ODE
\[ a y(x) y'(x)+f(x)+x y(x) y''(x)+x y'(x)^2=0 \] ODE Classification
[[_2nd_order, _exact, _nonlinear], [_2nd_order, _reducible, _mu_xy]]
Book solution method
TO DO
Mathematica ✓
cpu = 320.582 (sec), leaf count = 104
\[\left \{\left \{y(x)\to -\sqrt {2} \sqrt {\int _1^x K[3]^{-a} \left (\int _1^{K[3]} K[2]^{a-1} (-f(K[2])) \, dK[2]+c_1\right ) \, dK[3]+c_2}\right \},\left \{y(x)\to \sqrt {2} \sqrt {\int _1^x K[3]^{-a} \left (\int _1^{K[3]} K[2]^{a-1} (-f(K[2])) \, dK[2]+c_1\right ) \, dK[3]+c_2}\right \}\right \}\]
Maple ✓
cpu = 0.049 (sec), leaf count = 43
\[ \left \{ {\frac { \left ( -2\,\int \!{x}^{a-1}f \left ( x \right ) \,{\rm d}x-2\,{\it \_C1} \right ) {x}^{1-a}}{2}}+{\frac { \left ( a-1 \right ) \left ( y \left ( x \right ) \right ) ^{2}}{2}}+\int \!f \left ( x \right ) \,{\rm d}x+{\it \_C2}=0 \right \} \] Mathematica raw input
DSolve[f[x] + a*y[x]*y'[x] + x*y'[x]^2 + x*y[x]*y''[x] == 0,y[x],x]
Mathematica raw output
{{y[x] -> -(Sqrt[2]*Sqrt[C[2] + Integrate[(C[1] + Integrate[-(f[K[2]]*K[2]^(-1 + a)), {K[2], 1, K[3]}])/K[3]^a, {K[3], 1, x}]])}, {y[x] -> Sqrt[2]*Sqrt[C[2] + Integrate[(C[1] + Integrate[-(f[K[2]]*K[2]^(-1 + a)), {K[2], 1, K[3]}])/K[3]^a, {K[3], 1, x}]]}}
Maple raw input
dsolve(x*y(x)*diff(diff(y(x),x),x)+x*diff(y(x),x)^2+a*y(x)*diff(y(x),x)+f(x) = 0, y(x),'implicit')
Maple raw output
1/2*(-2*Int(x^(a-1)*f(x),x)-2*_C1)*x^(1-a)+1/2*(a-1)*y(x)^2+Int(f(x),x)+_C2 = 0