\[ \boxed { {\frac {\rm d}{{\rm d}x}}y \left ( x \right ) -{x}^{a} \left ( y \left ( x \right ) \right ) ^{3}+3\, \left ( y \left ( x \right ) \right ) ^{2}-{x}^{-a}y \left ( x \right ) -{x}^{-2\,a}+a{x}^{-a-1}=0} \]
Mathematica: cpu = 0.227529 (sec), leaf count = 258 \[ \left \{\left \{y(x)\to x^{-a}-\frac {e^{-\frac {2 x^{1-a}}{1-a}}}{\sqrt {c_1-\frac {2 x \left (\frac {4^{\frac {a+1}{a-1}} x \left (\frac {x^{1-a}}{1-a}\right )^{\frac {2}{a-1}} \Gamma \left (-\frac {2}{a-1},-\frac {4 x^{1-a}}{a-1}\right )}{a-1}+e^{\frac {4 x^{1-a}}{a-1}} x^a\right )}{a+1}}}\right \},\left \{y(x)\to \frac {e^{-\frac {2 x^{1-a}}{1-a}}}{\sqrt {c_1-\frac {2 x \left (\frac {4^{\frac {a+1}{a-1}} x \left (\frac {x^{1-a}}{1-a}\right )^{\frac {2}{a-1}} \Gamma \left (-\frac {2}{a-1},-\frac {4 x^{1-a}}{a-1}\right )}{a-1}+e^{\frac {4 x^{1-a}}{a-1}} x^a\right )}{a+1}}}+x^{-a}\right \}\right \} \]
Maple: cpu = 0.062 (sec), leaf count = 1052 \[ \left \{ y \left ( x \right ) =-{1{{\rm e}^{2\,{\frac {x}{ \left ( a-1 \right ) {x}^{a}}}}}{\frac {1}{\sqrt {{\it \_C1}-2\,{\frac {1}{1-a}{2} ^{-2\,{\frac {a}{1-a}}-2\, \left ( 1-a \right ) ^{-1}} \left ( \left ( 1- a \right ) ^{-1} \right ) ^{-{\frac {a}{1-a}}- \left ( 1-a \right ) ^{-1}} \left ( {\frac {1-a}{-3+a}{2}^{-3+2\,{\frac {a}{1-a}}+2\, \left ( 1-a \right ) ^{-1}+2\, \left ( a-1 \right ) ^{-1}}{x}^{-{\frac {{a}^{2}}{1-a }}+ \left ( 1-a \right ) ^{-1}-1+a} \left ( \left ( 1-a \right ) ^{-1} \right ) ^{{\frac {a}{1-a}}+ \left ( 1-a \right ) ^{-1}} \left ( -4\,{ \frac {{x}^{1-a}{a}^{2}}{1-a}}+8\,{\frac {a{x}^{1-a}}{1-a}}-4\,{\frac {{x}^{1-a}}{1-a}}+2\,a-2 \right ) \left ( {\frac {{x}^{1-a}}{1-a}} \right ) ^{ \left ( a-1 \right ) ^{-1}}{{\rm e}^{-2\,{\frac {{x}^{1-a}}{ 1-a}}}}{{\sl M}_{-{\frac {a+1}{a-1}}+ \left ( a-1 \right ) ^{-1},\,- \left ( a-1 \right ) ^{-1}+1/2}\left (4\,{\frac {{x}^{1-a}}{1-a}} \right )} \left ( {\frac {a}{1-a}}+ \left ( 1-a \right ) ^{-1} \right ) ^{- 1}}-{\frac {1-a}{-3+a}{2}^{-1+2\,{\frac {a}{1-a}}+2\, \left ( 1-a \right ) ^{-1}+2\, \left ( a-1 \right ) ^{-1}}{x}^{-{\frac {{a}^{2}}{1-a }}+ \left ( 1-a \right ) ^{-1}-1+a} \left ( \left ( 1-a \right ) ^{-1} \right ) ^{{\frac {a}{1-a}}+ \left ( 1-a \right ) ^{-1}} \left ( {\frac { {x}^{1-a}}{1-a}} \right ) ^{ \left ( a-1 \right ) ^{-1}}{{\rm e}^{-2\,{ \frac {{x}^{1-a}}{1-a}}}}{{\sl M}_{-{\frac {a+1}{a-1}}+ \left ( a-1 \right ) ^{-1}+1,\,- \left ( a-1 \right ) ^{-1}+1/2}\left (4\,{\frac {{x} ^{1-a}}{1-a}}\right )} \left ( {\frac {a}{1-a}}+ \left ( 1-a \right ) ^{-1 } \right ) ^{-1}} \right ) }}}}}+ \left ( {x}^{a} \right ) ^{-1},y \left ( x \right ) ={1{{\rm e}^{2\,{\frac {x}{ \left ( a-1 \right ) {x}^{a}}}}}{ \frac {1}{\sqrt {{\it \_C1}-2\,{\frac {1}{1-a}{2}^{-2\,{\frac {a}{1-a} }-2\, \left ( 1-a \right ) ^{-1}} \left ( \left ( 1-a \right ) ^{-1} \right ) ^{-{\frac {a}{1-a}}- \left ( 1-a \right ) ^{-1}} \left ( {\frac {1-a}{-3+a}{2}^{-3+2\,{\frac {a}{1-a}}+2\, \left ( 1-a \right ) ^{-1}+2 \, \left ( a-1 \right ) ^{-1}}{x}^{-{\frac {{a}^{2}}{1-a}}+ \left ( 1-a \right ) ^{-1}-1+a} \left ( \left ( 1-a \right ) ^{-1} \right ) ^{{\frac {a}{1-a}}+ \left ( 1-a \right ) ^{-1}} \left ( -4\,{\frac {{x}^{1-a}{a}^{ 2}}{1-a}}+8\,{\frac {a{x}^{1-a}}{1-a}}-4\,{\frac {{x}^{1-a}}{1-a}}+2\, a-2 \right ) \left ( {\frac {{x}^{1-a}}{1-a}} \right ) ^{ \left ( a-1 \right ) ^{-1}}{{\rm e}^{-2\,{\frac {{x}^{1-a}}{1-a}}}}{{\sl M}_{-{ \frac {a+1}{a-1}}+ \left ( a-1 \right ) ^{-1},\,- \left ( a-1 \right ) ^{- 1}+1/2}\left (4\,{\frac {{x}^{1-a}}{1-a}}\right )} \left ( {\frac {a}{1-a }}+ \left ( 1-a \right ) ^{-1} \right ) ^{-1}}-{\frac {1-a}{-3+a}{2}^{-1+ 2\,{\frac {a}{1-a}}+2\, \left ( 1-a \right ) ^{-1}+2\, \left ( a-1 \right ) ^{-1}}{x}^{-{\frac {{a}^{2}}{1-a}}+ \left ( 1-a \right ) ^{-1}- 1+a} \left ( \left ( 1-a \right ) ^{-1} \right ) ^{{\frac {a}{1-a}}+ \left ( 1-a \right ) ^{-1}} \left ( {\frac {{x}^{1-a}}{1-a}} \right ) ^{ \left ( a-1 \right ) ^{-1}}{{\rm e}^{-2\,{\frac {{x}^{1-a}}{1-a}}}}{ {\sl M}_{-{\frac {a+1}{a-1}}+ \left ( a-1 \right ) ^{-1}+1,\,- \left ( a- 1 \right ) ^{-1}+1/2}\left (4\,{\frac {{x}^{1-a}}{1-a}}\right )} \left ( { \frac {a}{1-a}}+ \left ( 1-a \right ) ^{-1} \right ) ^{-1}} \right ) }}}}} + \left ( {x}^{a} \right ) ^{-1} \right \} \]
Sage: cpu = 0 (sec), leaf count = 0 \[ \text {Maxima was unable to solve this ODE} \]