HA 6, problem 3c (sections 4.1 and 4.2)

[Shaghayegh A]

http://www.math.toronto.edu/courses/apm346h1/20169/PDE-textbook/Chapter4/S4.2.P.html#mjx-eqn-a

For 3c: I assume that M(y) and N(y) are two arbitrary eigenfunctions with the same eigenvalues $\omega$. Then, M and N satisfy
$$Y^{(4)} (y)=\omega^4 Y(y) \\
Y(-L)=Y_y (-L)=0 \\
Y(L)=Y_y(L)=0
$$ where I've switched coordinate systems so that $y=x-l/2=x-L$. I want to prove
$$\int_{-L}^{L} M(y) N(y) dy=0$$ but I'm not sure how to do that. Any advise?
Thank you

[Victor Ivrii]

Different eigenvalues. For the same eigenvalue it will be plain wrong