4.12.29 \(x (1-2 x y(x)) y'(x)+y(x) (2 x y(x)+1)=0\)

ODE
\[ x (1-2 x y(x)) y'(x)+y(x) (2 x y(x)+1)=0 \] ODE Classification

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

Book solution method
Change of Variable, Two new variables

Mathematica
cpu = 0.88013 (sec), leaf count = 32

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

Maple
cpu = 0.024 (sec), leaf count = 30

\[ \left \{ \ln \relax (x ) -{\it \_C1}-{\frac {2\,\ln \left (xy \relax (x ) \right ) xy \relax (x ) +1}{4\,xy \relax (x ) }}=0 \right \} \] Mathematica raw input

DSolve[y[x]*(1 + 2*x*y[x]) + x*(1 - 2*x*y[x])*y'[x] == 0,y[x],x]

Mathematica raw output

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

Maple raw input

dsolve(x*(1-2*x*y(x))*diff(y(x),x)+y(x)*(1+2*x*y(x)) = 0, y(x),'implicit')

Maple raw output

ln(x)-_C1-1/4*(2*ln(x*y(x))*x*y(x)+1)/x/y(x) = 0