Did I solve this diff eq substition problem properly?

[utnip123]

xy'=yln(xy)
xdy=yln(xy)dx
\frac{dy}{y}=\frac{ln(xy)dx}{x}

Substitution:

v=ln(xy)
dv = \frac{dy}{y}-\frac{dx}{x}
dv-\frac{dx}{x}=\frac{vdx}{x}

∫\frac{dv}{v+1}=∫\frac{dx}{x}

ln(v+1)=ln x + C

v+1 = Cx
ln(xy) +1 = Cx

Would that basically be the complete answer?

 

[mfb]

It might be interesting to solve this last equation for y=... and verify the answer with the initial equation.