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I'm trying to find a rectangular region $\mathcal{R}$ of the $ty$-plane for which the differential equation $(1+y^3)y' = t^2$ would have a unique solution through a point $(t_0, y_0)$ in $\mathcal{R}$.

When talking about uniqueness of solutions of DEs, though, the context in which I learned about them was of an initial value problem with a non-arbitrary point. Picard's Existence and Uniqueness theorem from my book has associated example problems which only show, well, existence and uniqueness of solutions. Additionally, the example problems from my book and lecture have an actual initial condition, so I'm not sure how to use Picard's theorem to determine the actual rectangular region $\mathcal{R}$. I hate to be that guy, but I'm sincerely unsure where to begin.

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Rewrite the equation as $$ y'=\frac{t^2}{1+y^3}. $$ The right hand side has a singularity at $y=-1$. Any rectangle $[a,b]\times[c,d]$ such that $-1\notin[c,d]$ will do. That the solution goes through $(x_0,y_0)$ translates into the initial condition $y(x_0)=y_0$.

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