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 = 0.357747 (sec), leaf count = 86
\[\left \{\left \{y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {1}{\exp \left (\int _1^{K[3]}-g(K[1])dK[1]\right ) c_1+\exp \left (\int _1^{K[3]}-g(K[1])dK[1]\right ) \int _1^{K[3]}-\exp \left (-\int _1^{K[2]}-g(K[1])dK[1]\right ) f(K[2])dK[2]}dK[3]\& \right ][x+c_2]\right \}\right \}\]
Maple ✓
cpu = 0.278 (sec), leaf count = 44
\[\left [y \left (x \right ) = \textit {\_C1}, \int _{}^{y \left (x \right )}\frac {{\mathrm e}^{\int g \left (\textit {\_g} \right )d \textit {\_g}}}{\int -f \left (\textit {\_g} \right ) {\mathrm e}^{\int g \left (\textit {\_g} \right )d \textit {\_g}}d \textit {\_g} +\textit {\_C1}}d \textit {\_g} -x -\textit {\_C2} = 0\right ]\] 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[Inactive[Integrate][(E^Inactive[Integrate][-g[K[1]], {K[1], 1, K[3]}]*C[1] + E^Inactive[Integrate][-g[K[1]], {K[1], 1, K[3]}]*Inactive[Integrate][-(f[K[2]]/E^Inactive[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))
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]