# Homework Help: Orthogonality functions

1. Mar 18, 2013

### dirk_mec1

1. The problem statement, all variables and given/known data
Prove that the eigenfunctions of (1) are orthogonal.

$$m y_{tt} + 2Uy_{tx} + U^2y_{xx} +EIy_{xxxx} =0$$ (1)

for

$$0<x<L, t>0$$

with

$$y(0,t) = 0$$$$y_x(x,t) = 0$$$$y_{xx}(L,t) = 0$$$$y_{xxx}(L,t) = 0$$
$$y(x,0) = f(x)$$$$y_t (x,0) = g(x)$$

3. The attempt at a solution
I substituted:

$$y= e^{-\lambda_n t} X_n(x)$$ and$$y= e^{-\lambda_m t} X_m(x)$$ and subtracted the results.

The problem lies in the mixed derative term, i can not seem to let it vanish with the help of the BC's:

$$\lambda_n X_m X'_n - \lambda_m X_n X'_m$$

How am I suppose to let that term vanish? I've tried IBP but iw will not vanish.