Consider this curve $f(x,y)=0$ given by

$$ f(x,y) := y^3 + y^2 x + x^4 =0.$$ Is it obvious that after a change of coordinates near the origin, this curve is equivalent to

$$ \hat{y}^2 \hat{x} + \hat{x}^4 = 0 $$

I think, these are both $D_5$ singularities. It seems like the change of coordinates that would achieve this is of the form

$$ x = \hat{x} - y + c_2 y^2 + c_3 y^3 + \ldots $$

where $x$ is an infinite power series. We can kill off the coefficients of $y^n$, for all $n$. This would give us a factor of $\hat{x}$, i.e we get something of the form $$ f = \hat{x}\cdot g $$ And then we can make another change of coordinate, so that $g$ becomes $$ g = \hat{y}^2 + \hat{x}^3.$$ Is there a simpler way to prove this? And aside from proving the power series converges, is there anything missing in the proof? Everything is over the complex numbers.