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Q.
If $y = y ( x )$ is the solution of the differential equation $2 x^{2} \frac{d y}{d x}-2 x y+3 y^{2}=0$ such that $y(e)=\frac{e}{3}$, then $y(1)$ is equal to
$\frac{ dy }{ dx }-\frac{ y }{ x }=-\frac{3}{2}\left(\frac{ y }{ x }\right)^{2}$
$y = vx$
$\frac{ dv }{ v ^{2}}=-\frac{3 dx }{2 x }$
$-\frac{1}{ v }=-\frac{3}{2} \ln | x |+ C$
$-\frac{ x }{ y }=\frac{-3}{2} \ln | x |+ C$
$x = e , y =\frac{ e }{3}$
$C =-\frac{3}{2}$
When $x =1, y =\frac{2}{3}$