Show this IVP is Lipschitz continuous

Question

I need to show that the IVP $y'=t^3, y(0)=0$ is Lipschitz continuous in $y$. I know I need to find some number $L$ so that $\forall y_1, y_2$ and $t\in[a,b]$, $|f(t,y_2)-f(t,y_1)|\leq L|y_2-y_1|$.

I'm not quite sure what to do with this function since there is no $y$. This is what I'm thinking so far: $|y_2^3-y_1^3|=|y_2-y_1|\cdot|y_2^2+y_1y_2+y_1^2|$. Technically we can bound $|y_2^2+y_1y_2+y_1^2|$ by some unknown real constant $M$, but I feel that's cheap with a lack of mathematical justification.

Any thoughts?