ODE
\[ a y(x) \left (1-y(x)^n\right )+x^2 y''(x)=0 \] ODE Classification
[[_2nd_order, _with_linear_symmetries]]
Book solution method
TO DO
Mathematica ✗
cpu = 33.6338 (sec), leaf count = 0 , could not solve
DSolve[a*y[x]*(1 - y[x]^n) + x^2*Derivative[2][y][x] == 0, y[x], x]
Maple ✓
cpu = 1.322 (sec), leaf count = 67
\[ \left \{ y \left ( x \right ) ={\it ODESolStruc} \left ( {\it \_a},[ \left \{ {\frac {\rm d}{{\rm d}{\it \_a}}}{\it \_b} \left ( {\it \_a} \right ) = \left ( {\it \_b} \left ( {\it \_a} \right ) \right ) ^{2} \left ( {\it \_b} \left ( {\it \_a} \right ) {\it \_a}\,a-{{\it \_a}}^{n+1}{\it \_b} \left ( {\it \_a} \right ) a-1 \right ) \right \} , \left \{ {\it \_a}=y \left ( x \right ) ,{\it \_b} \left ( {\it \_a} \right ) ={\frac {1}{x{\frac {\rm d}{{\rm d}x}}y \left ( x \right ) }} \right \} , \left \{ x={{\rm e}^{\int \!{\it \_b} \left ( {\it \_a} \right ) \,{\rm d}{\it \_a}+{\it \_C1}}},y \left ( x \right ) ={\it \_a} \right \} ] \right ) \right \} \] Mathematica raw input
DSolve[a*y[x]*(1 - y[x]^n) + x^2*y''[x] == 0,y[x],x]
Mathematica raw output
DSolve[a*y[x]*(1 - y[x]^n) + x^2*Derivative[2][y][x] == 0, y[x], x]
Maple raw input
dsolve(x^2*diff(diff(y(x),x),x)+a*y(x)*(1-y(x)^n) = 0, y(x),'implicit')
Maple raw output
y(x) = ODESolStruc(_a,[{diff(_b(_a),_a) = _b(_a)^2*(_b(_a)*_a*a-_a^(n+1)*_b(_a)*a-1)}, {_a = y(x), _b(_a) = 1/x/diff(y(x),x)}, {x = exp(Int(_b(_a),_a)+_C1), y(x) = _a}])