I am trying to express a function $f(x)$ in terms of a complete set of eigenfunctions found from a Sturm-Liouville boundary value problem:
$$2y''(x)+4y'(x)+\lambda y(x)=0$$ $$y(0)=0, y'(2)=0$$
For this i found that when $\lambda =2+2\beta^2$ i get the eigenfunction $y_n=e^{-x}\sin(\beta_n x)$ and when $\lambda=2-\beta^2$, $y_n=e^{-x}\sinh(\beta_n x)$.
Normally to express $f(x)$ in terms of the eigenfunctions i use: $$f(x)=\sum_{n=1}^\infty C_n y_n(x), a<x<b$$ where, $$C_n=\frac{I_1}{I_2}$$ and, $$I_1=\int_a^b r(x)f(x)y_n(x)dx$$ $$I_2=\int_a^b r(x)y_n^2(x)dx$$
But as i have different sets of eigenfunctions depending on my $\lambda$, which should i use for my $y_n(x)$?