I have the following differential equation,
$\frac{d}{dt}c_k(t) = \frac{\beta}{2} (c_{k-1}(t)-c_{k+1}(t))$
with the initial coefficient $c_k(t=0)= f(k)$
I know that,
$\frac{d}{dt}J_n(t) = \frac{1}{2} (J_{n-1}(t)-J_{n+1}(t))$
where $J_n$ is the Bessel function of the first kind. However I don't know how to take care of the additional factor $\beta$ to get a solution. Any suggestion would be greatly appreciated.
Thanks,