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

ODE
\[ f(x) y'(x)+g(y(x)) y'(x)^2+y''(x)=0 \] ODE Classification

[_Liouville, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]]

Book solution method
TO DO

Mathematica ✓
cpu = 0.231416 (sec), leaf count = 61

\[\left \{\left \{y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\exp \left (-\int _1^{K[4]}-g(K[1])dK[1]\right )dK[4]\& \right ]\left [\int _1^x-\exp \left (-\int _1^{K[5]}f(K[2])dK[2]\right ) c_1dK[5]+c_2\right ]\right \}\right \}\]

Maple ✓
cpu = 0.103 (sec), leaf count = 29

\[[\int _{}^{y \left (x \right )}{\mathrm e}^{\int g \left (\textit {\_b} \right )d \textit {\_b}}d \textit {\_b} -\textit {\_C1} \left (\int {\mathrm e}^{-\left (\int f \left (x \right )d x \right )}d x \right )-\textit {\_C2} = 0]\] Mathematica raw input

DSolve[f[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[4]}]), {K[4], 1, #1}] & ][C[2] + Inactive[Integrate][-(C[1]/E^Inacti
ve[Integrate][f[K[2]], {K[2], 1, K[5]}]), {K[5], 1, x}]]}}

Maple raw input

dsolve(diff(diff(y(x),x),x)+g(y(x))*diff(y(x),x)^2+f(x)*diff(y(x),x) = 0, y(x))

Maple raw output

[Intat(exp(Int(g(_b),_b)),_b = y(x))-_C1*Int(exp(-Int(f(x),x)),x)-_C2 = 0]