ODE
\[ f(x) y'(x)^k+g(x) y'(x)+y''(x)=0 \] ODE Classification
[[_2nd_order, _missing_y]]
Book solution method
TO DO
Mathematica ✓
cpu = 0.446836 (sec), leaf count = 84
\[\left \{\left \{y(x)\to \int _1^x\left (\exp \left (-\left ((k-1) \int _1^{K[3]}-g(K[1])dK[1]\right )\right ) \left (c_1-(k-1) \int _1^{K[3]}-\exp \left ((k-1) \int _1^{K[2]}-g(K[1])dK[1]\right ) f(K[2])dK[2]\right )\right ){}^{\frac {1}{1-k}}dK[3]+c_2\right \}\right \}\]
Maple ✓
cpu = 0.712 (sec), leaf count = 60
\[\left [y \left (x \right ) = \int \left (k \left (\int f \left (x \right ) {\mathrm e}^{-\left (\int g \left (x \right )d x \right ) \left (k -1\right )}d x \right )+\textit {\_C1} -\left (\int f \left (x \right ) {\mathrm e}^{-\left (\int g \left (x \right )d x \right ) \left (k -1\right )}d x \right )\right )^{-\frac {1}{k -1}} {\mathrm e}^{-\left (\int g \left (x \right )d x \right )}d x +\textit {\_C2}\right ]\] Mathematica raw input
DSolve[g[x]*y'[x] + f[x]*y'[x]^k + y''[x] == 0,y[x],x]
Mathematica raw output
{{y[x] -> C[2] + Inactive[Integrate][((C[1] - (-1 + k)*Inactive[Integrate][-(E^((-1 + k)*Inactive[Integrate][-g[K[1]], {K[1], 1, K[2]}])*f[K[2]]), {K[2], 1, K[3]}])/E^((-1 + k)*Inactive[Integrate][-g[K[1]], {K[1], 1, K[3]}]))^(1 - k)^(-1), {K[3], 1, x}]}}
Maple raw input
dsolve(diff(diff(y(x),x),x)+f(x)*diff(y(x),x)^k+g(x)*diff(y(x),x) = 0, y(x))
Maple raw output
[y(x) = Int((k*Int(f(x)*exp(-Int(g(x),x)*(k-1)),x)+_C1-Int(f(x)*exp(-Int(g(x),x)*(k-1)),x))^(-1/(k-1))*exp(-Int(g(x),x)),x)+_C2]