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heman
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Can anybody help me with the proof of Existence and Uniqueness Theorems.
heman said:Can anybody help me with the proof of Existence and Uniqueness Theorems.
saltydog said:A unique solution of this equation about the point [itex](x_0,y_0)[/itex] exists if f(x,y) and it's partial with respect to y is continuous about the point.
heman said:Thanks for Amazing Clarification Salty but my doubt is this that i want to know why the happening of continuity of F(x,y) and Fy(x,y) ensures the uniqueness.!
HallsofIvy said:Actually, continuity of Fy(x,y) is not necessary. What is necessary for uniqueness is that F be "Lipschitz" in y. A function of one variable is Lipschitz (on a set) if and only if [itex]|f(x)-f(y)|\leC|x-y|[/itex].
heman said:Salty,
Can you give me an online link on Differential eqns. which stresses on proof!
Actually my textbook Kreyzig sucks!
saltydog said:Heman, really I think only "text book in hand" is the best way to learn the proof. Go through each line carefully and make sure you understand every detail. Draw pictures, go back and review all the theorems that are quoted. Do examples.
Also, I have Kreyzig. It's a good book for Engineers but not for Mathematicians.
God I hope I don't get in trouble with the Engineers in here for saying that.
The Existence and Uniqueness Theorem is a mathematical theorem that guarantees the existence and uniqueness of a solution to a particular type of differential equation. It states that if certain conditions are met, a single solution to the differential equation can be found for all initial values within a given interval.
The conditions for the Existence and Uniqueness Theorem to be applicable are that the differential equation must be continuous, the initial values must be within a given interval, and the differential equation must satisfy the Lipschitz condition. The Lipschitz condition states that the slope of the differential equation cannot change too drastically within the given interval.
The Existence and Uniqueness Theorem has many real-life applications, particularly in the fields of physics and engineering. It is used to model and predict the behavior of systems that can be described by differential equations, such as the motion of objects or the flow of fluids. It is also used in weather forecasting, economic modeling, and many other areas of science and technology.
If the conditions for the Existence and Uniqueness Theorem are not met, then the theorem cannot be applied and there is no guarantee of the existence or uniqueness of a solution to the differential equation. In some cases, there may be no solution at all, or there may be multiple solutions. In these cases, alternative methods must be used to find a solution.
Yes, the Existence and Uniqueness Theorem can be extended to systems of differential equations. In this case, the conditions for the theorem to be applicable become more complex, but the basic idea remains the same. The theorem guarantees the existence and uniqueness of a solution to the system of equations for a given set of initial values within a given interval.