I have a second order linear differential equation of the form $u'' + k^2\epsilon(x)u = k^2q^2u$. Here, k is a known parameter, $\epsilon(x)$ is a known periodic function and $q^2$ is basically the eigenvalue that I need to solve for and plot as a function of $k_Bk$.
So, I know Bloch's theorem, but I am not sure how to go any further with it. Substituting the Bloch ansatz $u(x) = r(x)\exp\left(\iota k_Bkx\right)$ leads to another second order differential equation in $r(x)$ whose solution is periodic. I could not figure out how to proceed from here.
Most literature that I found was related to condensed matter physics, which is not the context in which I am trying to work and went over my head. Any help would be very appreciated.
Edit: $k$ is a constant while $k_B$ is varied, just a notational thing.