ODE
\[ y(x) y''(x)=-y(x) \left (f'(x)-y(x)^2 g'(x)\right )+y'(x) \left (f(x)+g(x) y(x)^2\right )+y'(x)^2 \] ODE Classification
(ODEtools/info) missing specification of intermediate function
Book solution method
TO DO
Mathematica ✗
cpu = 21.005 (sec), leaf count = 0 , could not solve
DSolve[y[x]*Derivative[2][y][x] == -(y[x]*(Derivative[1][f][x] - y[x]^2*Derivative[1][g][x])) + (f[x] + g[x]*y[x]^2)*Derivative[1][y][x] + Derivative[1][y][x]^2, y[x], x]
Maple ✓
cpu = 0.319 (sec), leaf count = 53
\[ \left \{ y \left ( x \right ) ={\it ODESolStruc} \left ( {\it \_b} \left ( {\it \_a} \right ) ,[ \left \{ {\frac {-g \left ( {\it \_a} \right ) \left ( {\it \_b} \left ( {\it \_a} \right ) \right ) ^{2}+{\it \_C1}\,{\it \_b} \left ( {\it \_a} \right ) +{\frac {\rm d}{{\rm d}{\it \_a}}}{\it \_b} \left ( {\it \_a} \right ) +f \left ( {\it \_a} \right ) }{{\it \_b} \left ( {\it \_a} \right ) }}=0 \right \} , \left \{ {\it \_a}=x,{\it \_b} \left ( {\it \_a} \right ) =y \left ( x \right ) \right \} , \left \{ x={\it \_a},y \left ( x \right ) ={\it \_b} \left ( {\it \_a} \right ) \right \} ] \right ) \right \} \] Mathematica raw input
DSolve[y[x]*y''[x] == -(y[x]*(f'[x] - y[x]^2*g'[x])) + (f[x] + g[x]*y[x]^2)*y'[x] + y'[x]^2,y[x],x]
Mathematica raw output
DSolve[y[x]*Derivative[2][y][x] == -(y[x]*(Derivative[1][f][x] - y[x]^2*Derivative[1][g][x])) + (f[x] + g[x]*y[x]^2)*Derivative[1][y][x] + Derivative[1][y][x]^2, y[x], x]
Maple raw input
dsolve(y(x)*diff(diff(y(x),x),x) = diff(y(x),x)^2+(f(x)+g(x)*y(x)^2)*diff(y(x),x)-y(x)*(diff(f(x),x)-diff(g(x),x)*y(x)^2), y(x),'implicit')
Maple raw output
y(x) = ODESolStruc(_b(_a),[{(-g(_a)*_b(_a)^2+_C1*_b(_a)+diff(_b(_a),_a)+f(_a))/_b(_a) = 0}, {_a = x, _b(_a) = y(x)}, {x = _a, y(x) = _b(_a)}])