Proving Existence and Uniqueness of Y(x) for 0<Y(x)<1

[cosmic_tears]

Hi! Thanks for reading! :)

Homework Statement

Y(x) is the solution of the next DFQ problem:
y' = [(y-1)*sin(xy)]/(1+x^2+y^2), y(0) = 1/2.
I need to prove that for all x (in Y(x)'s definition zone), 0<Y(x)<1.

Homework Equations

I just know that this exercise is under the title of "The existence and uniqueness theorem".

The Attempt at a Solution

I'm sorry to say I don't have much to show here. I just noticed that for y=0, y'=0, and for y=1, y'=0... but I can't progress any farther...
Moreover, I don't see how this exercise is relevant to the existence and uniqueness theorem, but it has to be...

Hints? Tips? Anything?
Thanks!

 

[HallsofIvy]

First, what is "Y(x)'s definition zone"?

Second, can you put upper and lower bounds on y' in that zone?

 

[cosmic_tears]

I think Y(x)'s definition zone is all of R, since Lipschitz law is being satisfied in every closed area in R^2, etc...

Well, I can see that y'(x)<=y-1, but I can't see where it leads...