#### 2.139   ODE No. 139

$a x^k-(b-1) b+x^2 \left (y'(x)+y(x)^2\right )=0$ Mathematica : cpu = 0.202571 (sec), leaf count = 821

$\left \{\left \{y(x)\to -\frac {a^{\frac {b}{k}+\frac {1}{2} \left (\frac {1}{k}-\frac {2 b}{k}\right )} \left (\frac {b}{k}+\frac {1}{2} \left (\frac {1}{k}-\frac {2 b}{k}\right )\right ) x^{k-1} \left (x^k\right )^{\frac {b}{k}+\frac {1}{2} \left (\frac {1}{k}-\frac {2 b}{k}\right )-1} J_{\frac {2 b-1}{k}}\left (\frac {2 \sqrt {a} \sqrt {x^k}}{k}\right ) \Gamma \left (\frac {2 b}{k}-\frac {1}{k}+1\right ) k^{1-\frac {1}{k}}+\frac {1}{2} a^{\frac {b}{k}+\frac {1}{2} \left (\frac {1}{k}-\frac {2 b}{k}\right )+\frac {1}{2}} x^{k-1} \left (x^k\right )^{\frac {b}{k}+\frac {1}{2} \left (\frac {1}{k}-\frac {2 b}{k}\right )-\frac {1}{2}} \left (J_{\frac {2 b-1}{k}-1}\left (\frac {2 \sqrt {a} \sqrt {x^k}}{k}\right )-J_{\frac {2 b-1}{k}+1}\left (\frac {2 \sqrt {a} \sqrt {x^k}}{k}\right )\right ) \Gamma \left (\frac {2 b}{k}-\frac {1}{k}+1\right ) k^{-1/k}+c_1 \left (a^{\frac {1-b}{k}+\frac {1}{2} \left (\frac {2 b}{k}-\frac {1}{k}\right )} \left (\frac {1-b}{k}+\frac {1}{2} \left (\frac {2 b}{k}-\frac {1}{k}\right )\right ) k^{-\frac {2 (1-b)}{k}-\frac {2 b}{k}+\frac {1}{k}+1} x^{k-1} J_{\frac {1-2 b}{k}}\left (\frac {2 \sqrt {a} \sqrt {x^k}}{k}\right ) \Gamma \left (-\frac {2 b}{k}+\frac {1}{k}+1\right ) \left (x^k\right )^{\frac {1-b}{k}+\frac {1}{2} \left (\frac {2 b}{k}-\frac {1}{k}\right )-1}+\frac {1}{2} a^{\frac {1-b}{k}+\frac {1}{2} \left (\frac {2 b}{k}-\frac {1}{k}\right )+\frac {1}{2}} k^{-\frac {2 (1-b)}{k}-\frac {2 b}{k}+\frac {1}{k}} x^{k-1} \left (J_{\frac {1-2 b}{k}-1}\left (\frac {2 \sqrt {a} \sqrt {x^k}}{k}\right )-J_{\frac {1-2 b}{k}+1}\left (\frac {2 \sqrt {a} \sqrt {x^k}}{k}\right )\right ) \Gamma \left (-\frac {2 b}{k}+\frac {1}{k}+1\right ) \left (x^k\right )^{\frac {1-b}{k}+\frac {1}{2} \left (\frac {2 b}{k}-\frac {1}{k}\right )-\frac {1}{2}}\right )}{-a^{\frac {1-b}{k}+\frac {1}{2} \left (\frac {2 b}{k}-\frac {1}{k}\right )} k^{-\frac {2 (1-b)}{k}-\frac {2 b}{k}+\frac {1}{k}} J_{\frac {1-2 b}{k}}\left (\frac {2 \sqrt {a} \sqrt {x^k}}{k}\right ) c_1 \Gamma \left (-\frac {2 b}{k}+\frac {1}{k}+1\right ) \left (x^k\right )^{\frac {1-b}{k}+\frac {1}{2} \left (\frac {2 b}{k}-\frac {1}{k}\right )}-a^{\frac {b}{k}+\frac {1}{2} \left (\frac {1}{k}-\frac {2 b}{k}\right )} k^{-1/k} J_{\frac {2 b-1}{k}}\left (\frac {2 \sqrt {a} \sqrt {x^k}}{k}\right ) \Gamma \left (\frac {2 b}{k}-\frac {1}{k}+1\right ) \left (x^k\right )^{\frac {b}{k}+\frac {1}{2} \left (\frac {1}{k}-\frac {2 b}{k}\right )}}\right \}\right \}$ Maple : cpu = 0.11 (sec), leaf count = 219

$\left \{ y \left ( x \right ) ={\frac {1}{2\,x} \left ( -2\,{{\sl J}_{{\frac {\sqrt { \left ( -1+2\,b \right ) ^{2}}+k}{k}}}\left (2\,{\frac {\sqrt {a}{x}^{k/2}}{k}}\right )}\sqrt {a}{x}^{k/2}-2\,{x}^{k/2}{{\sl Y}_{{\frac {\sqrt { \left ( -1+2\,b \right ) ^{2}}+k}{k}}}\left (2\,{\frac {\sqrt {a}{x}^{k/2}}{k}}\right )}\sqrt {a}{\it \_C1}+2\, \left ( {{\sl Y}_{{\frac {\sqrt { \left ( -1+2\,b \right ) ^{2}}}{k}}}\left (2\,{\frac {\sqrt {a}{x}^{k/2}}{k}}\right )}{\it \_C1}+{{\sl J}_{{\frac {\sqrt { \left ( -1+2\,b \right ) ^{2}}}{k}}}\left (2\,{\frac {\sqrt {a}{x}^{k/2}}{k}}\right )} \right ) \left ( 1/2+ \left ( b-1/2 \right ) {\it csgn} \left ( -1+2\,b \right ) \right ) \right ) \left ( {{\sl Y}_{{\frac {1}{k}\sqrt { \left ( -1+2\,b \right ) ^{2}}}}\left (2\,{\frac {\sqrt {a}{x}^{k/2}}{k}}\right )}{\it \_C1}+{{\sl J}_{{\frac {1}{k}\sqrt { \left ( -1+2\,b \right ) ^{2}}}}\left (2\,{\frac {\sqrt {a}{x}^{k/2}}{k}}\right )} \right ) ^{-1}} \right \}$