ODE
\[ f(x) y'(x)+g(x) y'(x)^2+y''(x)=0 \] ODE Classification
[[_2nd_order, _missing_y]]
Book solution method
TO DO
Mathematica ✓
cpu = 0.343443 (sec), leaf count = 66
\[\left \{\left \{y(x)\to \int _1^x\frac {\exp \left (\int _1^{K[3]}-f(K[1])dK[1]\right )}{c_1-\int _1^{K[3]}-\exp \left (\int _1^{K[2]}-f(K[1])dK[1]\right ) g(K[2])dK[2]}dK[3]+c_2\right \}\right \}\]
Maple ✓
cpu = 0.252 (sec), leaf count = 32
\[\left [y \left (x \right ) = \int \frac {{\mathrm e}^{-\left (\int f \left (x \right )d x \right )}}{\int {\mathrm e}^{-\left (\int f \left (x \right )d x \right )} g \left (x \right )d x +\textit {\_C1}}d x +\textit {\_C2}\right ]\] Mathematica raw input
DSolve[f[x]*y'[x] + g[x]*y'[x]^2 + y''[x] == 0,y[x],x]
Mathematica raw output
{{y[x] -> C[2] + Inactive[Integrate][E^Inactive[Integrate][-f[K[1]], {K[1], 1, K[3]}]/(C[1] - Inactive[Integrate][-(E^Inactive[Integrate][-f[K[1]], {K[1], 1, K[2]}]*g[K[2]]), {K[2], 1, K[3]}]), {K[3], 1, x}]}}
Maple raw input
dsolve(diff(diff(y(x),x),x)+f(x)*diff(y(x),x)+g(x)*diff(y(x),x)^2 = 0, y(x))
Maple raw output
[y(x) = Int(exp(-Int(f(x),x))/(Int(exp(-Int(f(x),x))*g(x),x)+_C1),x)+_C2]