ODE
\[ f(x) f'(x) y'(x)+f(x)^2 y''(x)=g\left (y(x),f(x) y'(x)\right ) \] ODE Classification
(ODEtools/info) missing specification of intermediate function
Book solution method
TO DO
Mathematica ✗
cpu = 0.973163 (sec), leaf count = 0 , could not solve
DSolve[f[x]*Derivative[1][f][x]*Derivative[1][y][x] + f[x]^2*Derivative[2][y][x] == g[y[x], f[x]*Derivative[1][y][x]], y[x], x]
Maple ✓
cpu = 0.319 (sec), leaf count = 68
\[ \left \{ y \left ( x \right ) ={\it ODESolStruc} \left ( {\it \_a},[ \left \{ {\frac {\rm d}{{\rm d}{\it \_a}}}{\it \_b} \left ( {\it \_a} \right ) =-g \left ( {\it \_a}, \left ( {\it \_b} \left ( {\it \_a} \right ) \right ) ^{-1} \right ) \left ( {\it \_b} \left ( {\it \_a} \right ) \right ) ^{3} \right \} , \left \{ {\it \_a}=y \left ( x \right ) ,{\it \_b} \left ( {\it \_a} \right ) ={\frac {1}{f \left ( x \right ) {\frac {\rm d}{{\rm d}x}}y \left ( x \right ) }} \right \} , \left \{ x={\it RootOf} \left ( \int \!{\it \_b} \left ( {\it \_a} \right ) \,{\rm d}{\it \_a}+{\it \_C1}-\int ^{{\it \_Z}}\! \left ( f \left ( {\it \_f} \right ) \right ) ^{-1}{d{\it \_f}} \right ) ,y \left ( x \right ) ={\it \_a} \right \} ] \right ) \right \} \] Mathematica raw input
DSolve[f[x]*f'[x]*y'[x] + f[x]^2*y''[x] == g[y[x], f[x]*y'[x]],y[x],x]
Mathematica raw output
DSolve[f[x]*Derivative[1][f][x]*Derivative[1][y][x] + f[x]^2*Derivative[2][y][x] == g[y[x], f[x]*Derivative[1][y][x]], y[x], x]
Maple raw input
dsolve(f(x)^2*diff(diff(y(x),x),x)+f(x)*diff(f(x),x)*diff(y(x),x) = g(y(x),f(x)*diff(y(x),x)), y(x),'implicit')
Maple raw output
y(x) = ODESolStruc(_a,[{diff(_b(_a),_a) = -g(_a,1/_b(_a))*_b(_a)^3}, {_a = y(x), _b(_a) = 1/f(x)/diff(y(x),x)}, {x = RootOf(Int(_b(_a),_a)+_C1-Intat(1/f(_f),_f = _Z)), y(x) = _a}])