4.37.14 \(f(x) y'(x)+g(x) y'(x)^2+y''(x)=0\)

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 = 9.9327 (sec), leaf count = 62

\[\left \{\left \{y(x)\to \int _1^x \frac {e^{\int _1^{K[3]} -f(K[1]) \, dK[1]}}{c_1-\int _1^{K[3]} g(K[2]) \left (-e^{\int _1^{K[2]} -f(K[1]) \, dK[1]}\right ) \, dK[2]} \, dK[3]+c_2\right \}\right \}\]

Maple ✓
cpu = 0.11 (sec), leaf count = 32

\[ \left \{ y \left ( x \right ) =\int \!{\frac {{{\rm e}^{-\int \!f \left ( x \right ) \,{\rm d}x}}}{\int \!{{\rm e}^{-\int \!f \left ( x \right ) \,{\rm d}x}}g \left ( x \right ) \,{\rm d}x+{\it \_C1}}}\,{\rm d}x+{\it \_C2} \right \} \] 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] + Integrate[E^Integrate[-f[K[1]], {K[1], 1, K[3]}]/(C[1] - Integr
ate[-(E^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),'implicit')

Maple raw output

y(x) = Int(exp(-Int(f(x),x))/(Int(exp(-Int(f(x),x))*g(x),x)+_C1),x)+_C2