Q:

Don't understand this Sturm - Liouville problem

All the SL problems I have seen before had $ \lambda$ in them, so finding the eigenvalues meant finding the values of $ \lambda$. However I don't know what I am supposed to do in the following:
Find the eigenvalues & normalised eigenvectors for $y''+4y'+(4+n^2 \pi^2)y=0,$ with conditions $y(0)=y(1)=0$, where $n$ is an integer.  
I suppose I should put it in the SL usual form but I don't know how to deal with the absense of  $ \lambda$.

A:

Giving an operator $\mathcal{D}(\cdot)$ their eigenvalues are all $\lambda$ such that
$$
\mathcal{D}(y)=\lambda y
$$
solving
$$
y''+4y'=\lambda y
$$
we have
$$
y = c_1 e^{(-2-\sqrt{\lambda+4})x}+ c_2e^{(-2+\sqrt{\lambda+4})x}
$$
here $\lambda = -(4+n^2\pi^2)$  and then
$$
y = c_1 e^{(-2-i n \pi)x}+ c_2e^{(-2+i n\pi)x}\Rightarrow y = \left(C_1 \cos(n\pi x)+C_2\sin(n\pi x)\right)e^{-2x}
$$
so due to the contour conditions
$$
y(0) = C_1 = 0\\
y(1) = C_1(-1)^ne^{-2} = 0
$$
then 
$$
y_n = C_2\sin(n\pi x)e^{-2x}
$$
are the eigenfunctions associated to the boundary conditions.