Eigenvalues

Lanboy88

New member
Joined
Jan 25, 2007
Messages
3
I need some help with eigenvalues...can someone show me the general strategy / steps to solving for the eigenvalues?

Right now I'm just interested in initial value problems of the form
y" + λy = 0 (with some initial condition and λ being the eigenvalue),
but any help with the general strategy and understanding of eigenvalue is appreciated as well!

Thanks!
 
The homogenous second order linear ODE above is a Sturm Liouville problem. To find the eigenvalues and eigenfunctions of this problem, you will have to consider three cases: \(\displaystyle \lambda = 0, \lambda < 0,\) and \(\displaystyle \lambda > 0\) with some Dirichlet, Neumann, or Robin boundary conditions. (sometimes mixed)

The Sturm Liouville problems come regular or singular and they have a beautiful Theorem (Theorems). If you can put the above ODE in Sturm Liouville form and know how to use the Theorem, you will know exactly where to look for the eigenvalues. In other words, you will know that some of the three cases above are not needed which will save you time to test only 1 case or mostly 2. The Theorem will also you give you a lot of information which are very useful in solving complicated problems such as Legendre's equation, Bessel's equation, or other equations where the eigenvalues are very difficult to find.

Because you are interested in solving these simple types of ODE (above), I think that you will not need the Theorem (Theorems).

Now let us present the above problem with boundary conditions (Neumann) and some interval so that we have a complete problem to solve.

\(\displaystyle y''(x) + \lambda y(x) = 0, \ \ \ \ 0 < x < a,\)
\(\displaystyle y'(0) = 0,\)
\(\displaystyle y'(a) = 0,\)

After solving this problem, you should get,

eigenvalues: \(\displaystyle \lambda_n = \left(\frac{n\pi}{a}\right)^2\)

eigenfunctions: \(\displaystyle y_n(x) = A_n\cos \left(\frac{n\pi}{a}\right)\)

\(\displaystyle n \geq 0\)

The general strategy to solve such problems is simple. You will have to consider three cases and in each one, you will have to test the boundary conditions. Remember this, we are looking for nontrivial solutions.

Case 1: \(\displaystyle \lambda = 0\)

\(\displaystyle y''(x) = 0\)
\(\displaystyle y(x) = A + Bx\)
After applying the boundary conditions, you will get the nontrivial solution:
\(\displaystyle y(x) = A\)


Case 2: \(\displaystyle \lambda < 0\)

Let \(\displaystyle \lambda = -\mu^2\), where \(\displaystyle \mu \neq 0\)

\(\displaystyle y''(x) - \mu^2 y(x) = 0\)
\(\displaystyle y(x) = A\cosh \mu x + B\sinh \mu x\)
After applying the boundary conditions, you will get only the trivial solution:
\(\displaystyle y(x) = 0\)
So there is no nontrivial solution in this case.


Case 3: \(\displaystyle \lambda > 0\)

Let \(\displaystyle \lambda = \mu^2\), where \(\displaystyle \mu \neq 0\)

\(\displaystyle y''(x) + \mu^2 y(x) = 0\)
\(\displaystyle y(x) = A\cos \mu x + B\sin \mu x\)
After applying the boundary conditions, you will get the nontrivial solution:
\(\displaystyle y(x) = A\cos \mu x\)

Case three is the heart of this problem. Solving it will give you \(\displaystyle \mu = \frac{n\pi}{a}\) and \(\displaystyle \lambda = \mu^2\), so we have \(\displaystyle \lambda = \left(\frac{n\pi}{a}\right)^2\).

Finally, after getting the eigenvalues and eigenfunctions, you can use the principle of super position to write the solution as sum.

You can do this:

\(\displaystyle y(x) = \sum_{n=0}^{\infty} A_n \cos \frac{n\pi x}{a}\)

Or this:

\(\displaystyle y(x) = A_0 + \sum_{n=1}^{\infty} A_n \cos \frac{n\pi x}{a}\)
 
Is there any particular reason for resurrecting this 16-year-old thread?
 
Top