
help with a problem
Hey guys, i just started a new course in differential equations and im kinda having a hard time. Heres one of the problems I am having difficulty with:
Verify that the piecewisedefined function
is a solution of the differential equations on
Thanks!

What have you tried so far?

Im not even sure how to start this problem. Is there a clue you can give me?

Well, you're not asked to solve the DE, but to verify the solution. So, I would plug the given solution into the DE and see if you get an equality. What do you get when you follow that path?

hmph. i just did:
if
Is this what its asking for?

Not quite. You cannot just choose two values of x. If x is any positive number, then so that y'= 2x. The equation xy'2y = 0 becomes so the equation is satisfied for any positive x. If x is negative, . Do the same with that. Most importantly, what happens at x= 0? In particular what is the derivative of y at x= 0? (You can't just differentiate and set x= 0 because the derivative involves a limit from both sides: If x= 0 and h is positive, then x+h= 0+ h= h is positive, if h is negative, x+ h= h is negative.
If those two limits are the same, that is the dervative at x= 0. If they are not, the function is not differentiable at x= 0.