Existence and Uniqueness Theorem

In summary: It is possible that P(x) could be considered a solution to the IVP even though y'=F(x,y) is undefined at the initial condition.
  • #1
Bipolarity
776
2
Suppose you have an ODE [itex] y' = F(x,y) [/itex] that is undefined at x=c but defined and continuous everywhere else. Now suppose you have an IVP at the point (c,y(c)). Then is it impossible for there to be a solution to this IVP on any interval containing c, given that the derivative of the function, i.e. y', does not even exist at that point?

So say you found a solution P(x) which does go through (c,y(c)) and satisfies the ODE [itex] y' = F(x,y) [/itex] everywhere where F(x,y) exists. Would P(x) be considered a solution of the IVP?

This is a rather technical issue. I would appreciate if anyone addressed it.
Thanks!

BiP
 
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  • #2
The "existance and uniqueness" theorems are all of the form "if **** is true, then there exists a unique solution". They do not say anything about what happens if **** is not true.
 
  • #3
HallsofIvy said:
The "existance and uniqueness" theorems are all of the form "if **** is true, then there exists a unique solution". They do not say anything about what happens if **** is not true.

I see. I guess my question is not so much about the E-U theorem, but really about what it means to "solve an IVP". To solve an IVP, certainly the function must pass through the initial condition. It must also certainly solve the ODE at points where both sides of the ODE are defined (i.e. exist). What about at points in the ODE for which y' = F(x,y) is not defined?

So if you have y' = F(x,y), with some initial condition and you claim that P(x) is a solution to the IVP, then first P(x) should certainly pass through the initial condition. But suppose that y'=F(x,y) is undefined at the initial condition. If P(x) satisfies the ODE everywhere, except at the initial condition where the ODE is not even defined, then can P(x) be considered a solution to the IVP?

BiP
 

FAQ: Existence and Uniqueness Theorem

1. What is the Existence and Uniqueness Theorem?

The Existence and Uniqueness Theorem is a widely-used theorem in mathematics that guarantees the existence and uniqueness of a solution to certain types of mathematical problems. It is particularly useful in differential equations and other equations involving initial value problems.

2. How does the Existence and Uniqueness Theorem work?

The theorem states that if a mathematical problem satisfies certain conditions, then there exists a unique solution to that problem. These conditions typically involve continuity and differentiability of the functions involved in the problem.

3. What types of problems can the Existence and Uniqueness Theorem be applied to?

The theorem can be applied to various types of problems, including differential equations, integral equations, and initial value problems. It is also used in other areas of mathematics, such as optimization and control theory.

4. Why is the Existence and Uniqueness Theorem important?

The theorem is important because it provides a powerful tool for proving the existence and uniqueness of solutions to mathematical problems. This can be used to verify the validity of solutions and to guide the development of new mathematical models.

5. Are there any limitations to the Existence and Uniqueness Theorem?

Yes, there are certain limitations to the theorem. It may not be applicable to all types of mathematical problems, and the conditions for its use can sometimes be difficult to verify. Additionally, the theorem only guarantees the existence and uniqueness of a solution within a certain domain, and solutions outside of this domain may not be covered by the theorem.

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