Orthogonality/infinite series solutions differential equation

[bossman007]

Homework Statement

This is the problem statement in the picture for exersize 14, it's rather long (pertaining to orthogonality - which I only understand what the definition of orthogonality is, which is the "(15)" on the side of the image below.

[PLAIN]http://postimage.org/image/oxhw2uf8p/ [/PLAIN]

Homework Equations

y'' + (k^2)y = 0

The Attempt at a Solution

My attempted solution attached below has a hint for exercise 14 which I will write out at the top of the page

 

[gabbagabbahey]

When you multiply both sides of the differential equation $y_n''(x)+k_n^2y_n(x)=0$ by $y_m(x)$, you don't get $y_m''(x)+k_n^2y_m(x)=0$. That's not how multiplication works.

 

[bossman007]

Oh wow, sorry careless error!

I got this far and don't know what I'm supposed to do after this:

 

[gabbagabbahey]

Oh wow, sorry careless error!

I got this far and don't know what I'm supposed to do after this:

Does $k_n=k_m$ for all possible values of $m$ and $n$? If not, how do you justify cancelling out the 2 terms that you cancelled?

Rather, you should have $(k_m^2-k_n^2)y_m y_n = \frac{d}{dx}\left( y_my_n' - y_n y_m' \right)$. As for what to do next, just follow the hint (it is very explicit in its instructions) and integrate both sides of the equation over a full period of $y_n$ (you'll probably want to start by figuring out what the period of $y_n$ is )...what do you get?

 

[bossman007]

Thanks so much, i got the final answer !