ODE
\[ a y'(x)^2+f(x) y(x) y'(x)+g(x) y(x)^2+y(x) y''(x)=0 \] ODE Classification
[[_2nd_order, _with_linear_symmetries]]
Book solution method
TO DO
Mathematica ✗
cpu = 41.8484 (sec), leaf count = 0 , could not solve
DSolve[g[x]*y[x]^2 + f[x]*y[x]*Derivative[1][y][x] + a*Derivative[1][y][x]^2 + y[x]*Derivative[2][y][x] == 0, y[x], x]
Maple ✓
cpu = 2.915 (sec), leaf count = 70
\[ \left \{ y \left ( x \right ) ={\it ODESolStruc} \left ( {{\rm e}^{\int \!{\it \_b} \left ( {\it \_a} \right ) \,{\rm d}{\it \_a}+{\it \_C1}}},[ \left \{ {\frac {\rm d}{{\rm d}{\it \_a}}}{\it \_b} \left ( {\it \_a} \right ) = \left ( -1-a \right ) \left ( {\it \_b} \left ( {\it \_a} \right ) \right ) ^{2}-f \left ( {\it \_a} \right ) {\it \_b} \left ( {\it \_a} \right ) -g \left ( {\it \_a} \right ) \right \} , \left \{ {\it \_a}=x,{\it \_b} \left ( {\it \_a} \right ) ={\frac {{\frac {\rm d}{{\rm d}x}}y \left ( x \right ) }{y \left ( x \right ) }} \right \} , \left \{ x={\it \_a},y \left ( x \right ) ={{\rm e}^{\int \!{\it \_b} \left ( {\it \_a} \right ) \,{\rm d}{\it \_a}+{\it \_C1}}} \right \} ] \right ) \right \} \] Mathematica raw input
DSolve[g[x]*y[x]^2 + f[x]*y[x]*y'[x] + a*y'[x]^2 + y[x]*y''[x] == 0,y[x],x]
Mathematica raw output
DSolve[g[x]*y[x]^2 + f[x]*y[x]*Derivative[1][y][x] + a*Derivative[1][y][x]^2 + y[x]*Derivative[2][y][x] == 0, y[x], x]
Maple raw input
dsolve(y(x)*diff(diff(y(x),x),x)+a*diff(y(x),x)^2+f(x)*y(x)*diff(y(x),x)+g(x)*y(x)^2 = 0, y(x),'implicit')
Maple raw output
y(x) = ODESolStruc(exp(Int(_b(_a),_a)+_C1),[{diff(_b(_a),_a) = (-1-a)*_b(_a)^2-f(_a)*_b(_a)-g(_a)}, {_a = x, _b(_a) = diff(y(x),x)/y(x)}, {x = _a, y(x) = exp(Int(_b(_a),_a)+_C1)}])