= Solution
Use the two charts of the <blowup of the affine plane at the origin>. In the chart $y=xv$, the total transform is
$$
x^2-y^3=x^2(1-xv^3),
$$
so the <proper transform> has equation $1-xv^3=0$. It misses the exceptional divisor $x=0$ and is smooth, since its derivative with respect to $x$ is $-v^3\ne0$ there.
In the chart $x=yu$, the total transform is $y^2(u^2-y)$, so the proper transform is
$$
\boxed{y=u^2,\qquad x=u^3.}
$$
Its equation $u^2-y=0$ has partial derivative $-1$ with respect to $y$, so it is smooth even at its unique intersection $(u,y)=(0,0)$ with the exceptional divisor. The two chart descriptions agree on their overlap. Thus \b[one blowup makes the cusp nonsingular], although it is tangent to the exceptional divisor there.
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