I am trying to find the regions for the differential equation $\frac{dy}{dx}=\sqrt{xy} $ for which the solution is unique, with graph passing through
the point $(x_0, y_0)$ in these regions.
I tried to applied Picard theorem for existence and uniqueness. The function $\frac{\partial f}{\partial y}(x,y) = \frac{\sqrt x}{2 \sqrt y}$ is defined when $xy\geq 0$ and $y \neq 0$ thus we have existence and uniqueness in the
regions $R = [0, \infty)\times (0, \infty)$ and $R = (-\infty, 0]\times (-\infty, 0)$ for any $(x_0, y_0) \in R$.
However, answer is $R = (0, \infty)\times (0, \infty)$ and $R = (-\infty, 0)\times (-\infty, 0)$.
My question is why not $x=0$ is included in the region of existence and uniquness?
Thank you
Best Answer
Your analysis is correct. The only reason to exclude $x=0$ would be that the requested regions have to be open sets by the definition of the term "region" or "domain of the ODE".
This could be also be implied in
because to "pass through" there needs to be a left and right of $x_0$, resp. $x_0$ needs to be an inner point of the domain of the solution, which is not given if $x_0=0$