Ιnitial Value Problem

Question

Let the initial value problem :

$$y' = x\ln y, y(1)=1$$

Study the existence of solutions in a domain around the initial value and study its uniqueness too.

I know that an initial value problem has a unique solution if the function completes the Lipshitz Condition, but I am not sure on how to show it. I do not know how to proceed on the first part, studying the existence of the solutions.

We've only approached theorems and their proofs so I have completely no experience over exercises, I would really appreciate a thorough solution or explanation on how to handle such problems, because I really have an issue regarding exercises since we haven't been given books/leaflets yet.

Thanks in advance !

Answer 1

HINT: write $$\frac{dy}{\ln(y)}=xdx$$