consider the differential equation
$$y''+\lambda y = 0$$ $$y(0) = 0 , y(π) = 0$$
find the eigenvalues and corresponding eigenfunctions, @mathematics

Question

consider the differential equation 
$$y''+\lambda y = 0$$ $$y(0) = 0 , y(π) = 0$$
find the eigenvalues and corresponding eigenfunctions, @mathematics

jamesj · Answer

What are you finding hard about this?  Apart from the eigensomething language, you've solved this problem a few dozen times by now, haven't you?

unklerhaukus · Answer

well i have forgotten how to begin, and how to finish, Ill probably be able to do the mathematics but i am struggling to turn the question into equations

jamesj · Answer

Consider three cases in turn.  lambda > 0, = 0, < 0.  Find the general solution for each such lambda; see if it can be made to meet the initial conditions.  

The lambda for which you can satisfy those initial conditions are the eigenvalues and the corresponding solutions the eigenfunctions.

jamesj · Answer

btw, you're kind of stingy in acknowledging help.  Not that I'm neurotically checking, really I'm not, but the occasional medal for my amazing insights would be appreciated. ;-)

unklerhaukus · Answer

What are you talking about i always reward a good answer with a medal.

jamesj · Answer

Ok then.  Anyway, analyze this as I suggest.  Basically what you're doing is finding the eigenvectors of the operator $ D^2 $ acting on the $ C^\infty(\mathbb{R}) $.  Lambda here is the negative of the eigenvalues.

I.e., if nu is an eigenvalue, then
$$ D^2 = \nu \iota $$
where iota is the identity function, and thus $ \lambda = -\nu $.

jamesj · Answer

The set of eigenvalues an operator such as D^2 has is called the spectrum of the operator.  And once you learn to think about this property of operators, it opens up rooms and rooms of insight.

unklerhaukus · Answer

well i am currently fresh outta insight 
and stocked up on confusion and malaise 
i will do this problem someothertime