ODE
\[ f(y(x)) y'(x)+g(y(x)) y'(x)^2+y''(x)=0 \] ODE Classification
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_xy]]
Book solution method
TO DO
Mathematica ✓
cpu = 45.286 (sec), leaf count = 81
\[\left \{\left \{y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}} \frac {1}{c_1 e^{\int _1^{K[3]} -g(K[1]) \, dK[1]}+e^{\int _1^{K[3]} -g(K[1]) \, dK[1]} \int _1^{K[3]} f(K[2]) \left (-e^{-\int _1^{K[2]} -g(K[1]) \, dK[1]}\right ) \, dK[2]} \, dK[3]\& \right ]\left [c_2+x\right ]\right \}\right \}\]
Maple ✓
cpu = 0.173 (sec), leaf count = 44
\[ \left \{ \int ^{y \left ( x \right ) }\!{\frac {1}{ \left ( \int \!-f \left ( {\it \_g} \right ) {{\rm e}^{\int \!g \left ( {\it \_g} \right ) \,{\rm d}{\it \_g}}}\,{\rm d}{\it \_g}+{\it \_C1} \right ) {{\rm e}^{\int \!-g \left ( {\it \_g} \right ) \,{\rm d}{\it \_g}}}}}{d{\it \_g}}-x-{\it \_C2}=0,y \left ( x \right ) ={\it \_C1} \right \} \] Mathematica raw input
DSolve[f[y[x]]*y'[x] + g[y[x]]*y'[x]^2 + y''[x] == 0,y[x],x]
Mathematica raw output
{{y[x] -> InverseFunction[Integrate[(E^Integrate[-g[K[1]], {K[1], 1, K[3]}]*C[1] + E^Integrate[-g[K[1]], {K[1], 1, K[3]}]*Integrate[-(f[K[2]]/E^Integrate[-g[K[1]], {K[1], 1, K[2]}]), {K[2], 1, K[3]}])^(-1), {K[3], 1, #1}] & ][x + C[2]]}}
Maple raw input
dsolve(diff(diff(y(x),x),x)+g(y(x))*diff(y(x),x)^2+f(y(x))*diff(y(x),x) = 0, y(x),'implicit')
Maple raw output
y(x) = _C1, Intat(1/(Int(-f(_g)*exp(Int(g(_g),_g)),_g)+_C1)/exp(Int(-g(_g),_g)),_g = y(x))-x-_C2 = 0