#### 2.25   ODE No. 25

$a y(x)^2-b x^{2 \nu }-c x^{\nu -1}+y'(x)=0$ Mathematica : cpu = 0.331444 (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 = 0.272 (sec), leaf count = 348

$\left \{ y \left ( x \right ) =-{\frac {1}{2\,ax} \left ( \left ( \left ( -\nu -2 \right ) {b}^{{\frac {3}{2}}}+\sqrt {a}bc \right ) {{\sl M}_{-{\frac {1}{2\,\nu +2} \left ( \left ( -2\,\nu -2 \right ) \sqrt {b}+\sqrt {a}c \right ) {\frac {1}{\sqrt {b}}}},\, \left ( 2\,\nu +2 \right ) ^{-1}}\left (2\,{\frac {\sqrt {a}\sqrt {b}{x}^{\nu +1}}{\nu +1}}\right )}+2\,{b}^{3/2}{\it \_C1}\, \left ( \nu +1 \right ) {{\sl W}_{-{\frac { \left ( -2\,\nu -2 \right ) \sqrt {b}+\sqrt {a}c}{\sqrt {b} \left ( 2\,\nu +2 \right ) }},\, \left ( 2\,\nu +2 \right ) ^{-1}}\left (2\,{\frac {\sqrt {a}\sqrt {b}{x}^{\nu +1}}{\nu +1}}\right )}+ \left ( {b}^{{\frac {3}{2}}}\nu -2\,\sqrt {a} \left ( {x}^{\nu +1}b+c/2 \right ) b \right ) \left ( {{\sl W}_{-{\frac {c}{2\,\nu +2}\sqrt {a}{\frac {1}{\sqrt {b}}}},\, \left ( 2\,\nu +2 \right ) ^{-1}}\left (2\,{\frac {\sqrt {a}\sqrt {b}{x}^{\nu +1}}{\nu +1}}\right )}{\it \_C1}+{{\sl M}_{-{\frac {c}{2\,\nu +2}\sqrt {a}{\frac {1}{\sqrt {b}}}},\, \left ( 2\,\nu +2 \right ) ^{-1}}\left (2\,{\frac {\sqrt {a}\sqrt {b}{x}^{\nu +1}}{\nu +1}}\right )} \right ) \right ) {b}^{-{\frac {3}{2}}} \left ( {{\sl W}_{-{\frac {c}{2\,\nu +2}\sqrt {a}{\frac {1}{\sqrt {b}}}},\, \left ( 2\,\nu +2 \right ) ^{-1}}\left (2\,{\frac {\sqrt {a}\sqrt {b}{x}^{\nu +1}}{\nu +1}}\right )}{\it \_C1}+{{\sl M}_{-{\frac {c}{2\,\nu +2}\sqrt {a}{\frac {1}{\sqrt {b}}}},\, \left ( 2\,\nu +2 \right ) ^{-1}}\left (2\,{\frac {\sqrt {a}\sqrt {b}{x}^{\nu +1}}{\nu +1}}\right )} \right ) ^{-1}} \right \}$

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\left ( x\right ) +Q\left ( x\right ) y+R\left ( x\right ) y^{2}\nonumber \end {align}

This is Riccati ﬁrst order non-linear ODE with $$P\left ( x\right ) =bx^{v}+cx^{v-1},Q\left ( x\right ) =0,R\left ( x\right ) =-a$$.

Need to do this later.