 Solution

Use the alternative affine coordinatesso that $x=t/w$ and $y=-1/w$. The equation becomesRecursive coefficient comparison gives a unique series $w(t)\in t^{3}k[[t]]$. Since $v(w(t))=3v(t)$ and hence $v(x)=-2v(t)$, $v(y)=-3v(t)$, this gives the formal coordinates on a short Weierstrass curve identification