# Orthogonality and Weighting Function of Sturm-Liouville Equation

1. Oct 30, 2012

### JordanGo

1. The problem statement, all variables and given/known data
A set of eigenfunctions yn(x) satisfies the following Sturm-Liouville equation:
$\frac{d(f(x)*y'_{m})}{dx}+\lambda*\omega*y_{m}=0$
with following boundary conditions:
$\alpha_{1}y+\beta_{1}y'=0$
at x=a
$\alpha_{2}y+\beta_{2}y'=0$
at x=b
Show that the derivatives un(x)=y'n(x) are orthogonal functions.
Determine the weighting function for these functions.
What boundary conditions are required for orthogonality?

2. Relevant equations

Orthogonal functions:
$\int(dx*\omega*y_{n}(x)*y_{m}(x)=0$
Integrate from a to b.
3. The attempt at a solution
I'm not sure how to start this problem, can someone point me in the right direction?

2. Oct 30, 2012

### Staff: Mentor

Problems that involve differential equations should be posted in the Calculus & Beyond section, not in the Precalculus Math section. I am moving this thread to that section.

3. Oct 30, 2012

### JordanGo

Anyway, well I found a way to prove orthogonality and ended up with:
$(\lambda_{m}-\lambda_{n})\int(w*y_{n}*y_{m}) =0$
(integral from a to b)
Now how do I find the weighting function?

4. Oct 30, 2012

### LCKurtz

Of course you must mean the definite integral$$(\lambda_{m}-\lambda_{n})\int_a^b(w*y_{n}*y_{m}) =0$$That is a standard result in S-L theory, and the weight function is the $w$ in the integrand. However, your original post asked you to show the derivatives of the $y_n$ were orthogonal. I wondered when I saw your OP whether that was a typo or whether it was true. In any case, it isn't what you found the proof for.

Last edited: Oct 30, 2012
5. Oct 30, 2012

### JordanGo

I don't quiet understand what it means then to show if the derivatives are orthogonal...

6. Oct 30, 2012

### LCKurtz

Above, if $\lambda_m\ne \lambda_n$ then $\int_a^b(w\cdot y_{n}\cdot y_{m})\, dx =0$, which is what it means for $y_m$ and $y_n$ to be orthogonal with respect to the weight function $w$. For the derivatives to be orthogonal with respect to some weight function $f(x)$would mean $\int_a^bf\cdot y'_{n}\cdot y'_{m}\, dx =0$ if $\lambda_m\ne \lambda_n$.

7. Oct 30, 2012

### JordanGo

Ok, so looking at the equation:
$(\lambda_{m}-\lambda_{n})\int(f(x))y'_{n}y'_{m}=0$
the only possibility is f(x)=0 because no matter what I do, I can't get terms to separate and moved to the right hand side.

8. Oct 30, 2012

### LCKurtz

Assuming your mathematics is at an advanced enough level to be studying S-L problems and orthogonality, that comment is just silly. Above you stated that you understand how to get the orthogonality of the eigenfunctions $y_n(x)$ so presumably you have some idea of what is involved to prove orthogonality.

I have shown you what you need to prove. I don't know offhand how to solve your problem and I'm not inclined to spend the evening figuring it out. It might even be easy; I don't know. But I'm sure you need to use the given DE and boundary conditions somehow and maybe even use the orthogonality of the $y_n(x)$ themselves. You don't start with the conclusion. Good luck with it.

[Edit, added]:You might try mimicking the standard proof but changing it where required because your boundary conditions are different.