So I know that the maximal solution to an ODE is the biggest interval where the solution is defined, but imagine you are working with this DE:
$$ x(t)\frac{dx}{dt}=4t $$ To solve this I used the separable variables method where the IVP was defined as $$x(t_0)=x_0$$ $$ x(t)\sqrt{4t^2-4t_0^2+x_0} $$
Now I went to check where this solution is valid, which led me to $$t\geq\frac{\sqrt{x_0-4t_0}}2$$ But now at the same time I have to take in consideration that $$t_0\leq\frac{x_0}{4}$$
How do I find the maximal solution, with all this information?