4.12.23 \(x (3-x y(x)) y'(x)=y(x) (x y(x)-1)\)

ODE
\[ x (3-x y(x)) y'(x)=y(x) (x y(x)-1) \] ODE Classification

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

Book solution method
Exact equation, integrating factor

Mathematica
cpu = 4.95322 (sec), leaf count = 30

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

Maple
cpu = 0.011 (sec), leaf count = 21

\[ \left \{ \ln \relax (x ) -{\it \_C1}+{\frac {xy \relax (x ) }{2}}-{\frac {3\,\ln \left (xy \relax (x ) \right ) }{2}}=0 \right \} \] Mathematica raw input

DSolve[x*(3 - x*y[x])*y'[x] == y[x]*(-1 + x*y[x]),y[x],x]

Mathematica raw output

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

Maple raw input

dsolve(x*(3-x*y(x))*diff(y(x),x) = y(x)*(x*y(x)-1), y(x),'implicit')

Maple raw output

ln(x)-_C1+1/2*x*y(x)-3/2*ln(x*y(x)) = 0