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 = 0.244315 (sec), leaf count = 108
\[\left \{\left \{y(x)\to -\sqrt {2} \sqrt {\int _1^x-K[2]^{-a} \left (c_1+\int _1^{K[2]}f(K[1]) K[1]^{a-1}dK[1]\right )dK[2]+c_2}\right \},\left \{y(x)\to \sqrt {2} \sqrt {\int _1^x-K[2]^{-a} \left (c_1+\int _1^{K[2]}f(K[1]) K[1]^{a-1}dK[1]\right )dK[2]+c_2}\right \}\right \}\]
Maple ✓
cpu = 0.419 (sec), leaf count = 114
\[\left [y \left (x \right ) = \frac {\sqrt {2}\, \sqrt {\left (a -1\right ) \left (x^{1-a} \left (\int \frac {x^{a} f \left (x \right )}{x}d x \right )+x^{1-a} \textit {\_C1} -\left (\int f \left (x \right )d x \right )-\textit {\_C2} \right )}}{a -1}, y \left (x \right ) = -\frac {\sqrt {2}\, \sqrt {\left (a -1\right ) \left (x^{1-a} \left (\int \frac {x^{a} f \left (x \right )}{x}d x \right )+x^{1-a} \textit {\_C1} -\left (\int f \left (x \right )d x \right )-\textit {\_C2} \right )}}{a -1}\right ]\] 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] + Inactive[Integrate][-((C[1] + Inactive[Integrate][f[K[1]]*K[1]^(-1 + a), {K[1], 1, K[2]}])/K[2]^a), {K[2], 1, x}]])}, {y[x] -> Sqrt[2]*Sqrt[C[2] + Inactive[Integrate][-((C[1] + Inactive[Integrate][f[K[1]]*K[1]^(-1 + a), {K[1], 1, K[2]}])/K[2]^a), {K[2], 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))
Maple raw output
[y(x) = 1/(a-1)*2^(1/2)*((a-1)*(x^(1-a)*Int(x^a/x*f(x),x)+x^(1-a)*_C1-Int(f(x),x)-_C2))^(1/2), y(x) = -1/(a-1)*2^(1/2)*((a-1)*(x^(1-a)*Int(x^a/x*f(x),x)+x^(1-a)*_C1-Int(f(x),x)-_C2))^(1/2)]