$x^{3} dy+x y \,dx=x^{2} \,dy+2y \,dx$
$\Rightarrow dy\left(x^{3}-x^{2}\right)=dx (2 y-x y)$
$\Rightarrow -\int \frac{1}{y} dy=\int \frac{x-2}{x^{2}(x-1)} dx$
$\Rightarrow \quad-\ell ny =\int\left(\frac{ A }{ x }+\frac{ B }{ x ^{2}}+\frac{ C }{( x -1)}\right) dx$
Where $A =1, B =+2, C =-1$
$\Rightarrow -\ell n y=\ln x-\frac{2}{x}-\ln (x-1)+\lambda$
$ \Rightarrow y(2)=e$
$\Rightarrow -1=\ln 2-1-0+\lambda $
$ \therefore \lambda=-\ln 2$
$\Rightarrow \ell n y =-\ln x +\frac{2}{ x }+\ln ( x -1)+\ln 2$
Now put $x=4$ in equation
$\Rightarrow \ell ny =-\ln 4+\frac{1}{2}+\ln 3+\ln 2$
$\Rightarrow \quad \ell n y =\ln \left(\frac{3}{2}\right)+\frac{1}{2} \ell ne$
$\Rightarrow \quad y=\frac{3}{2} \sqrt{ e }$