Find all integer solutions $(x,y)$ such that $2x^2 + y^2 = 3x^2y$ 
Find all integer solutions $(x,y)$ such that $2x^2 + y^2 = 3x^2y$.

We can rearrange the given equation to $$y^2 = x^2(3y-2)\tag1$$ Thus $3y-2$ must be a perfect square and so $3y-2 = k^2$.
How can we continue?

You have that $3y-2$ divides $y^2$.
Notice that $(3y-2,y^2)| (3y-2,y)^2$.
On the other hand, if $d$ divides $3y-2$ and $y$ we have $d|2$. So $(3y-2,y^2)| 4$.
Therefore $3y-2|4$
Therefore $y=0,1$ or $2$.
If $y=0$ we have $x=0$.
If $y=1$ we have $x=\pm 1$
If $y=2$ we have $x=\pm 1$
