Determine the Eigen values and Eigen functions of the problem $y''+μy=0$; $y(0)=0$,$y'(L)=0$

Question

I've not come across this type of problem before, nor does the chapter I'm working through contain similar problems. Can someone help me understand how I can pull out eigenvalues from the equation above, please?

My instinct says to find a characteristic equation, therefore, I have:

$$r^2+μ=0$$ solving I get: $$r=\pm i\sqrt{\mu}$$

giving me a characteristic equation of:

$$y_c(x)=A\cos(\sqrt{\mu}x)+B\sin(\sqrt{\mu}x)$$

$$y'_c(x)=-\sqrt{\mu}A\sin(\sqrt{\mu}x)+\sqrt{\mu}B\cos(\sqrt{\mu}x)$$

If I plug in a value thats given, such as $y(0)=0$, then that gives me $A=0$.

Now I don't know what to do or understand what I'm looking at. Do I use that other value given, $y'(L)=0$?

If I do I get $$\sqrt{\mu}B\cos(\sqrt{\mu}L)=0$$ and that's all I got.

Any help would be greatly appreciated.

Thank you

Answer 1

There are two ways that the expression $B \cos \sqrt{\mu} L = 0$. One way is to take $B=0$ but this gives the trivial solution $y(x) \equiv 0$ (you've already found $A=0$). If you put $\cos \sqrt{\mu} L = 0$ this gives restrictions on the allowed values of $\mu$.