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 = 25.6333 (sec), leaf count = 80
\[\left \{\left \{y(x)\to \int _1^x \left (\exp \left (-(k-1) \int _1^{K[3]} -g(K[1]) \, dK[1]\right ) \left (c_1-(k-1) \int _1^{K[3]} f(K[2]) \left (-e^{(k-1) \int _1^{K[2]} -g(K[1]) \, dK[1]}\right ) \, dK[2]\right )\right ){}^{\frac {1}{1-k}} \, dK[3]+c_2\right \}\right \}\]
Maple ✓
cpu = 0.243 (sec), leaf count = 45
\[ \left \{ y \left ( x \right ) =\int \! \left ( \left ( k-1 \right ) \int \!f \left ( x \right ) {{\rm e}^{-\int \!g \left ( x \right ) \,{\rm d}x \left ( k-1 \right ) }}\,{\rm d}x+{\it \_C1} \right ) ^{- \left ( k-1 \right ) ^{-1}}{{\rm e}^{-\int \!g \left ( x \right ) \,{\rm d}x}}\,{\rm d}x+{\it \_C2} \right \} \] 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] + Integrate[((C[1] - (-1 + k)*Integrate[-(E^((-1 + k)*Integrate[-g[K[1]], {K[1], 1, K[2]}])*f[K[2]]), {K[2], 1, K[3]}])/E^((-1 + k)*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),'implicit')
Maple raw output
y(x) = Int(((k-1)*Int(f(x)*exp(-Int(g(x),x)*(k-1)),x)+_C1)^(-1/(k-1))*exp(-Int(g(x),x)),x)+_C2