Green's Functions: Solving Sturm-Liouville Problems

[the_dialogue]

Homework Statement

I'm learning Sturm-Liouville theory and currently studying Green's function.

In the following image, the author states that $$y_n(x)$$ satisfy the "same" boundary conditions as $$y(x)$$.

Homework Equations

http://img2.pict.com/e9/87/37/3735148/0/1277836655.jpg

The Attempt at a Solution

How can I prove this? This is strictly for pedagogical purposes.

The closest I can get to solving this problem is considering the boundary conditions for Regular Sturm-Liouville problems, which is:
http://img2.pict.com/d8/f9/a0/3735157/0/1277836732.jpg

Any help appreciated!

 

[LCKurtz]

How can you prove what? I don't see anything to prove. It looks to me like the author is simply saying he wants to solve the NH differential equation having the same form as the SL problem satisfying the same boundary conditions. k may or may not be an eigenvalue of the associated homogeneous equation.

 

[the_dialogue]

I apologize.

The author states that we may express equation 3.21 with the equation that follows, since the y_n and y satisfy the same boundary conditions. To me, however, it seems like they are 2 different equations. How can we say that they satisfy the same boundary conditions, in order to justify expressing 3.21 in the different form?

Thanks!

 

[LCKurtz]

I apologize.

The author states that we may express equation 3.21 with the equation that follows, since the y_n and y satisfy the same boundary conditions. To me, however, it seems like they are 2 different equations. How can we say that they satisfy the same boundary conditions, in order to justify expressing 3.21 in the different form?

Thanks!

I think you are correct; they are two different equations. In the latter equation he is just re-iterating the equation that the eigenfunctions of the homogeneous equation satisfy. If I were you I would just read on. The author is probably going to show how to come to the solution of the NH equation.

 

[the_dialogue]

Well all the author does is solve for 'y_n' (or rather a normalized form of y_n) and then use that term to substitute into 3.21.

In other words the author is basing his future derivation on the presumption that the 2nd differential equation and the 1st (original) have as their base y or y_n that both satisfy some kind of boundary condition.

Can someone shed some light on this?

Here are the pages:

http://img2.pict.com/d6/7f/14/3735488/0/1277843810.jpg

http://img2.pict.com/e7/3c/fe/3735494/0/1277843841.jpg

 

[LCKurtz]

What the author is doing is using the Sturm-Liouville theory that tells you the normalized eigenfunctions form a complete orthonormal set. Because of that, the solution y(x), which is given to satisfy the boundary conditions, can be expressed in terms of the eigenfunctions as a generalized Fourier series

$$y(x) = \sum_{n=1}^\infty c_n\phi_n(x)$$

He then goes on to plug this into the equation and use the orthogonality properties to solve for the c_n in terms of the eigenvalues, eigenfunctions, and f(x).

This gives a formula for the solution of the NH equation. The summation in that final integral is what he is going to introduce to you as Green's function. It has important properties you are about to learn.

 

[the_dialogue]

Thanks LCKurtz.

Refer to the bottom of the 1st page ("Since..."). How can we use an expression and plug it into the problem, when that expression is taken from an equation (2nd expression on 1st page) that is not equivalent to the original differential equation? The author says this is justified because both expressions satisfy the same boundary conditions, but how do we prove this?

Sorry if I'm missing something.

 

[LCKurtz]

Thanks LCKurtz.

Refer to the bottom of the 1st page ("Since..."). How can we use an expression and plug it into the problem, when that expression is taken from an equation (2nd expression on 1st page) that is not equivalent to the original differential equation? The author says this is justified because both expressions satisfy the same boundary conditions, but how do we prove this?

Sorry if I'm missing something.

You are missing the fact that the normalized eigenfunctions to a S_L problem form a complete orthonormal set. This means that any [continuous, I think, I don't have it right in front of me] function which satisfies the S_L boundary conditions can be expressed in terms of the eigenfunctions. In particular, the y(x) that satisfies the NH equation is such a function. That is how you know y can be expressed as the generalized FS. He then puts that expression back in the NH equation and uses properties the eigenfunctions must have given where they came from.

It's like when you have the {sin(nx),cos(nx}} solutions to y''+ n²y=0 with periodic boundary conditions, you can express other periodic functions in terms of them, wherever they came from.

Hope that helps -- got to run for now.

[Edit] Corrected typos.