Existence and Uniqueness theorem for 1st order ODEs

In summary, the given initial value problem, dy/dx = sin(y), with the initial condition y(X) = Y, has a unique solution. This is because both f(x,y) and df(x,y)/dy are continuous for all values of x and y, ensuring the existence of exactly one solution. However, it cannot be guaranteed for all values of x as the solution involves a negative number on the RHS if x < -C. This makes it non-computable. It is also important to verify that the existence of a unique solution is guaranteed.
  • #1
Silversonic
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1

Homework Statement



Consider the IVP compromising the ODE.

dy/dx = sin(y)

subject to the initial condition y(X) = Y

Without solving the problem, decide if this initial value problem is guaranteed to have a unique solution. If it does, determine whether the existence of that solution is guaranteed for all values of x.

I'm not sure how to answer this. f(x,y) and df(x,y)/dy are both continuous for all values of x and y. This means there is exactly one solution to the IVP.

Now working out what the solution is, we get;

In(1-cos(y)/sin(y)) = x + C.

What I don't get is whether it's guaranteed for all values of x? I don't believe it is, as because if x < -C then we get a negative number on the RHS. This is not computable. HOWEVER, if I'm right, how was I meant to work that out without working out the solution?

I also need to check that I'm correct in saying that that existence of a unique solution is guaranteed.
 
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1. What is the Existence and Uniqueness theorem for 1st order ODEs?

The Existence and Uniqueness theorem for 1st order ODEs is a mathematical theorem that guarantees the existence and uniqueness of a solution to a first order ordinary differential equation (ODE) under certain conditions. It is an important tool in analyzing and solving ODEs in various fields of science and engineering.

2. What are the conditions for the Existence and Uniqueness theorem to hold?

The conditions for the Existence and Uniqueness theorem to hold are that the ODE must be in the standard form of y' = f(x,y), the function f(x,y) must be continuous in some region R, and it must satisfy a Lipschitz condition in that region. This means that there exists a constant K such that |f(x,y1) - f(x,y2)| ≤ K|y1 - y2| for all (x,y1), (x,y2) in R.

3. What does the Existence and Uniqueness theorem guarantee?

The Existence and Uniqueness theorem guarantees that there is at least one solution that satisfies the given initial condition for the ODE. It also guarantees that this solution is unique, meaning that no other solution can exist with the same initial condition. This theorem is important in ensuring the validity and accuracy of solutions to ODEs in various applications.

4. How is the Existence and Uniqueness theorem applied in real-world problems?

The Existence and Uniqueness theorem is applied in various fields of science and engineering where ODEs are used to model real-world problems. For example, it is used in physics to study the motion of particles under the influence of forces, in biology to model population growth and decay, and in engineering to analyze the behavior of electrical circuits and mechanical systems.

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

Yes, there are limitations to the Existence and Uniqueness theorem. It only guarantees the existence and uniqueness of a solution in a specific region where the conditions are satisfied. If the ODE does not meet these conditions, the theorem may not hold and there may not be a unique solution. Additionally, the theorem does not provide a method for finding the solution, it only guarantees its existence and uniqueness.

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