A DFQ problem.

1. May 26, 2008

cosmic_tears

1. The problem statement, all variables and given/known data
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.

2. Relevant equations
I just know that this excercise is under the title of "The existence and uniqueness theorem".

3. 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 excercise is relevant to the existence and uniqueness theorem, but it has to be...

Hints? Tips? Anything?
Thanks!

2. May 26, 2008

HallsofIvy

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

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

3. May 26, 2008

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....