I would separate variables right away. Let (for short, ).
So we end up with
Since each side of the equation is defined in a different variable when compared to one another, we can conclude that LHS = RHS only when they're equal to the same constant, call it .
Thus, we can rewrite this as two different ODEs:
Consider the first equation. To make life easier, let . Then we see that the equation becomes
As TES mentioned in his post, you're going to get different solutions depending on the value of . I will do the case and leave the other two cases for you to work out.
So the characteristic equation for (*) is . Assuming , we see that . So, we see that .
Now, we do something similar for the second equation. The nice thing is that this equation is separable:
So we see that (aside question: is it ok to remove absolute values at this point?)
Thus, your solution should be .
Reiterating what TES said, you will get other solutions for testing the other cases.
Hopefully this makes some sense.