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

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SUMMARY

The discussion focuses on proving the existence and uniqueness of the solution Y(x) for the differential equation y' = [(y-1)*sin(xy)]/(1+x^2+y^2) with the initial condition y(0) = 1/2. Participants emphasize the relevance of the existence and uniqueness theorem, highlighting that Y(x) remains bounded between 0 and 1 for all x in its definition zone. The Lipschitz condition is noted as being satisfied in R², which is crucial for establishing the solution's properties.

PREREQUISITES
  • Understanding of differential equations, specifically first-order ordinary differential equations (ODEs).
  • Familiarity with the existence and uniqueness theorem for ODEs.
  • Knowledge of Lipschitz continuity and its implications for solution behavior.
  • Basic trigonometric functions and their properties, particularly sine functions.
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  • Study the existence and uniqueness theorem for first-order ODEs in detail.
  • Explore Lipschitz continuity and its role in proving the boundedness of solutions.
  • Learn about upper and lower bounds for derivatives in the context of differential equations.
  • Investigate specific examples of differential equations that demonstrate the application of these concepts.
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Mathematics students, educators, and researchers focusing on differential equations, particularly those interested in the existence and uniqueness of solutions in applied mathematics.

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

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

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