Question

Determine an appropriate solution to solve xdy/dx=yln(xy)

Answer 1

$x\dfrac{\mathrm dy}{\mathrm dx}=y\ln(xy)$

Let $v=xy$, so that $\dfrac{\mathrm dv}{\mathrm dx}=y+x\dfrac{\mathrm dy}{\mathrm dx}$, or $\dfrac{\mathrm dv}{\mathrm dx}-\dfrac vx=x\dfrac{\mathrm dy}{\mathrm dx}$. Then you have

$\dfrac{\mathrm dv}{\mathrm dx}-\dfrac vx=\dfrac vx\ln v$
$\dfrac{\mathrm dv}{v(\ln v+1)}=\dfrac{\mathrm dx}x$

Integrate both sides to get

$\ln|\ln v+1|=\ln|x|+C$
$\ln v+1=Cx$
$v=e^{Cx-1}$

then back-substitute to find the solution for $y$.

$xy=e^{Cx-1}\implies y=\dfrac{e^{Cx-1}}x$