4.12.20 \(x (1-x y(x)) y'(x)+y(x) (x y(x)+1)=0\)

ODE
\[ x (1-x y(x)) y'(x)+y(x) (x y(x)+1)=0 \] ODE Classification

[[_homogeneous, `class G`], _rational, [_Abel, `2nd type``class B`]]

Book solution method
Exact equation, integrating factor

Mathematica
cpu = 0.98151 (sec), leaf count = 30

\[\left \{\left \{y(x)\to -\frac {1}{x W\left (\frac {e^{\frac {9 c_1}{2^{2/3}}-1}}{x^2}\right )}\right \}\right \}\]

Maple
cpu = 0.017 (sec), leaf count = 29

\[ \left \{ \ln \relax (x ) -{\it \_C1}-{\frac {\ln \left (xy \relax (x ) \right ) xy \relax (x ) +1}{2\,xy \relax (x ) }}=0 \right \} \] Mathematica raw input

DSolve[y[x]*(1 + x*y[x]) + x*(1 - x*y[x])*y'[x] == 0,y[x],x]

Mathematica raw output

{{y[x] -> -(1/(x*ProductLog[E^(-1 + (9*C[1])/2^(2/3))/x^2]))}}

Maple raw input

dsolve(x*(1-x*y(x))*diff(y(x),x)+(1+x*y(x))*y(x) = 0, y(x),'implicit')

Maple raw output

ln(x)-_C1-1/2*(ln(x*y(x))*x*y(x)+1)/x/y(x) = 0