ODE
\[ y(x) y''(x)=f(x) y(x) y'(x)+g(x) y(x)^2+y'(x)^2 \] ODE Classification
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_xy]]
Book solution method
TO DO
Mathematica ✓
cpu = 0.232971 (sec), leaf count = 61
\[\left \{\left \{y(x)\to c_2 \exp \left (\int _1^x\exp \left (\int _1^{K[3]}f(K[1])dK[1]\right ) \left (c_1+\int _1^{K[3]}\exp \left (-\int _1^{K[2]}f(K[1])dK[1]\right ) g(K[2])dK[2]\right )dK[3]\right )\right \}\right \}\]
Maple ✓
cpu = 1.874 (sec), leaf count = 61
\[[y \left (x \right ) = {\mathrm e}^{\left (\int {\mathrm e}^{\int f \left (x \right )d x}d x \right ) \left (\int {\mathrm e}^{\int -f \left (x \right )d x} g \left (x \right )d x \right )} {\mathrm e}^{\int -\textit {\_C1} \,{\mathrm e}^{\int f \left (x \right )d x}d x} {\mathrm e}^{\int \left (\int -{\mathrm e}^{\int f \left (x \right )d x}d x \right ) {\mathrm e}^{\int -f \left (x \right )d x} g \left (x \right )d x} \textit {\_C2}]\] Mathematica raw input
DSolve[y[x]*y''[x] == g[x]*y[x]^2 + f[x]*y[x]*y'[x] + y'[x]^2,y[x],x]
Mathematica raw output
{{y[x] -> E^Inactive[Integrate][E^Inactive[Integrate][f[K[1]], {K[1], 1, K[3]}]*(C[1] + Inactive[Integrate][g[K[2]]/E^Inactive[Integrate][f[K[1]], {K[1], 1, K[2]}], {K[2], 1, K[3]}]), {K[3], 1, x}]*C[2]}}
Maple raw input
dsolve(y(x)*diff(diff(y(x),x),x) = diff(y(x),x)^2+f(x)*y(x)*diff(y(x),x)+g(x)*y(x)^2, y(x))
Maple raw output
[y(x) = exp(Int(exp(Int(f(x),x)),x)*Int(1/exp(Int(f(x),x))*g(x),x))/exp(_C1*Int(exp(Int(f(x),x)),x))/exp(Int(Int(exp(Int(f(x),x)),x)/exp(Int(f(x),x))*g(x),x))*_C2]