$ Problem $ Deduce conditions on $ a (x) $, $ b (x) $, $ c (x) $ so that $$ y ^ \prime = a (x) y ^ 2 + b (x) y + c (x), y (x_0) = y_0 $$ have a unique solution for $ (x_0, y_0) \in R = [x_0- \alpha, x_0 + \alpha] \times [y_0- \beta, y_0 + \beta] $
$ Solution $: Applying the Picard-Lindelöf theorem, if we make $ f (x, y) = a (x) y ^ 2 + b (x) y + c (x) $, the Cauchy problem has a unique solution if $ f (x, y) $ is continuous in $ R $ and satisfies the condition of Lipschitz in the variable "$y$" in $ R $. That is, we need in the first instance that $ a (x) $, $ b (x) $, $ c (x) $ be continuous in $ [x_0- \alpha, x_0 + \alpha] $, and for the Lipschitz condition to be verified, $a(x)$ and $b(x)$ must be delimited in $ [x_0- \alpha, x_0 + \alpha] $. Is my argument correct? Thanks for reading everything. Any suggestion is welcome