#### 2.107   ODE No. 107

$a x^{\alpha } y(x)^2+b y(x)-c x^{\beta }+x y'(x)=0$ Mathematica : cpu = 0.326367 (sec), leaf count = 1415

$\left \{\left \{y(x)\to \frac {x^{1-\alpha } \left ((-1)^{\frac {\alpha -b}{\alpha +\beta }} a^{\frac {\alpha -b}{\alpha +\beta }+\frac {1}{2} \left (\frac {b}{\alpha +\beta }-\frac {\alpha }{\alpha +\beta }\right )} (\alpha +\beta )^{\frac {\alpha }{\alpha +\beta }-\frac {b}{\alpha +\beta }+1} \left (\alpha ^2+2 \beta \alpha +\beta ^2\right )^{-\frac {\alpha -b}{\alpha +\beta }} \left (\frac {\alpha -b}{\alpha +\beta }+\frac {1}{2} \left (\frac {b}{\alpha +\beta }-\frac {\alpha }{\alpha +\beta }\right )\right ) c^{\frac {\alpha -b}{\alpha +\beta }+\frac {1}{2} \left (\frac {b}{\alpha +\beta }-\frac {\alpha }{\alpha +\beta }\right )} \left (x^{\alpha +\beta }\right )^{\frac {\alpha -b}{\alpha +\beta }+\frac {1}{2} \left (\frac {b}{\alpha +\beta }-\frac {\alpha }{\alpha +\beta }\right )-1} I_{\frac {\alpha -b}{\alpha +\beta }}\left (\frac {2 \sqrt {a} \sqrt {c} \sqrt {x^{\alpha +\beta }}}{\sqrt {\alpha ^2+2 \beta \alpha +\beta ^2}}\right ) \Gamma \left (\frac {\alpha }{\alpha +\beta }-\frac {b}{\alpha +\beta }+1\right ) x^{\alpha +\beta -1}+\frac {1}{2} (-1)^{\frac {\alpha -b}{\alpha +\beta }} a^{\frac {\alpha -b}{\alpha +\beta }+\frac {1}{2} \left (\frac {b}{\alpha +\beta }-\frac {\alpha }{\alpha +\beta }\right )+\frac {1}{2}} (\alpha +\beta )^{\frac {\alpha }{\alpha +\beta }-\frac {b}{\alpha +\beta }+1} \left (\alpha ^2+2 \beta \alpha +\beta ^2\right )^{-\frac {\alpha -b}{\alpha +\beta }-\frac {1}{2}} c^{\frac {\alpha -b}{\alpha +\beta }+\frac {1}{2} \left (\frac {b}{\alpha +\beta }-\frac {\alpha }{\alpha +\beta }\right )+\frac {1}{2}} \left (x^{\alpha +\beta }\right )^{\frac {\alpha -b}{\alpha +\beta }+\frac {1}{2} \left (\frac {b}{\alpha +\beta }-\frac {\alpha }{\alpha +\beta }\right )-\frac {1}{2}} \left (I_{\frac {\alpha -b}{\alpha +\beta }-1}\left (\frac {2 \sqrt {a} \sqrt {c} \sqrt {x^{\alpha +\beta }}}{\sqrt {\alpha ^2+2 \beta \alpha +\beta ^2}}\right )+I_{\frac {\alpha -b}{\alpha +\beta }+1}\left (\frac {2 \sqrt {a} \sqrt {c} \sqrt {x^{\alpha +\beta }}}{\sqrt {\alpha ^2+2 \beta \alpha +\beta ^2}}\right )\right ) \Gamma \left (\frac {\alpha }{\alpha +\beta }-\frac {b}{\alpha +\beta }+1\right ) x^{\alpha +\beta -1}+c_1 \left (\frac {1}{2} a^{\frac {1}{2} \left (\frac {\alpha }{\alpha +\beta }-\frac {b}{\alpha +\beta }\right )} (\alpha +\beta )^{-\frac {\alpha }{\alpha +\beta }+\frac {b}{\alpha +\beta }+1} \left (\frac {\alpha }{\alpha +\beta }-\frac {b}{\alpha +\beta }\right ) c^{\frac {1}{2} \left (\frac {\alpha }{\alpha +\beta }-\frac {b}{\alpha +\beta }\right )} \left (x^{\alpha +\beta }\right )^{\frac {1}{2} \left (\frac {\alpha }{\alpha +\beta }-\frac {b}{\alpha +\beta }\right )-1} I_{\frac {b-\alpha }{\alpha +\beta }}\left (\frac {2 \sqrt {a} \sqrt {c} \sqrt {x^{\alpha +\beta }}}{\sqrt {\alpha ^2+2 \beta \alpha +\beta ^2}}\right ) \Gamma \left (\frac {b}{\alpha +\beta }+\frac {\beta }{\alpha +\beta }\right ) x^{\alpha +\beta -1}+\frac {a^{\frac {1}{2} \left (\frac {\alpha }{\alpha +\beta }-\frac {b}{\alpha +\beta }\right )+\frac {1}{2}} (\alpha +\beta )^{-\frac {\alpha }{\alpha +\beta }+\frac {b}{\alpha +\beta }+1} c^{\frac {1}{2} \left (\frac {\alpha }{\alpha +\beta }-\frac {b}{\alpha +\beta }\right )+\frac {1}{2}} \left (x^{\alpha +\beta }\right )^{\frac {1}{2} \left (\frac {\alpha }{\alpha +\beta }-\frac {b}{\alpha +\beta }\right )-\frac {1}{2}} \left (I_{\frac {b-\alpha }{\alpha +\beta }-1}\left (\frac {2 \sqrt {a} \sqrt {c} \sqrt {x^{\alpha +\beta }}}{\sqrt {\alpha ^2+2 \beta \alpha +\beta ^2}}\right )+I_{\frac {b-\alpha }{\alpha +\beta }+1}\left (\frac {2 \sqrt {a} \sqrt {c} \sqrt {x^{\alpha +\beta }}}{\sqrt {\alpha ^2+2 \beta \alpha +\beta ^2}}\right )\right ) \Gamma \left (\frac {b}{\alpha +\beta }+\frac {\beta }{\alpha +\beta }\right ) x^{\alpha +\beta -1}}{2 \sqrt {\alpha ^2+2 \beta \alpha +\beta ^2}}\right )\right )}{a \left ((-1)^{\frac {\alpha -b}{\alpha +\beta }} a^{\frac {\alpha -b}{\alpha +\beta }+\frac {1}{2} \left (\frac {b}{\alpha +\beta }-\frac {\alpha }{\alpha +\beta }\right )} (\alpha +\beta )^{\frac {\alpha }{\alpha +\beta }-\frac {b}{\alpha +\beta }} c^{\frac {\alpha -b}{\alpha +\beta }+\frac {1}{2} \left (\frac {b}{\alpha +\beta }-\frac {\alpha }{\alpha +\beta }\right )} \left (x^{\alpha +\beta }\right )^{\frac {\alpha -b}{\alpha +\beta }+\frac {1}{2} \left (\frac {b}{\alpha +\beta }-\frac {\alpha }{\alpha +\beta }\right )} I_{\frac {\alpha -b}{\alpha +\beta }}\left (\frac {2 \sqrt {a} \sqrt {c} \sqrt {x^{\alpha +\beta }}}{\sqrt {\alpha ^2+2 \beta \alpha +\beta ^2}}\right ) \Gamma \left (\frac {\alpha }{\alpha +\beta }-\frac {b}{\alpha +\beta }+1\right ) \left (\alpha ^2+2 \beta \alpha +\beta ^2\right )^{-\frac {\alpha -b}{\alpha +\beta }}+a^{\frac {1}{2} \left (\frac {\alpha }{\alpha +\beta }-\frac {b}{\alpha +\beta }\right )} (\alpha +\beta )^{\frac {b}{\alpha +\beta }-\frac {\alpha }{\alpha +\beta }} c^{\frac {1}{2} \left (\frac {\alpha }{\alpha +\beta }-\frac {b}{\alpha +\beta }\right )} \left (x^{\alpha +\beta }\right )^{\frac {1}{2} \left (\frac {\alpha }{\alpha +\beta }-\frac {b}{\alpha +\beta }\right )} I_{\frac {b-\alpha }{\alpha +\beta }}\left (\frac {2 \sqrt {a} \sqrt {c} \sqrt {x^{\alpha +\beta }}}{\sqrt {\alpha ^2+2 \beta \alpha +\beta ^2}}\right ) c_1 \Gamma \left (\frac {b}{\alpha +\beta }+\frac {\beta }{\alpha +\beta }\right )\right )}\right \}\right \}$ Maple : cpu = 0.194 (sec), leaf count = 174

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