where
with
and
are differentiable functions and
. A non zero solution of the system defined by (1) and (2) exists in general for a specified . The functions corresponding to that specified are called eigenfunctions.
More generally, if is some linear differential operator, and
and is a function such that
then we say is an eigenfunction of with eigenvalue.
![$\displaystyle \frac{d}{dx}\left[p(x)\frac{dy}{dx}\right]+q(x)y+\lambda r(x)y=0\;\;\;\;\;\;a\leq x\leq b$](http://images.planetmath.org:8080/cache/objects/3117/l2h/img1.png "$\displaystyle \frac{d}{dx}\left[p(x)\frac{dy}{dx}\right]+q(x)y+\lambda r(x)y=0\;\;\;\;\;\;a\leq x\leq b$")
