#### 2.144   ODE No. 144

$x^2 \left (a y(x)^2+y'(x)\right )+b x^{\alpha }+c=0$ Mathematica : cpu = 0.217838 (sec), leaf count = 1787

$\left \{\left \{y(x)\to \frac {a^{\frac {i \sqrt {4 a c-1} \alpha +\alpha }{2 \alpha ^2}-\frac {\sqrt {\alpha ^2-4 a \alpha ^2 c}}{2 \alpha ^2}} \alpha ^{-\frac {i \sqrt {4 a c-1} \alpha +\alpha }{\alpha ^2}+\frac {\sqrt {\alpha ^2-4 a \alpha ^2 c}}{\alpha ^2}+1} b^{\frac {i \sqrt {4 a c-1} \alpha +\alpha }{2 \alpha ^2}-\frac {\sqrt {\alpha ^2-4 a \alpha ^2 c}}{2 \alpha ^2}} \left (\frac {i \sqrt {4 a c-1} \alpha +\alpha }{2 \alpha ^2}-\frac {\sqrt {\alpha ^2-4 a \alpha ^2 c}}{2 \alpha ^2}\right ) \left (x^{\alpha }\right )^{\frac {i \sqrt {4 a c-1} \alpha +\alpha }{2 \alpha ^2}-\frac {\sqrt {\alpha ^2-4 a \alpha ^2 c}}{2 \alpha ^2}-1} J_{\frac {\sqrt {\alpha ^2-4 a \alpha ^2 c}}{\alpha ^2}}\left (\frac {2 \sqrt {a} \sqrt {b} \sqrt {x^{\alpha }}}{\alpha }\right ) \Gamma \left (\frac {\sqrt {1-4 a c}}{\alpha }+1\right ) x^{\alpha -1}+\frac {1}{2} a^{\frac {i \sqrt {4 a c-1} \alpha +\alpha }{2 \alpha ^2}-\frac {\sqrt {\alpha ^2-4 a \alpha ^2 c}}{2 \alpha ^2}+\frac {1}{2}} \alpha ^{\frac {\sqrt {\alpha ^2-4 a \alpha ^2 c}}{\alpha ^2}-\frac {i \sqrt {4 a c-1} \alpha +\alpha }{\alpha ^2}} b^{\frac {i \sqrt {4 a c-1} \alpha +\alpha }{2 \alpha ^2}-\frac {\sqrt {\alpha ^2-4 a \alpha ^2 c}}{2 \alpha ^2}+\frac {1}{2}} \left (x^{\alpha }\right )^{\frac {i \sqrt {4 a c-1} \alpha +\alpha }{2 \alpha ^2}-\frac {\sqrt {\alpha ^2-4 a \alpha ^2 c}}{2 \alpha ^2}-\frac {1}{2}} \left (J_{\frac {\sqrt {\alpha ^2-4 a \alpha ^2 c}}{\alpha ^2}-1}\left (\frac {2 \sqrt {a} \sqrt {b} \sqrt {x^{\alpha }}}{\alpha }\right )-J_{\frac {\sqrt {\alpha ^2-4 a \alpha ^2 c}}{\alpha ^2}+1}\left (\frac {2 \sqrt {a} \sqrt {b} \sqrt {x^{\alpha }}}{\alpha }\right )\right ) \Gamma \left (\frac {\sqrt {1-4 a c}}{\alpha }+1\right ) x^{\alpha -1}+c_1 \left (a^{\frac {\alpha -i \alpha \sqrt {4 a c-1}}{2 \alpha ^2}+\frac {\sqrt {\alpha ^2-4 a \alpha ^2 c}}{2 \alpha ^2}} \alpha ^{-\frac {\alpha -i \alpha \sqrt {4 a c-1}}{\alpha ^2}-\frac {\sqrt {\alpha ^2-4 a \alpha ^2 c}}{\alpha ^2}+1} b^{\frac {\alpha -i \alpha \sqrt {4 a c-1}}{2 \alpha ^2}+\frac {\sqrt {\alpha ^2-4 a \alpha ^2 c}}{2 \alpha ^2}} \left (\frac {\alpha -i \alpha \sqrt {4 a c-1}}{2 \alpha ^2}+\frac {\sqrt {\alpha ^2-4 a \alpha ^2 c}}{2 \alpha ^2}\right ) \left (x^{\alpha }\right )^{\frac {\alpha -i \alpha \sqrt {4 a c-1}}{2 \alpha ^2}+\frac {\sqrt {\alpha ^2-4 a \alpha ^2 c}}{2 \alpha ^2}-1} J_{-\frac {\sqrt {\alpha ^2-4 a \alpha ^2 c}}{\alpha ^2}}\left (\frac {2 \sqrt {a} \sqrt {b} \sqrt {x^{\alpha }}}{\alpha }\right ) \Gamma \left (1-\frac {\sqrt {1-4 a c}}{\alpha }\right ) x^{\alpha -1}+\frac {1}{2} a^{\frac {\alpha -i \alpha \sqrt {4 a c-1}}{2 \alpha ^2}+\frac {\sqrt {\alpha ^2-4 a \alpha ^2 c}}{2 \alpha ^2}+\frac {1}{2}} \alpha ^{-\frac {\alpha -i \alpha \sqrt {4 a c-1}}{\alpha ^2}-\frac {\sqrt {\alpha ^2-4 a \alpha ^2 c}}{\alpha ^2}} b^{\frac {\alpha -i \alpha \sqrt {4 a c-1}}{2 \alpha ^2}+\frac {\sqrt {\alpha ^2-4 a \alpha ^2 c}}{2 \alpha ^2}+\frac {1}{2}} \left (x^{\alpha }\right )^{\frac {\alpha -i \alpha \sqrt {4 a c-1}}{2 \alpha ^2}+\frac {\sqrt {\alpha ^2-4 a \alpha ^2 c}}{2 \alpha ^2}-\frac {1}{2}} \left (J_{-\frac {\sqrt {\alpha ^2-4 a \alpha ^2 c}}{\alpha ^2}-1}\left (\frac {2 \sqrt {a} \sqrt {b} \sqrt {x^{\alpha }}}{\alpha }\right )-J_{1-\frac {\sqrt {\alpha ^2-4 a \alpha ^2 c}}{\alpha ^2}}\left (\frac {2 \sqrt {a} \sqrt {b} \sqrt {x^{\alpha }}}{\alpha }\right )\right ) \Gamma \left (1-\frac {\sqrt {1-4 a c}}{\alpha }\right ) x^{\alpha -1}\right )}{a \left (a^{\frac {\alpha -i \alpha \sqrt {4 a c-1}}{2 \alpha ^2}+\frac {\sqrt {\alpha ^2-4 a \alpha ^2 c}}{2 \alpha ^2}} b^{\frac {\alpha -i \alpha \sqrt {4 a c-1}}{2 \alpha ^2}+\frac {\sqrt {\alpha ^2-4 a \alpha ^2 c}}{2 \alpha ^2}} \left (x^{\alpha }\right )^{\frac {\alpha -i \alpha \sqrt {4 a c-1}}{2 \alpha ^2}+\frac {\sqrt {\alpha ^2-4 a \alpha ^2 c}}{2 \alpha ^2}} J_{-\frac {\sqrt {\alpha ^2-4 a \alpha ^2 c}}{\alpha ^2}}\left (\frac {2 \sqrt {a} \sqrt {b} \sqrt {x^{\alpha }}}{\alpha }\right ) c_1 \Gamma \left (1-\frac {\sqrt {1-4 a c}}{\alpha }\right ) \alpha ^{-\frac {\alpha -i \alpha \sqrt {4 a c-1}}{\alpha ^2}-\frac {\sqrt {\alpha ^2-4 a \alpha ^2 c}}{\alpha ^2}}+a^{\frac {i \sqrt {4 a c-1} \alpha +\alpha }{2 \alpha ^2}-\frac {\sqrt {\alpha ^2-4 a \alpha ^2 c}}{2 \alpha ^2}} b^{\frac {i \sqrt {4 a c-1} \alpha +\alpha }{2 \alpha ^2}-\frac {\sqrt {\alpha ^2-4 a \alpha ^2 c}}{2 \alpha ^2}} \left (x^{\alpha }\right )^{\frac {i \sqrt {4 a c-1} \alpha +\alpha }{2 \alpha ^2}-\frac {\sqrt {\alpha ^2-4 a \alpha ^2 c}}{2 \alpha ^2}} J_{\frac {\sqrt {\alpha ^2-4 a \alpha ^2 c}}{\alpha ^2}}\left (\frac {2 \sqrt {a} \sqrt {b} \sqrt {x^{\alpha }}}{\alpha }\right ) \Gamma \left (\frac {\sqrt {1-4 a c}}{\alpha }+1\right ) \alpha ^{\frac {\sqrt {\alpha ^2-4 a \alpha ^2 c}}{\alpha ^2}-\frac {i \sqrt {4 a c-1} \alpha +\alpha }{\alpha ^2}}\right )}\right \}\right \}$ Maple : cpu = 0.095 (sec), leaf count = 219

$\left \{ y \left ( x \right ) ={\frac {1}{2\,ax} \left ( -2\, \left ( {{\sl Y}_{{\frac {\sqrt {-4\,ac+1}+\alpha }{\alpha }}}\left (2\,{\frac {\sqrt {ab}{x}^{\alpha /2}}{\alpha }}\right )}{\it \_C1}+{{\sl J}_{{\frac {\sqrt {-4\,ac+1}+\alpha }{\alpha }}}\left (2\,{\frac {\sqrt {ab}{x}^{\alpha /2}}{\alpha }}\right )} \right ) \sqrt {ab}{x}^{\alpha /2}+ \left ( \sqrt {-4\,ac+1}+1 \right ) \left ( {{\sl Y}_{{\frac {1}{\alpha }\sqrt {-4\,ac+1}}}\left (2\,{\frac {\sqrt {ab}{x}^{\alpha /2}}{\alpha }}\right )}{\it \_C1}+{{\sl J}_{{\frac {1}{\alpha }\sqrt {-4\,ac+1}}}\left (2\,{\frac {\sqrt {ab}{x}^{\alpha /2}}{\alpha }}\right )} \right ) \right ) \left ( {{\sl Y}_{{\frac {1}{\alpha }\sqrt {-4\,ac+1}}}\left (2\,{\frac {\sqrt {ab}{x}^{\alpha /2}}{\alpha }}\right )}{\it \_C1}+{{\sl J}_{{\frac {1}{\alpha }\sqrt {-4\,ac+1}}}\left (2\,{\frac {\sqrt {ab}{x}^{\alpha /2}}{\alpha }}\right )} \right ) ^{-1}} \right \}$