ODE
\[ y''(x)=x^{n-2} f\left (x^{-n} y(x),x^{1-n} y'(x)\right ) \] ODE Classification
[[_2nd_order, _with_linear_symmetries]]
Book solution method
TO DO
Mathematica ✗
cpu = 7.7034 (sec), leaf count = 0 , could not solve
DSolve[Derivative[2][y][x] == x^(-2 + n)*f[y[x]/x^n, x^(1 - n)*Derivative[1][y][x]], y[x], x]
Maple ✓
cpu = 0.938 (sec), leaf count = 125
\[ \left \{ y \left ( x \right ) ={\it ODESolStruc} \left ( {\frac {{\it \_a}}{{{\rm e}^{- \left ( \int \!{\it \_b} \left ( {\it \_a} \right ) \,{\rm d}{\it \_a}+{\it \_C1} \right ) n}}}},[ \left \{ {\frac {\rm d}{{\rm d}{\it \_a}}}{\it \_b} \left ( {\it \_a} \right ) = \left ( -{\it \_b} \left ( {\it \_a} \right ) f \left ( {\it \_a},{\frac {{\it \_a}\,{\it \_b} \left ( {\it \_a} \right ) n+1}{{\it \_b} \left ( {\it \_a} \right ) }} \right ) +n{\it \_a}\, \left ( n-1 \right ) {\it \_b} \left ( {\it \_a} \right ) +2\,n-1 \right ) \left ( {\it \_b} \left ( {\it \_a} \right ) \right ) ^{2} \right \} , \left \{ {\it \_a}=y \left ( x \right ) {x}^{-n},{\it \_b} \left ( {\it \_a} \right ) ={\frac {1}{{x}^{-n} \left ( x{\frac {\rm d}{{\rm d}x}}y \left ( x \right ) -ny \left ( x \right ) \right ) }} \right \} , \left \{ x={{\rm e}^{\int \!{\it \_b} \left ( {\it \_a} \right ) \,{\rm d}{\it \_a}+{\it \_C1}}},y \left ( x \right ) ={\frac {{\it \_a}}{{{\rm e}^{- \left ( \int \!{\it \_b} \left ( {\it \_a} \right ) \,{\rm d}{\it \_a}+{\it \_C1} \right ) n}}}} \right \} ] \right ) \right \} \] Mathematica raw input
DSolve[y''[x] == x^(-2 + n)*f[y[x]/x^n, x^(1 - n)*y'[x]],y[x],x]
Mathematica raw output
DSolve[Derivative[2][y][x] == x^(-2 + n)*f[y[x]/x^n, x^(1 - n)*Derivative[1][y][x]], y[x], x]
Maple raw input
dsolve(diff(diff(y(x),x),x) = x^(n-2)*f(y(x)/(x^n),diff(y(x),x)/(x^(n-1))), y(x),'implicit')
Maple raw output
y(x) = ODESolStruc(_a/exp(-(Int(_b(_a),_a)+_C1)*n),[{diff(_b(_a),_a) = (-_b(_a)*f(_a,(_a*_b(_a)*n+1)/_b(_a))+n*_a*(n-1)*_b(_a)+2*n-1)*_b(_a)^2}, {_a = y(x)*x^(-n), _b(_a) = 1/(x^(-n))/(x*diff(y(x),x)-n*y(x))}, {x = exp(Int(_b(_a),_a)+_C1), y(x) = _a/exp(-(Int(_b(_a),_a)+_C1)*n)}])