#### 2.99   ODE No. 99

$a y(x)^2-b y(x)-c x^{\beta }+x y'(x)=0$ Mathematica : cpu = 0.155337 (sec), leaf count = 244

$\left \{\left \{y(x)\to -\frac {\sqrt {-a} \sqrt {c} x^{\beta /2} \left (c_1 J_{1-\frac {b}{\beta }}\left (\frac {2 \sqrt {-a} \sqrt {c} x^{\beta /2}}{\beta }\right )-c_1 J_{-\frac {b+\beta }{\beta }}\left (\frac {2 \sqrt {-a} \sqrt {c} x^{\beta /2}}{\beta }\right )-2 J_{\frac {b}{\beta }-1}\left (\frac {2 \sqrt {-a} \sqrt {c} x^{\beta /2}}{\beta }\right )\right )-b c_1 J_{-\frac {b}{\beta }}\left (\frac {2 \sqrt {-a} \sqrt {c} x^{\beta /2}}{\beta }\right )}{2 a \left (c_1 J_{-\frac {b}{\beta }}\left (\frac {2 \sqrt {-a} \sqrt {c} x^{\beta /2}}{\beta }\right )+J_{\frac {b}{\beta }}\left (\frac {2 \sqrt {-a} \sqrt {c} x^{\beta /2}}{\beta }\right )\right )}\right \}\right \}$ Maple : cpu = 0.089 (sec), leaf count = 171

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