When I looked at this example, my first instinct was to homogenize only the generators of $I=\langle f_1 := y-x^2,f_2:=z-x^3\rangle$ in a new variable $w$. But then, I realized that I would miss some elements since $\tilde I \ni w^2z-xy, xz-y^2 \not \in \langle \tilde f_1,\tilde f_2 \rangle =\langle wy-x^2,w^2z-x^3\rangle$.
According to my source, one can use the colon ideal $(I:w^\infty):=\{f\mid w^nf\in I \text{ for some }n\}$ and Gröbner basis to check that $\tilde I$ is actually just $\langle wy-x^2,w^2z-xy, xz-y^2\rangle$.
Could someone provide me with a reference or work out this method here?