# What are the eigenvalues and eigenfunctions for this Sturm-Liouville problem?

• angelas
In summary, the Sturm-Liouville Problem is a mathematical problem involving finding the solution to a second-order linear differential equation with boundary conditions. It has many applications in physics, engineering, and other fields and is named after mathematicians J.C.F. Sturm and Joseph Liouville. The boundary conditions in this problem can be either homogeneous or non-homogeneous. The problem is typically solved using a technique called separation of variables and the solutions are characterized by eigenvalues and eigenfunctions. These play a crucial role in understanding the applications of the Sturm-Liouville Problem.
angelas
Hi everyone,
I would really appreciate if any of you can help me solve this problem:

[Sinx y']' + [Cosx+ lambda Sinx] y = 0 ; 1<x<2; y(1) = y(2) = 0

this is a regular sturm-liouville problem. I need to find the eigenvalues and eigenfunctions of this problem.

What have you tried? Have you tried finding the Green's function?

Thanks for your reply. No I haven't. I don't know how to do that.

## 1. What is the Sturm-Liouville Problem?

The Sturm-Liouville Problem is a mathematical problem that involves finding the solution to a second-order linear differential equation with boundary conditions. It is named after mathematicians J.C.F. Sturm and Joseph Liouville, who independently studied this problem in the 19th century.

## 2. What is the significance of the Sturm-Liouville Problem?

The Sturm-Liouville Problem is significant because it has many applications in physics, engineering, and other fields. It is used to model physical phenomena such as heat transfer, vibrations, and fluid flow, and it also has applications in quantum mechanics and signal processing.

## 3. What are the boundary conditions in the Sturm-Liouville Problem?

The boundary conditions in the Sturm-Liouville Problem are the conditions that must be satisfied by the solution to the differential equation at the boundaries of the interval in which it is defined. These conditions can be either homogeneous (the solution and its derivative are zero at the boundary) or non-homogeneous (the solution and/or its derivative have a specific value at the boundary).

## 4. How is the Sturm-Liouville Problem solved?

The Sturm-Liouville Problem is typically solved using a technique called separation of variables, which involves writing the solution as a product of two functions and substituting it into the differential equation. This results in two separate differential equations, each of which can be solved separately. The solutions to these equations are then combined to form the general solution to the original problem.

## 5. What are the eigenvalues and eigenfunctions in the Sturm-Liouville Problem?

The solutions to the Sturm-Liouville Problem are characterized by eigenvalues and eigenfunctions. The eigenvalues are the values of a parameter that satisfy the boundary conditions, and the corresponding eigenfunctions are the solutions to the differential equation for each eigenvalue. These eigenvalues and eigenfunctions play a crucial role in solving the Sturm-Liouville Problem and understanding its applications.

• Calculus and Beyond Homework Help
Replies
11
Views
3K
• Calculus and Beyond Homework Help
Replies
2
Views
1K
• Calculus and Beyond Homework Help
Replies
5
Views
562
• Calculus and Beyond Homework Help
Replies
4
Views
1K
• Calculus and Beyond Homework Help
Replies
8
Views
2K
• Calculus and Beyond Homework Help
Replies
2
Views
2K
• Calculus and Beyond Homework Help
Replies
10
Views
535
• Calculus and Beyond Homework Help
Replies
2
Views
839
• Calculus and Beyond Homework Help
Replies
2
Views
578
• Calculus and Beyond Homework Help
Replies
11
Views
1K