\[ a y(x)^2-b x^{2 \nu }-c x^{\nu -1}+y'(x)=0 \] ✓ Mathematica : cpu = 0.365001 (sec), leaf count = 1835
\[\left \{\left \{y(x)\to \frac {-2^{\frac {\nu }{2 (\nu +1)}-1} e^{-\frac {\sqrt {a} \sqrt {b} x^{\nu +1}}{\sqrt {\nu ^2+2 \nu +1}}} \nu \left (x^{\nu +1}\right )^{\frac {\nu }{2 (\nu +1)}} L_{-\frac {\frac {\sqrt {a} \sqrt {b} \nu c}{\sqrt {(\nu +1)^2}}+\frac {\sqrt {a} \sqrt {b} c}{\sqrt {(\nu +1)^2}}+b \nu }{2 (\nu b+b)}}^{\frac {\nu }{\nu +1}-1}\left (\frac {2 \sqrt {a} \sqrt {b} x^{\nu +1}}{\sqrt {\nu ^2+2 \nu +1}}\right ) x^{-\frac {\nu }{2}-1}-\frac {2^{\frac {\nu }{2 (\nu +1)}} \sqrt {a} \sqrt {b} e^{-\frac {\sqrt {a} \sqrt {b} x^{\nu +1}}{\sqrt {\nu ^2+2 \nu +1}}} (\nu +1) \left (x^{\nu +1}\right )^{\frac {\nu }{2 (\nu +1)}} L_{-\frac {\frac {\sqrt {a} \sqrt {b} \nu c}{\sqrt {(\nu +1)^2}}+\frac {\sqrt {a} \sqrt {b} c}{\sqrt {(\nu +1)^2}}+b \nu }{2 (\nu b+b)}}^{\frac {\nu }{\nu +1}-1}\left (\frac {2 \sqrt {a} \sqrt {b} x^{\nu +1}}{\sqrt {\nu ^2+2 \nu +1}}\right ) x^{\nu /2}}{\sqrt {\nu ^2+2 \nu +1}}+2^{\frac {\nu }{2 (\nu +1)}-1} e^{-\frac {\sqrt {a} \sqrt {b} x^{\nu +1}}{\sqrt {\nu ^2+2 \nu +1}}} \nu \left (x^{\nu +1}\right )^{\frac {\nu }{2 (\nu +1)}-1} L_{-\frac {\frac {\sqrt {a} \sqrt {b} \nu c}{\sqrt {(\nu +1)^2}}+\frac {\sqrt {a} \sqrt {b} c}{\sqrt {(\nu +1)^2}}+b \nu }{2 (\nu b+b)}}^{\frac {\nu }{\nu +1}-1}\left (\frac {2 \sqrt {a} \sqrt {b} x^{\nu +1}}{\sqrt {\nu ^2+2 \nu +1}}\right ) x^{\nu /2}-\frac {2^{\frac {\nu }{2 (\nu +1)}+1} \sqrt {a} \sqrt {b} e^{-\frac {\sqrt {a} \sqrt {b} x^{\nu +1}}{\sqrt {\nu ^2+2 \nu +1}}} (\nu +1) \left (x^{\nu +1}\right )^{\frac {\nu }{2 (\nu +1)}} L_{-\frac {\frac {\sqrt {a} \sqrt {b} \nu c}{\sqrt {(\nu +1)^2}}+\frac {\sqrt {a} \sqrt {b} c}{\sqrt {(\nu +1)^2}}+b \nu }{2 (\nu b+b)}-1}^{\frac {\nu }{\nu +1}}\left (\frac {2 \sqrt {a} \sqrt {b} x^{\nu +1}}{\sqrt {\nu ^2+2 \nu +1}}\right ) x^{\nu /2}}{\sqrt {\nu ^2+2 \nu +1}}+c_1 \left (-2^{\frac {\nu }{2 (\nu +1)}-1} e^{-\frac {\sqrt {a} \sqrt {b} x^{\nu +1}}{\sqrt {\nu ^2+2 \nu +1}}} \nu \left (x^{\nu +1}\right )^{\frac {\nu }{2 (\nu +1)}} U\left (\frac {\frac {\sqrt {a} \sqrt {b} \nu c}{\sqrt {(\nu +1)^2}}+\frac {\sqrt {a} \sqrt {b} c}{\sqrt {(\nu +1)^2}}+b \nu }{2 (\nu b+b)},\frac {\nu }{\nu +1},\frac {2 \sqrt {a} \sqrt {b} x^{\nu +1}}{\sqrt {\nu ^2+2 \nu +1}}\right ) x^{-\frac {\nu }{2}-1}-\frac {2^{\frac {\nu }{2 (\nu +1)}} \sqrt {a} \sqrt {b} e^{-\frac {\sqrt {a} \sqrt {b} x^{\nu +1}}{\sqrt {\nu ^2+2 \nu +1}}} (\nu +1) \left (x^{\nu +1}\right )^{\frac {\nu }{2 (\nu +1)}} U\left (\frac {\frac {\sqrt {a} \sqrt {b} \nu c}{\sqrt {(\nu +1)^2}}+\frac {\sqrt {a} \sqrt {b} c}{\sqrt {(\nu +1)^2}}+b \nu }{2 (\nu b+b)},\frac {\nu }{\nu +1},\frac {2 \sqrt {a} \sqrt {b} x^{\nu +1}}{\sqrt {\nu ^2+2 \nu +1}}\right ) x^{\nu /2}}{\sqrt {\nu ^2+2 \nu +1}}+2^{\frac {\nu }{2 (\nu +1)}-1} e^{-\frac {\sqrt {a} \sqrt {b} x^{\nu +1}}{\sqrt {\nu ^2+2 \nu +1}}} \nu \left (x^{\nu +1}\right )^{\frac {\nu }{2 (\nu +1)}-1} U\left (\frac {\frac {\sqrt {a} \sqrt {b} \nu c}{\sqrt {(\nu +1)^2}}+\frac {\sqrt {a} \sqrt {b} c}{\sqrt {(\nu +1)^2}}+b \nu }{2 (\nu b+b)},\frac {\nu }{\nu +1},\frac {2 \sqrt {a} \sqrt {b} x^{\nu +1}}{\sqrt {\nu ^2+2 \nu +1}}\right ) x^{\nu /2}-\frac {2^{\frac {\nu }{2 (\nu +1)}} \sqrt {a} \sqrt {b} e^{-\frac {\sqrt {a} \sqrt {b} x^{\nu +1}}{\sqrt {\nu ^2+2 \nu +1}}} (\nu +1) \left (\frac {\sqrt {a} \sqrt {b} \nu c}{\sqrt {(\nu +1)^2}}+\frac {\sqrt {a} \sqrt {b} c}{\sqrt {(\nu +1)^2}}+b \nu \right ) \left (x^{\nu +1}\right )^{\frac {\nu }{2 (\nu +1)}} U\left (\frac {\frac {\sqrt {a} \sqrt {b} \nu c}{\sqrt {(\nu +1)^2}}+\frac {\sqrt {a} \sqrt {b} c}{\sqrt {(\nu +1)^2}}+b \nu }{2 (\nu b+b)}+1,\frac {\nu }{\nu +1}+1,\frac {2 \sqrt {a} \sqrt {b} x^{\nu +1}}{\sqrt {\nu ^2+2 \nu +1}}\right ) x^{\nu /2}}{(\nu b+b) \sqrt {\nu ^2+2 \nu +1}}\right )}{a \left (2^{\frac {\nu }{2 (\nu +1)}} e^{-\frac {\sqrt {a} \sqrt {b} x^{\nu +1}}{\sqrt {\nu ^2+2 \nu +1}}} \left (x^{\nu +1}\right )^{\frac {\nu }{2 (\nu +1)}} c_1 U\left (\frac {\frac {\sqrt {a} \sqrt {b} \nu c}{\sqrt {(\nu +1)^2}}+\frac {\sqrt {a} \sqrt {b} c}{\sqrt {(\nu +1)^2}}+b \nu }{2 (\nu b+b)},\frac {\nu }{\nu +1},\frac {2 \sqrt {a} \sqrt {b} x^{\nu +1}}{\sqrt {\nu ^2+2 \nu +1}}\right ) x^{-\nu /2}+2^{\frac {\nu }{2 (\nu +1)}} e^{-\frac {\sqrt {a} \sqrt {b} x^{\nu +1}}{\sqrt {\nu ^2+2 \nu +1}}} \left (x^{\nu +1}\right )^{\frac {\nu }{2 (\nu +1)}} L_{-\frac {\frac {\sqrt {a} \sqrt {b} \nu c}{\sqrt {(\nu +1)^2}}+\frac {\sqrt {a} \sqrt {b} c}{\sqrt {(\nu +1)^2}}+b \nu }{2 (\nu b+b)}}^{\frac {\nu }{\nu +1}-1}\left (\frac {2 \sqrt {a} \sqrt {b} x^{\nu +1}}{\sqrt {\nu ^2+2 \nu +1}}\right ) x^{-\nu /2}\right )}\right \}\right \}\] ✓ Maple : cpu = 0.319 (sec), leaf count = 348
\[y \relax (x ) = -\frac {\left (\left (-\nu -2\right ) b^{\frac {3}{2}}+\sqrt {a}\, b c \right ) \WhittakerM \left (-\frac {\left (-2 \nu -2\right ) \sqrt {b}+\sqrt {a}\, c}{\sqrt {b}\, \left (2 \nu +2\right )}, \frac {1}{2 \nu +2}, \frac {2 \sqrt {a}\, \sqrt {b}\, x^{\nu +1}}{\nu +1}\right )+2 b^{\frac {3}{2}} c_{1} \left (\nu +1\right ) \WhittakerW \left (-\frac {\left (-2 \nu -2\right ) \sqrt {b}+\sqrt {a}\, c}{\sqrt {b}\, \left (2 \nu +2\right )}, \frac {1}{2 \nu +2}, \frac {2 \sqrt {a}\, \sqrt {b}\, x^{\nu +1}}{\nu +1}\right )+\left (b^{\frac {3}{2}} \nu -2 \left (x^{\nu +1} b +\frac {c}{2}\right ) b \sqrt {a}\right ) \left (\WhittakerW \left (-\frac {\sqrt {a}\, c}{\sqrt {b}\, \left (2 \nu +2\right )}, \frac {1}{2 \nu +2}, \frac {2 \sqrt {a}\, \sqrt {b}\, x^{\nu +1}}{\nu +1}\right ) c_{1}+\WhittakerM \left (-\frac {\sqrt {a}\, c}{\sqrt {b}\, \left (2 \nu +2\right )}, \frac {1}{2 \nu +2}, \frac {2 \sqrt {a}\, \sqrt {b}\, x^{\nu +1}}{\nu +1}\right )\right )}{2 b^{\frac {3}{2}} \left (\WhittakerW \left (-\frac {\sqrt {a}\, c}{\sqrt {b}\, \left (2 \nu +2\right )}, \frac {1}{2 \nu +2}, \frac {2 \sqrt {a}\, \sqrt {b}\, x^{\nu +1}}{\nu +1}\right ) c_{1}+\WhittakerM \left (-\frac {\sqrt {a}\, c}{\sqrt {b}\, \left (2 \nu +2\right )}, \frac {1}{2 \nu +2}, \frac {2 \sqrt {a}\, \sqrt {b}\, x^{\nu +1}}{\nu +1}\right )\right ) a x}\]
Hand solution
\begin {align} y^{\prime }+ay^{2}-bx^{2v}-cx^{v-1} & =0\nonumber \\ y^{\prime } & =bx^{v}+cx^{v-1}-ay^{2}\tag {1}\\ & =P\relax (x) +Q\relax (x) y+R\relax (x) y^{2}\nonumber \end {align}
This is Riccati first order non-linear ODE with \(P\relax (x) =bx^{v}+cx^{v-1},Q\relax (x) =0,R\relax (x) =-a\).
Need to do this later.