(\dst{y_1=x^{1/5}\sum_{n=0}^\infty{(-1)^n\prod_{j=1}^n(5j+1)\over 125^n(n!)^2} x^n}\); 

       \(\dst{y_2= y_1\ln x-x^{1/5}\sum_{n=1}^\infty{(-1)^n\prod_{j=1}^n(5j+1)\over125^n(n!)^2} \left(\sum_{j=1}^n{5j+2\over j(5j+1)}\right)x^n}\)

\(\dst{y_1=x^{1/2}\sum_{n=0}^\infty{(-1)^n\prod_{j=1}^n(2j-3)\over4^nn!} x^n}\); 

       \(\dst{y_2=y_1\ln x+3x^{1/2}\sum_{n=1}^\infty {(-1)^n\prod_{j=1}^n(2j-3)\over4^nn!}\left(\sum_{j=1}^n{1\over j(2j-3)}\right)x^n}\)

\(\dst{y_1=x^{1/3}\sum_{n=0}^\infty{(-1)^n\prod_{j=1}^n(6j-7)^2\over 81^n(n!)^2} x^n}\); 

       \(\dst{y_2=y_1\ln x+14x^{1/3}\sum_{n=1}^\infty{(-1)^n\prod_{j=1}^n(6j-7)^2\over 81^n(n!)^2}\left(\sum_{j=1}^n{1\over j(6j-7)})\right)x^n}\)

\(\dst{y_1=x^2\sum_{n=0}^\infty{(-1)^n\prod_{j=1}^n(2j+5)\over(n!)^2} x^n}\); 

       \(\dst{y_2= y_1\ln x-2x^2\sum_{n=1}^\infty{(-1)^n\prod_{j=1}^n(2j+5)\over(n!)^2} \left(\sum_{j=1}^n{(j+ 5)\over j(2j+ 5)}\right)x^n}\)

\(\dst{y_1={1\over x}\sum_{n=0}^\infty {2^n\prod_{j=1}^n(2j-1)\over n!} x^n}\); 

       \(\dst{y_2=y_1\ln x+{1\over x}\sum_{n=1}^\infty{2^n\prod_{j=1}^n(2j-1)\over n!} \left(\sum_{j=1}^n{1\over j(2j-1)}\right)x^n}\)

\(\dst{y_1={1\over x}\sum_{n=0}^\infty{(-1)^n\prod_{j=1}^n(2j-5)\over n!}x^n}\); 

       \(\dst{y_2=y_1\ln x+{5\over x}\sum_{n=1}^\infty{(-1)^n\prod_{j=1}^n(2j-5)\over n!} \left(\sum_{j=1}^n{1\over j(2j-5)}\right)x^n}\)

\(\dst{y_1=x^2\sum_{n=0}^\infty {(-1)^n\prod_{j=1}^n(2j+3)\over2^nn!}x^n}\); 

       \(\dst{y_2=y_1\ln x-3x^2\sum_{n=0}^\infty {(-1)^n\prod_{j=1}^n(2j+3)\over2^nn!} \left(\sum_{j=1}^n{1\over j(2j+ 3)}\right)x^n}\) 

\(\dst{y_1=x^{-2}\left(1+3x+{3\over2}x^2-{1\over2}x^3+\cdots\right)}\);  \(\dst{y_2=y_1\ln x-5x^{-1}\left(1+{5\over4}x-{1\over4}x^2+\cdots\right)}\)

\(\dst{y_1=x^3(1+20x+180x^2+1120x^3+\cdots}\);  \(\dst{y_2=y_1\ln x-x^4\left(26+324x+{6968\over3}x^2+\cdots\right)}\)

\(\dst{y_1=x\left(1-5x+{85\over4}x^2-{3145\over36}x^3+\cdots\right)}\);  \(\dst{y_2= y_1\ln x+x^2\left(2-{39\over4}x+{4499\over108}x^2+\cdots\right)}\)

\(\dst{y_1=1-x+{3\over4}x^2-{7\over12}x^3+\cdots}\);  \(\dst{y_2=y_1\ln x+x\left(1-{3\over4}x+{5\over9}x^2+\cdots\right)}\)

\(\dst{y_1=x^{-3}(1+16x+36x^2+16x^3+\cdots)}\);  \(\dst{y_2=y_1\ln x-x^{-2}\left(40+150x+{280\over3}x^2+\cdots\right)}\)

\(\dst{y_1=x\sum_{m=0}^\infty{(-1)^m\over2^mm!}x^{2m}}\);  \(\dst{y_2=y_1\ln x-{x\over2}\sum_{m=1}^\infty{(-1)^m\over 2^mm!}\left(\sum_{j=1}^m{1\over j}\right)x^{2m}}\)

\(\dst{y_1=x^2\sum_{m=0}^\infty(-1)^m(m+1)x^{2m}}\);  \(\dst{y_2=y_1\ln x-{x^2\over2}\sum_{m=1}^\infty(-1)^mmx^{2m}}\)

\(\dst{y_1=x^{1/2}\sum_{m=0}^\infty{(-1)^m\over4^mm!}x^{2m}}\);  \(\dst{y_2=y_1\ln x-{x^{1/2}\over2}\sum_{m=1}^\infty{(-1)^m\over 4^mm!}\left(\sum_{j=1}^m{1\over j}\right) x^{2m}}\)

\(\dst{y_1=x\sum_{m=0}^\infty{(-1)^m\prod_{j=1}^m(2j-1)\over2^mm!} x^{2m}}\); 

       \(\dst{y_2=y_1\ln x+{x\over2}\sum_{m=1}^\infty{(-1)^m\prod_{j=1}^m(2j-1)\over2^mm!} \left(\sum_{j=1}^m{1\over j(2j-1)}\right)x^{2m}}\)

\(\dst{y_1=x^{1/2}\sum_{m=0}^\infty{(-1)^m\prod_{j=1}^m(4j-1)\over8^mm!} x^{2m}}\); 

       \(\dst{y_2=y_1\ln x+{x^{1/2}\over2}\sum_{m=1}^\infty{(-1)^m\prod_{j=1}^m(4j-1)\over8^mm!} \left(\sum_{j=1}^m{1\over j(4j-1)}\right)x^{2m}}\)

\(\dst{y_1=x\sum_{m=0}^\infty{(-1)^m\prod_{j=1}^m(2j+1)\over2^mm!} x^{2m}}\); 

       \(\dst{y_2=y_1\ln x-{x\over2}\sum_{m=1}^\infty{(-1)^m\prod_{j=1}^m(2j+1)\over2^mm!} \left(\sum_{j=1}^m{1\over j(2j+ 1)}\right)x^{2m}}\)

\(\dst{y_1=x^{-1/4}\sum_{m=0}^\infty{(-1)^m\prod_{j=1}^m(8j-13)\over(32)^mm!} x^{2m}}\); 

       \(\dst{y_2=y_1\ln x+{13\over2}x^{-1/4}\sum_{m=1}^\infty{(-1)^m\prod_{j=1}^m(8j-13)\over(32)^mm!} \left(\sum_{j=1}^m{1\over j(8j-13)}\right)x^{2m}}\)

\(\dst{y_1=x^{1/3}\sum_{m=0}^\infty{(-1)^m\prod_{j=1}^m(3j-1)\over9^mm!} x^{2m}}\); 

       \(\dst{y_2=y_1\ln x+{x^{1/3}\over2}\sum_{m=1}^\infty{(-1)^m\prod_{j=1}^m(3j-1)\over9^mm!} \left(\sum_{j=1}^m{1\over j(3j-1)}\right)x^{2m}}\)

\(\dst{y_1=x^{1/2}\sum_{m=0}^\infty{(-1)^m\prod_{j=1}^m(4j-3)(4j-1)\over4^m(m!)^2} x^{2m}}\); 

       \(\dst{y_2=y_1\ln x+x^{1/2}\sum_{m=1}^\infty{(-1)^m\prod_{j=1}^m(4j-3)(4j-1)\over4^m(m!)^2} \left(\sum_{j=1}^m{8j-3\over j(4j-3) (4j-1)}\right)x^{2m}}\)

\(\dst{y_1=x^{5/3}\sum_{m=0}^\infty{(-1)^m\over3^mm!}x^{2m}}\);  \(\dst{y_2=y_21\ln x-{x^{5/3}\over2}\sum_{m=1}^\infty{(-1)^m\over3^mm!}\left(\sum_{j=1}^m{1\over j}\right)x^{2m}}\) 

\(\dst{y_1={1\over x }\sum_{m=0}^\infty{(-1)^m\prod_{j=1}^m(4j-7)\over2^mm!} x^{2m}}\); 

       \(\dst{y_2=y_1\ln x+{7\over2x}\sum_{m=1}^\infty{(-1)^m\prod_{j=1}^m(4j-7)\over2^mm!} \left(\sum_{j=1}^m{1\over j(4j-7)}\right)x^{2m}}\)

\(\dst{y_1=x^{-1}\left(1-{3\over2}x^2+{15\over8}x^4-{35\over16}x^6+\cdots\right)}\)

 ;      \(\dst{y_2=y_1\ln x+x\left({1\over4}-{13\over32}x^2+{101\over192}x^4 +\cdots\right)}\)

\(\dst{y_1=x\left(1-{1\over2}x^2+{1\over8}x^4-{1\over48}x^6+\cdots\right)}\);  \(\dst{y_2=y_1 \ln x+x^3\left({1\over4}-{3\over32}x^2+ {11\over576}x^4+\cdots\right)}\)

\(\dst{y_1=x^{-2}\left(1-{3\over4}x^2-{9\over64}x^4-{25\over256}x^6 +\cdots\right)}\);  \(\dst{y_2=y_1\ln x+{1\over2}-{21\over128}x^2-{215\over1536}x^4+\cdots}\)

\(\dst{y_1=x^{-3}\left(1-{17\over8}x^2+{85\over256}x^4-{85\over18432}x^6+\cdots\right)}\);  \(\dst{y_2=y_1\ln x +x^{-1}\left({25\over8}-{471\over512}x^2+{1583\over110592}x^4 +\cdots\right)}\)

\(\dst{y_1=x^{-1}\left(1-{3\over4}x^2+{45\over64}x^4-{175\over256}x^6+\cdots \right)}\);  \(\dst{y_2=y_1\ln x-x\left({1\over4}-{33\over128}x^2+{395\over1536}x^4 +\cdots\right)}\)

\(\dst{y_1={1\over x}}\);  \(\dst{y_2=y_1\ln x-6+6x-{8\over3}x^2}\)

\(\dst{y_1=1-x}\);  \(\dst{y_2=y_1\ln x+4x}\)

\(\dst{y_1={(x-1)^2\over x}}\);  \(\dst{y_2=y_1\ln x+3-3x+2\sum_{n=2}^\infty {1\over n(n^2-1)}x^n}\)

\(\dst{y_1=x^{1/2}(x+1)^2}\);  \(\dst{y_2=y_1\ln x-x^{3/2}\left( 3+3x+2\sum_{n=2}^\infty{(-1)^n\over n(n^2-1)}x^n\right)}\)

\(\dst{y_1=x^2(1-x)^3}\);  \(\dst{y_2=y_1\ln x+x^3\left(4-7x+{11\over3}x^2-6\sum_{n=3}^\infty{1\over n(n-2)(n^2-1)}x^n\right)}\)

\(\dst{y_1=x-4x^3+x^5}\);  \(\dst{y_2=y_1\ln x+6x^3-3x^5}\)

\(\dst{y_1=x^{1/3}\left(1-{1\over6}x^2\right)}\);  \(\dst{y_2=y_1\ln x +x^{7/3}\left({1\over4}-{1\over12}\sum_{m=1}^\infty {1\over6^mm(m+1)(m+1)!}x^{2m}\right)}\)

\(\dst{y_1=(1+x^2)^2}\);  \(\dst{y_2=y_1\ln x-{3\over2}x^2-{3\over2}x^4 +\sum_{m=3}^\infty{(-1)^m\over m(m-1)(m-2)}x^{2m}}\)

\(\dst{y_1=x^{-1/2}\left(1-{1\over2}x^2+{1\over32}x^4\right)}\);  \(\dst{y_2=y_1\ln x+x^{3/2}\left({5\over8}-{9\over128}x^2 +\sum_{m=2}^\infty{1\over4^{m+1}(m-1)m(m+1)(m+1)!}x^{2m}\right)}\).

\(\dst{y_1=\sum_{m=0}^\infty{(-1)^m\over4^m(m!)^2}x^{2m}}\);  \(\dst{y_2=y_1\ln x-\sum_{m=1}^\infty{(-1)^m\over4^m(m!)^2}\left(\sum_{j=1}^m{1\over j}\right)x^{2m}}\)

\(\dst{x^{1/2}\over1+x}\);  \(\dst{x^{1/2}\ln x\over1+x}\)

\(\dst{x^{1/3}\over3+x}\);  \(\dst{x^{1/3}\ln x\over3+x}\)

\(\dst{x\over2-x^2}\);  \(\dst{x\ln x\over2-x^2}\)

\(\dst{x^{1/4}\over1+x^2}\);  \(\dst{x^{1/4}\ln x\over1+x^2}\)

\(\dst{x\over4+3x}\);  \(\dst{x\ln x\over4+3x}\)

\(\dst{x^{1/2}\over1+3x+x^2}\);  \(\dst{x^{1/2}\ln x\over1+3x+x^2}\)

\(\dst{x\over(1-x)^2}\);  \(\dst{x\ln x\over(1-x)^2}\)

\(\dst{x^{1/3}\over1+x+x^2}\);  \(\dst{x^{1/3}\ln x\over1+x+x^2}\)

\(\dst{y_1=2x^3\sum_{n=0}^\infty {(-4)^n\over n!(n+2)!}x^n}\);  \(\dst{y_2=x+4x^2-8\left(y_1\ln x-4\sum_{n=1}^\infty{(-4)^n\over n!(n+2)!}\left(\sum_{j=1}^n{j+1\over j(j+2)}\right)x^n\right)}\)

\(\dst{y_1=x\sum_{n=0}^\infty{(-1)^n\over n!(n+1)!}x^n}\);  \(\dst{y_2=1-y_1\ln x +x\sum_{n=1}^\infty{(-1)^n\over n!(n+1)!}\left(\sum_{j=1}^n{2j+1\over j(j+1)}\right)x^n}\)

\(y_1=x^{1/2}\);  \(\dst{y_2=x^{-1/2}+y_1\ln x+x^{1/2}\sum_{n=1}^\infty{(-1)^n\over n}x^n}\)

\(\dst{y_1=x\sum_{n=0}^\infty{(-1)^n\over n!}x^n=xe^{-x}}\);  \(\dst{y_2=1-y_1\ln x+x\sum_{n=1}^\infty{(-1)^n\over n!}\left(\sum_{j=1}^n{1\over j}\right)x^n}\) 

\(\dst{y_1=x^{1/2}\sum_{n=0}^\infty\left(-{3\over4}\right)^n{\prod_{j=1}^n(2j+1) \over n!}x^n}\); 

     \(\dst{y_2=x^{-1/2}-{3\over4}\left(y_1\ln x-x^{1/2}\sum_{n=1}^\infty\left(-{3\over4}\right)^n{\prod_{j=1}^n(2j+1)\over n!}\left(\sum_{j=1}^n{1\over j(2j+1)}\right)x^n\right)}\)

\(\dst{y_1=x\sum_{n=0}^\infty {(-1)^n\over n!}x^n=xe^{-x}}\);  \(\dst{y_2=x^{-2}\left(1+{1\over2}x+{1\over2}x^2\right)-{1\over2} \left(y_1\ln x-x\sum_{n=1}^\infty{(-1)^n\over n!}\left(\sum_{j=1}^n{1\over j}\right)x^n\right)}\) 

\(\dst{y_1=6x^{3/2}\sum_{n=0}^\infty {(-1)^n\over4^nn!(n+3)!}x^n}\); 

     \(\dst{y_2=x^{-3/2}\left(1+{1\over8}x+{1\over64}x^2\right) -{1\over768}\left(y_1\ln x-6x^{3/2}\sum_{n=1}^\infty {(-1)^n\over4^nn!(n+3)!} \left(\sum_{j=1}^n{2j+3\over j(j+3)}\right)x^n\right)}\)

\(\dst{y_1={120\over x^2}\sum_{n=0}^\infty{(-1)^n\over n!(n+5)!}x^n}\); 

     \(\dst{y_2=x^{-7}\left(1+{1\over4}x+{1\over24}x^2+{1\over144}x^3 +{1\over576}x^4\right)-{1\over2880}\left(y_1\ln x-{120\over x^2} \sum_{n=1}^\infty{(-1)^n\over n!(n+5)!}\left(\sum_{j=1}^n{2j+5\over j(j+5)}\right)x^n\right)}\)

\(\dst{y_1={x^{1/2}\over6}\sum_{n=0}^\infty(-1)^n(n+1)(n+2)(n+3)x^n}\); 

     \(\dst{y_2=x^{-5/2}\left(1+{1\over2}x+x^2\right)-3y_1\ln x +{3\over2}x^{1/2}\sum_{n=1}^\infty(-1)^n(n+1)(n+2)(n+3) \left(\sum_{j=1}^n{1\over j(j+3)}\right)x^n}\)

\(\dst{y_1=x^4\left(1-{2\over5}x\right)}\)  \(\dst{y_2=1+10x+50x^2+200x^3-300\left(y_1\ln x+{27\over25}x^5-{1\over30}x^6\right)}\)

\(y_1=x^3\);  \(y_2=\dst{x^{-3}\left(1-{6\over5}x+{3\over4}x^2-{1\over3}x^3 +{1\over8}x^4-{1\over20}x^5\right) -{1\over120}\left(y_1\ln x+x^3\sum_{n=1}^\infty{(-1)^n6!\over n(n+6)!} x^n\right)}\)

\(\dst{y_1=x^2\sum_{n=0}^\infty{1\over n!}\left(\prod_{j=1}^n{2j+3\over j+4}\right)x^n}\); 

     \(\dst{y_2=x^{-2}\left(1+x+{1\over4}x^2-{1\over12}x^3\right)-{1\over16} y_1\ln x+ {x^2\over8}\sum_{n=1}^\infty{1\over n!}\left(\prod_{j=1}^n{2j+3\over j+4} \right)\left(\sum_{j=1}^n{(j^2+3j+6)\over j(j+4)(2j+3)}\right)x^n}\)

\(y_1=\dst{x^5\sum_{n=0}^\infty(-1)^n(n+1)(n+2)x^n}\);  \(y_2=\dst{1-{x\over2}+{x^2\over6}}\)

\(y_1=\dst{{1\over x}\sum_{n=0}^\infty{(-1)^n\over n!}\left(\prod_{j=1}^n{(j+3)(2j-3)\over j+6}\right)x^n}\);  \(y_2=\dst{x^{-7}\left(1+{26\over5}x+{143\over20}x^2\right)}\)

\(y_1=\dst{x^{7/2}\sum_{n=0}^\infty{(-1)^n\over2^n(n+4)!}x^n}\);  \(y_2=\dst{x^{-1/2}\left(1-{1\over2}x+{1\over8}x^2-{1\over48}x^3\right)}\)

\(y_1=\dst{x^{10/3}\sum_{n=0}^\infty{(-1)^n(n+1)\over9^n}\left(\prod_{j=1}^n {3j+7\over j+4}\right)x^n}\);  \(y_2=\dst{x^{-2/3}\left(1+{4\over27}x-{1\over243}x^2\right)}\)

\(y_1=\dst{x^3\sum_{n=0}^7(-1)^n(n+1)\left(\prod_{j=1}^n {j-8\over j+6}\right)x^n}\);  \(y_2=\dst x^{-3}\left(1+{52\over5}x+{234\over5}x^2+{572\over5}x^3+ 143x^4\right)\)

\(y_1=\dst{x^3\sum_{n=0}^\infty{(-1)^n\over n!}\left(\prod_{j=1}^n{(j+3)^2\over j+5}\right)x^n}\);    \(y_2=\dst{x^{-2}\left(1+{1\over4}x\right)}\)

\(y_1=\dst{x^6\sum_{n=0}^4(-1)^n2^n\left(\prod_{j=1}^n {j-5\over j+5}\right) x^n}\);  \(y_2=x(1+18x+144x^2+672x^3+2016x^4)\)

\(y_1=\dst{x^6\left(1+{2\over3}x+{1\over7}x^2\right)}\);  \(y_2=\dst{x\left(1+{21\over4}x+{21\over2}x^2+{35\over4}x^3\right)}\)

\(y_1=\dst{x^{7/2}\sum_{n=0}^\infty(-1)^n(n+1)x^n}\);  \(y_2=x^{-7/2} \dst\left(1-{5\over6}x+{2\over3}x^2- {1\over2}x^3+{1\over3}x^4-{1\over6}x^5\right)\)

\(y_1=\dst{{x^{10}\over6}\sum_{n=0}^\infty(-1)^n2^n(n+1)(n+2)(n+3)x^n}\); 

\(y_2=\dst\left(1-{4\over3}x+{5\over3}x^2-{40\over21}x^3 +{40\over21}x^4-{32\over21}x^5+{16\over21}x^6\right)\)

\(\dst{y_1=x^6\sum_{m=0}^\infty{(-1)^m\prod_{j=1}^m(2j+5)\over2^mm!}x^{2m}}\); 

     \(\dst{y_2=x^2\left(1+{3\over2}x^2\right)-{15\over2}y_1\ln x+ {75\over2}x^6\sum_{m=1}^\infty {(-1)^m\prod_{j=1}^m(2j+5)\over2^{m+1}m!} \left(\sum_{j=1}^m{1\over j(2j+5)}\right) x^{2m}}\)

\(\dst{y_1=x^6\sum_{m=0}^\infty{(-1)^m\over2^mm!}x^{2m}=x^6e^{-x^2/2}}\); 

     \(\dst{y_2=x^2\left(1+{1\over2}x^2\right)-{1\over2}y_1\ln x+{x^6\over4}\sum_{m=1}^\infty{(-1)^m\over2^mm!}\left(\sum_{j=1}^m{1\over j}\right)x^{2m}}\)

\(\dst{y_1=6x^6\sum_{m=0}^\infty{(-1)^m\over4^mm!(m+3)!}x^{2m}}\); 

\(\dst{y_2=1+{1\over8}x^2+{1\over64}x^4-{1\over384}\left( y_1\ln x-3x^6\sum_{m=1}^\infty{(-1)^m\over4^mm!(m+3)!} \left(\sum_{j=1}^m{2j+3\over j(j+3)}\right)x^{2m}\right)}\)

\(\dst{y_1={x\over2}\sum_{m=0}^\infty{(-1)^m(m+2)\over m!}x^{2m}}\); 

     \(\dst{y_2=x^{-1}-4y_1\ln x+x\sum_{m=1}^\infty{(-1)^m(m+2)\over m!}\left(\sum_{j=1}^m{j^2+4j+2\over j(j+1)(j+2)} \right)x^{2m}}\)

\(\dst{y_1=2x^3\sum_{m=0}^\infty{(-1)^m\over4^mm!(m+2)!}x^{2m}}\); 

     \(\dst{y_2=x^{-1}\left(1+{1\over4}x^2\right)-{1\over16}\left( y_1\ln x-2x^3\sum_{m=1}^\infty{(-1)^m\over4^mm!(m+2)!} \left(\sum_{j=1}^m{j+1\over j(j+2)} \right)x^{2m}\right)}\)

\(\dst{y_1=x^{-1/2}\sum_{m=0}^\infty{(-1)^m\prod_{j=1}^m(2j-1)\over8^mm!(m+1)!} x^{2m}}\); 

     \(\dst{y_2=x^{-5/2}+{1\over4}y_1\ln x -x^{-1/2}\sum_{m=1}^\infty{(-1)^m\prod_{j=1}^m(2j-1)\over8^{m+1}m!(m+1)!} \left(\sum_{j=1}^m{2j^2-2j-1\over j(j+1)(2j-1)}\right) x^{2m}}\)

\(\dst{y_1=x\sum_{m=0}^\infty{(-1)^m\over2^mm!}x^{2m}=xe^{-x^2/2}}\);  \(\dst{y_2=x^{-1}-y_1\ln x+{x\over2} \sum_{m=1}^\infty{(-1)^m\over2^mm!}\left(\sum_{j=1}^m{1\over j}\right)x^{2m}}\)

\(\dst{y_1=x^2\sum_{m=0}^\infty{1\over m!}x^{2m}=x^2e^{x^2}}\);  \(\dst{y_2=x^{-2}(1-x^2)-2y_1\ln x+x^2\sum_{m=1}^\infty{1\over m!}\left(\sum_{j=1}^m{1\over j}\right)x^{2m}}\)

\(\dst{y_1=6x^{5/2}\sum_{m=0}^\infty{(-1)^m\over16^mm!(m+3)!}x^{2m}}\); 

     \(\dst{y_2=x^{-7/2}\left(1+{1\over32}x^2+{1\over1024}x^4\right) -{1\over24576}\left(y_1\ln x-3x^{5/2} \sum_{m=1}^\infty{(-1)^m\over16^mm!(m+3)!} \left(\sum_{j=1}^m{2j+3\over j(j+3)}\right)x^{2m}\right)}\)

\(\dst{y_1=2x^{13/3}\sum_{m=0}^\infty{\prod_{j=1}^m(3j+1) \over9^mm!(m+2)!}x^{2m}}\); 

     \(\dst{y_2=x^{1/3}\left(1+{2\over9}x^2\right)+{2\over81}\left( y_1\ln x-x^{13/3}\sum_{m=0}^\infty{\prod_{j=1}^m(3j+1)\over9^mm!(m+2)!} \left(\sum_{j=1}^m{3j^2+2j+2\over j(j+2)(3j+1)}\right)x^{2m}\right)}\)

\(\dst{y_1=x^2}\);  \(\dst{y_2=x^{-2}(1+2x^2)-2\left(y_1\ln x+x^2\sum_{m=1}^\infty{1\over m(m+2)!}x^{2m}\right)}\)

\(\dst{y_1=x^2\left(1-{1\over2}x^2\right)}\);  \(\dst{y_2=x^{-2}\left(1+{9\over2}x^2\right)-{27\over2}\left( y_1\ln x+{7\over12}x^4-x^2\sum_{m=2}^\infty{\left(3\over2\right)^m \over m(m-1)(m+2)!}x^{2m}\right)}\)

\(y_1=\dst{\sum_{m=0}^\infty(-1)^m(m+1)x^{2m}}\);  \(y_2=\dst{x^{-4}}\)

\(y_1=\dst{x^{5/2}\sum_{m=0}^\infty{(-1)^m\over(m+1)(m+2)(m+3)}x^{2m}}\);  \(y_2=\dst{x^{-7/2}(1+x^2)^2}\)

\(y_1=\dst {x^7\over5}\sum_{m=0}^\infty(-1)^m(m+5)x^{2m}\);  \(y_2=\dst{x^{-1}\left(1-2x^2+3x^4-4x^6\right)}\)

\(y_1=\dst{x^3\sum_{m=0}^\infty(-1)^m{m+1\over2^m}\left(\prod_{j=1}^m{2j+1\over j+5}\right)x^{2m}}\);  \(y_2=\dst{x^{-7}\left(1+{21\over8}x^2+{35\over16}x^4+{35\over64}x^6\right)}\)

\(y_1=\dst{2x^4\sum_{m=0}^\infty(-1)^m{\prod_{j=1}^m(4j+5)\over2^m(m+2)!}x^{2m}}\);  \(y_2=\dst{1-{1\over2}x^2}\)

\(y_1=\dst{x^{3/2}\sum_{m=0}^\infty{(-1)^m \prod_{j=1}^m(2j-1)\over2^{m-1}(m+2)!}x^{2m}}\);  \(y_2=\dst{x^{-5/2}\left(1+{3\over2}x^2\right)}\)

\(\dst{y_1=x^\nu\sum_{m=0}^\infty{(-1)^m\over4^mm! \prod_{j=1}^m(j+\nu)}x^{2m}}\); 

     \(\dst{y_2=x^{-\nu}\sum_{m=0}^{\nu-1}{(-1)^m\over4^mm!\prod_{j=1}^m(j-\nu)}x^{2m} -{2\over4^\nu\nu!(\nu-1)!}\left(y_1 \ln x-{x^\nu\over2} \sum_{m=1}^\infty{(-1)^m\over4^mm!\prod_{j=1}^m(j+\nu)} \left(\sum_{j=1}^m{2j+\nu\over j(j+\nu)}\right) x^{2m}\right)}\)