Finding Eigen Values/functions

[ShowerHead]

Homework Statement

I have to find the eigenvalues&function of the eqn:
$$y''+\lambda y =0$$
With the boundary conditions:
$$y(0)+y'(0) = 0$$ and $$y(\pi) =0$$

Homework Equations

The Attempt at a Solution

I get the general equations, okay, but am having trouble due to the boundary conditions.
Assuming $$\lambda>0$$, then get the general solution:
$$y(x)=A\sin(x\sqrt{\lambda})+B\cos(x\sqrt{\lambda})$$

The best i can do now is that:
$$y(\pi)=A\sin(\sqrt{\lambda}\pi)+B\cos(\sqrt{\lambda}\pi)=0$$
Which can only be valid if B = 0 and $$\sqrt{\lambda} = n$$ where n is a positive integer.
Now I run into problems, the second bc gives:
$$y(0)=A\sin(n 0)=0$$
$$y'(0)=nA\cos( n 0)=nA$$
So:
$$y(0)+y'(0) = 0 + nA = 0$$

Am I on the right track, or can anyone see where I am messing up?
Thanks.

 

[gabbagabbahey]

$$y(\pi)=A\sin(\sqrt{\lambda}\pi)+B\cos(\sqrt{\lambda}\pi)=0$$
Which can only be valid if B = 0 and $$\sqrt{\lambda} = n$$ where n is a positive integer.

No, as a simple counterexample consider $\lambda=\frac{1}{16}$ and $A=-B$. If $\sqrt{\lambda}$ had to be an integer, then $B$ would have to be zero; but there's no reason to assume $\sqrt{\lambda}$ must be an integer.