Using the existence-uniqueness theorem

Question

Given an IVP: $\frac{dx}{dt}=\sqrt{xt}$ where $x(0)=1$, can the existence-uniqueness theorem be used since the theorem requires $x(0)=1$ to be in an open rectangle.

Answer 1

You can't use the existence/uniqueness theorem for the reason you described.

To see this, separate variables to get a solution for the ODE, bearing in mind that we restrict to the positive solution of $x(t)$: $$\frac{dx}{\sqrt{x}} = \sqrt{t}dt \implies 2\sqrt{x(t)} = \frac{2}{3}t^{3/2} + 2C \implies x(t) = \left( \frac{1}{3}t^{3/2} + C\right)^2.$$ We can pick either $C = 1$ or $C = -1$ to satisfy the initial condition $x(0) = 1$.