ODE/PDE- eighenvalues question

  • Thread starter Roni1985
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In summary, the conversation is about solving a differential equation with boundary conditions. The general solution is X(x) = C1 + C2*x, but there is a question about the value of C2 when solving for the specific case of \lambda = 0. The solution with \lambda\neq 0 approaches the solution for \lambda = 0 as \lambda \rightarrow 0.
  • #1
Roni1985
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Homework Statement


X''+[tex]\lambda[/tex]*X=0 0<x<2pi

X(0)=X(2pi) and X'(0)=X'(2pi)


Homework Equations





The Attempt at a Solution


My only problem is the case when [tex]\lambda[/tex]=0
in such case the general solution is X(x)= C1+C2*x

Now, after applying the BCs, this is what I get:

C1=C2*2pi
and

C2=C2

now, what should C2 be ? zero or any value ?
if C2 is zero, we have the trivial solution only. However, if C2 is any number, we have a nontrivial solution.

how can I solve this question?


Thanks.
 
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  • #2
After applying the first BC, I get C1=C1+C2*2*pi, not C1=C2*2*pi.
 
  • #3
Roni1985 said:
C1=C2*2pi

This should be

[tex]C_1 = C_1 + 2\pi C_2.[/tex]

How does the solution with [tex]\lambda\neq 0[/tex] behave in the limit as [tex]\lambda \rightarrow 0[/tex]? Is this consistent with your solution for [tex]\lambda = 0[/tex]?
 

Related to ODE/PDE- eighenvalues question

1. What is an eigenvalue?

An eigenvalue is a number that represents a special characteristic of a matrix or operator. It is often denoted by the Greek letter lambda (λ) and is used in the study of linear algebra and differential equations.

2. How do eigenvalues relate to ODE/PDEs?

Eigenvalues are crucial in solving ODEs and PDEs as they help to determine the behavior of the system. They allow us to find the solution to the differential equation by providing important information about the system's dynamics.

3. What is the process for finding eigenvalues?

The process for finding eigenvalues involves solving the characteristic equation of the matrix or operator. This equation is obtained by subtracting the eigenvalue from the diagonal elements of the matrix and setting the determinant equal to zero. The resulting roots of the equation are the eigenvalues.

4. Can a matrix have more than one eigenvalue?

Yes, a matrix can have multiple eigenvalues. In fact, the number of eigenvalues is equal to the number of rows or columns in the matrix. However, some eigenvalues may have multiplicity, meaning they have more than one corresponding eigenvector.

5. What is the significance of eigenvalues in real-world applications?

Eigenvalues have a wide range of applications in fields such as physics, engineering, and computer science. They are used to solve differential equations, analyze the stability of systems, and perform data analysis and compression. They also play a crucial role in the development of quantum mechanics and signal processing techniques.

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