Find the explicit solution and the maximal interval of existence for the initial value problem $x' = tx^3$ , $x(0) = x_0$. Recall that the maximal interval of existence depends on $x_0$, is open, and must contain $0$. Compare the maximal interval of existence to that guaranteed by the existence theorem.
Okay so for my explicit solution I got $\sqrt \frac{(x_0)^2}{1-t^2x^2}$
With this solution, the maximal interval of existence would be ($-\infty$, $\frac {1}{x_0}$)
Then we have to compute the maximal domain of existence using Picard-Lindelof Existence and Uniqueness Theorem. In class we learned how to do it for autonomous equations (our example was $x'=x^2 , x(0)=x_0$). We never did nonautonomous equations though.