Existence and Uniqueness of a solution for ordinary DE

In summary, the conversation discusses the importance of showing the existence and uniqueness of solutions to ODEs, as it provides a general setting in which a solution can be guaranteed. This is often done through the use of fixed point problems and the Banach contraction mapping principle. The purpose of this is to establish a unique solution for a class of ODEs, although there may be some restrictions. It is noted that in physics courses, the proof of uniqueness is often deferred in order to focus on finding solutions and justifying methods.
  • #1
O.J.
199
0
I just don't understand the idea behind it. I hate it when they throw these theories at us without proofs or elaborate explanations and just ask us to accept and applym mthem. Anyone care to enlighten me?
 
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  • #2
Well, sometimes it's hard (or annoying) to find solutions to ODEs by hand and, hence, it is easier to show that there exists a (unique) solution to an initial value problem depending on the information you're given.
However, in a lot of cases, existence/uniqueness is established in an attempt to formulate a general setting in which you are guaranteed to have a solution to an ODE, which is unique.
For example, the classical analysis problem: Show that there exists a solution to [tex]y'=f(x,y)[/tex], [tex]y(0)=y_0[/tex], [tex]\|f(x,y)-f(x,z)\| \le k\|y-z\|[/tex] in some interval [tex][0,a][/tex], where [tex]x\in[0,a][/tex], [tex]y:[0,a]\rightarrow\mathbb{R}[/tex] and [tex]f \in C([0,a]\times\mathbb{R})[/tex].
This could be formulated as a fixed point problem-- i.e. fixed points of:
[tex]Ty=y_0 + \int_{0}^{x}f(t,y(t))dt[/tex], [tex]x\in[0,a][/tex], would satisfy the ODE above.
Hence, one may use the Banach contraction mapping principle after showing that [tex]Ty[/tex] is a contraction mapping (in particular instances), and hence conclude that there exists a unique solution [tex]y[/tex] to the problem [tex]Ty=y[/tex] and hence, existence and uniqueness is established for this class of ODEs. (You would end up with some sort of restriction on [tex]a[/tex] though)
I have no idea why you would be exposed to these sorts of ideas without proofs, since that would sort of defy the purpose.
 
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  • #3
Quite often, in a physics course, uniqueness of the solutions to Poisson's equation or Laplace's equation are used to find solutions based on guesses and to justify the method of images. Proof is usually deferred.
 

1) What is the definition of "existence and uniqueness" in the context of ordinary differential equations (ODEs)?

Existence refers to the fact that there is at least one solution to the ODE that satisfies all initial conditions. Uniqueness means that there is only one such solution for a given set of initial conditions.

2) How is the existence and uniqueness of a solution for an ODE determined?

The existence and uniqueness of a solution for an ODE can be determined by using the Picard-Lindelöf theorem, which states that if the ODE is continuously differentiable and the initial conditions are well-defined, then there exists a unique solution.

3) What happens if an ODE does not have a unique solution?

If an ODE does not have a unique solution, it means that there are multiple solutions that satisfy the given initial conditions. This can happen if the ODE is not continuously differentiable or the initial conditions are not well-defined.

4) Can an ODE have multiple solutions that satisfy different initial conditions?

Yes, it is possible for an ODE to have multiple solutions that satisfy different initial conditions. This can happen if the ODE is not continuously differentiable or the initial conditions are not well-defined.

5) How does the concept of existence and uniqueness of a solution for an ODE relate to real-world applications?

The concept of existence and uniqueness of a solution for an ODE is crucial in real-world applications, especially in fields such as physics and engineering. It allows us to predict and model the behavior of systems described by ODEs, and ensures that our solutions are accurate and well-defined.

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