ODE
\[ y''(x)=(f(x)-2 y(x)) y'(x)+f(x) y(x)^2+g(x) \] ODE Classification
[[_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]]
Book solution method
TO DO
Mathematica ✗
cpu = 0.306519 (sec), leaf count = 0 , could not solve
DSolve[Derivative[2][y][x] == g[x] + f[x]*y[x]^2 + (f[x] - 2*y[x])*Derivative[1][y][x], y[x], x]
Maple ✓
cpu = 0.631 (sec), leaf count = 72
\[ \left \{ y \left ( x \right ) ={\it ODESolStruc} \left ( {\it \_b} \left ( {\it \_a} \right ) ,[ \left \{ {\frac {1}{{{\rm e}^{\int \!f \left ( {\it \_a} \right ) \,{\rm d}{\it \_a}}}} \left ( {\it \_C1}\,{{\rm e}^{\int \!f \left ( {\it \_a} \right ) \,{\rm d}{\it \_a}}}+ \left ( {\it \_b} \left ( {\it \_a} \right ) \right ) ^{2}-\int \!{\frac {g \left ( {\it \_a} \right ) }{{{\rm e}^{\int \!f \left ( {\it \_a} \right ) \,{\rm d}{\it \_a}}}}}\,{\rm d}{\it \_a}{{\rm e}^{\int \!f \left ( {\it \_a} \right ) \,{\rm d}{\it \_a}}}+{\frac {\rm d}{{\rm d}{\it \_a}}}{\it \_b} \left ( {\it \_a} \right ) \right ) }=0 \right \} , \left \{ {\it \_a}=x,{\it \_b} \left ( {\it \_a} \right ) =y \left ( x \right ) \right \} , \left \{ x={\it \_a},y \left ( x \right ) ={\it \_b} \left ( {\it \_a} \right ) \right \} ] \right ) \right \} \] Mathematica raw input
DSolve[y''[x] == g[x] + f[x]*y[x]^2 + (f[x] - 2*y[x])*y'[x],y[x],x]
Mathematica raw output
DSolve[Derivative[2][y][x] == g[x] + f[x]*y[x]^2 + (f[x] - 2*y[x])*Derivative[1][y][x], y[x], x]
Maple raw input
dsolve(diff(diff(y(x),x),x) = (f(x)-2*y(x))*diff(y(x),x)+g(x)+f(x)*y(x)^2, y(x),'implicit')
Maple raw output
y(x) = ODESolStruc(_b(_a),[{(_C1*exp(Int(f(_a),_a))+_b(_a)^2-Int(1/exp(Int(f(_a),_a))*g(_a),_a)*exp(Int(f(_a),_a))+diff(_b(_a),_a))/exp(Int(f(_a),_a)) = 0}, {_a = x, _b(_a) = y(x)}, {x = _a, y(x) = _b(_a)}])