Are the Eigenfunctions of a Differential Equation Orthogonal?

[dirk_mec1]

Homework Statement

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

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

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.

 

[mighty2000]

Thank you for your post. I am a scientist and I would like to help you with your proof. The eigenfunctions of equation (1) are orthogonal because of the following reasons:

1. The boundary conditions: The boundary conditions given in the problem statement are essential for the orthogonality of the eigenfunctions. These conditions ensure that the eigenfunctions are defined only on a specific interval and are zero at the boundaries. This allows for the integral of the product of two different eigenfunctions to be zero, making them orthogonal.

2. Properties of eigenfunctions: The eigenfunctions of a differential equation are unique and independent of each other. This means that the product of two different eigenfunctions will always result in a new eigenfunction, and not a combination of the two. Therefore, the integral of the product of two different eigenfunctions will always be zero, making them orthogonal.

3. Orthogonality condition: The orthogonality of the eigenfunctions can also be proved by using the orthogonality condition. This condition states that for a set of eigenfunctions, the integral of the product of any two different eigenfunctions is zero, except when the eigenfunctions are the same. This condition holds true for the eigenfunctions of equation (1), making them orthogonal.

Therefore, we can conclude that the eigenfunctions of equation (1) are orthogonal. I hope this helps with your proof. If you have any further questions, please feel free to ask.