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 = 9.98744 (sec), leaf count = 57
\[\left \{\left \{y(x)\to c_2 \exp \left (\int _1^x e^{\int _1^{K[3]} f(K[1]) \, dK[1]} \left (\int _1^{K[3]} g(K[2]) e^{-\int _1^{K[2]} f(K[1]) \, dK[1]} \, dK[2]+c_1\right ) \, dK[3]\right )\right \}\right \}\]
Maple ✓
cpu = 0.135 (sec), leaf count = 61
\[ \left \{ \int \!{{\rm e}^{\int \!f \left ( x \right ) \,{\rm d}x}}\,{\rm d}x\int \!{{\rm e}^{-\int \!f \left ( x \right ) \,{\rm d}x}}g \left ( x \right ) \,{\rm d}x-{\it \_C1}\,\int \!{{\rm e}^{\int \!f \left ( x \right ) \,{\rm d}x}}\,{\rm d}x-\ln \left ( y \left ( x \right ) \right ) -\int \!\int \!{{\rm e}^{\int \!f \left ( x \right ) \,{\rm d}x}}\,{\rm d}x{{\rm e}^{-\int \!f \left ( x \right ) \,{\rm d}x}}g \left ( x \right ) \,{\rm d}x+{\it \_C2}=0 \right \} \] 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^Integrate[E^Integrate[f[K[1]], {K[1], 1, K[3]}]*(C[1] + Integrate[g[K[2]]/E^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),'implicit')
Maple raw output
Int(exp(Int(f(x),x)),x)*Int(exp(-Int(f(x),x))*g(x),x)-_C1*Int(exp(Int(f(x),x)),x)-ln(y(x))-Int(Int(exp(Int(f(x),x)),x)*exp(-Int(f(x),x))*g(x),x)+_C2 = 0