- #1

Conor_McF

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## Homework Statement

A set of eigenfunctions

*y*satisfies the Sturm-Liouville equation

_{n}(x)**#1**with boundary conditions

**#2**. The function

*g(x) = 0*. Show that the derivatives

*u*are also orthogonal functions. Determine the weighting function

_{n}(x) = y_{n}^{'}(x)*w(x)*for these functions. What boundary conditions are required for orthogonality?

## Homework Equations

**#1**: [tex]\frac{d}{dx}[/tex]

*f(x)*[tex]\frac{dy}{dx}[/tex] -

*g(x)*[tex]\frac{dy}{dx}[/tex] + [tex]\lambda[/tex]

*w(x)y = 0*where

*w(x) [tex]\geq[/tex] 0*and

*a [tex]\leq[/tex] x [tex]\leq[/tex] b*

**#2**: [tex]\alpha[/tex]

_{1}

*y*+ [tex]\beta[/tex]

_{1}[tex]\frac{dy}{dx}[/tex] = 0 at

*x = a*and [tex]\alpha[/tex]

_{2}

*y*+ [tex]\beta[/tex]

_{2}[tex]\frac{dy}{dx}[/tex] = 0 at

*x = b*

[tex]\alpha[/tex] and [tex]\beta[/tex] are both constants and cannot both equal 0.

There is also the orthogonality relation [tex]\int y_{n}w(x)y_{m}dx = 0[/tex]

## The Attempt at a Solution

I guess what I'm most confused about here is what the problem is asking me to show. By saying the set "

*u*are orthogonal functions", what exactly are they orthogonal to? I'm assuming it means that [tex]\int u_{n}w(x)u_{m}dx = 0[/tex], provided that

_{n}(x) = y_{n}^{'}(x)*m*[tex]\neq[/tex]

*n*. Is this correct or am I way off?

Since

*g(x) = 0*, the DE becomes [tex]\frac{d}{dx}[/tex]

*f(x)*[tex]\frac{dy}{dx}[/tex] + [tex]\lambda[/tex]

*w(x)y = 0*. From the boundary conditions I can tell that

*y(a)*= - [tex]\frac{\alpha_{1}}{\beta_{1}}[/tex]

*u(a)*and

*y(b)*= - [tex]\frac{\alpha_{2}}{\beta_{2}}[/tex]

*u(b)*, but I don't know how this would help me at all. Can somebody please point me in the right direction?

Thanks; Conor.