I have the following system of coupled differential equations:
$$ r \frac{d}{dr}\left(r \frac{d \varphi_{m}}{dr} \right) - m^2 \varphi_{m} + r^2(F_0 \varphi_{m}+F_1 \varphi_{m-1}+F_2 \varphi_{m-2})=0, $$ where $m$ is an integer that ranges from $(m_0-k)$ to $(m_0+k)$. The $F$'s are known and I have some boundary conditions that depend on $m$.
The $\varphi$'s vanish for $m<-k$ or $m>k$. So the equation for $m=m_0-k$ is simply: $$ r \frac{d}{dr}\left(r \frac{d \varphi_{m_0-k}}{dr} \right) - (m_0-k)^2 \varphi_{m_0-k} + r^2F_0 \varphi_{m_0-k}=0. $$ So I need to know $\varphi_{m-2}$ and $\varphi_{m-1}$ to be able to solve the equation for $\varphi_m$ and iterate up to $(m_0+k)$.
I was thinking of having a list of InterpolatingFunction's. However, I don't know to handle the indices in a neat way.
Can someone please suggest an approach or point me some directions?