For $f: \mathbb R \times \mathbb R \to \mathbb R$, $(t,z) \to t^3z^2$, consider the initial value problem $\dot x(t) = f(t,x(t)) = t^3(x(t))^2, x(\tau)=\xi$ where $(\tau, \xi) \in \mathbb R \times \mathbb R$.
How can I find $\tau \in \mathbb R$ and $\xi \in \mathbb R$ such that the initial value problem has a maximal solution with a bounded maximal interval of existence?
The general solution to the ODE is given by \begin{equation} x(t) = \frac{1}{c - \frac{1}{4} t^4}, \tag{1} \end{equation} where $c$ is determined by the initial conditons. The solution is thus not defined if \begin{equation} t^4 = 4 c. \end{equation} From this, you immediately see that if $c<0$, the solution $(1)$ is defined for all $t$. Since you're looking for a solution which only exists on a bounded interval, choosing $c<0$ is not a good option. If we now choose $c>0$, at which points is the solution not defined? Does that give you an idea what a suitable bounded maximal interval could be? And how can you express $c$ in terms of $\xi$ and $\tau$? Using the answers to these questions, can you derive a relation between $\xi$ and $\tau$?