I am trying to solve for the time dynamics for a simple quantum system (two-site system with sinusoidal coupling and a decay parameter on one site) and the math is looking not so simple.
Here is the integro-differential equation I end up with for the time dynamics on one site:
$$y'(t) = \left(\gamma + \delta\sin{\left(\epsilon t\right)}\right)\exp{\left(-\alpha t\right)}\int_0^tdw\, \left(\gamma + \delta\sin{\left(\epsilon w\right)}\right)\exp{\left(-\beta w\right)}y(w)$$
$$y(0) = 1$$
My question is whether I should expect this equation to have a solution, and what types of methods I should pursue if it is soluble. My higher math background is somewhat limited (I get a little uneasy attempting any complex analysis).
My first thought was Laplace transforms, but looking through Schaum's tables of identities I am not sure if I can massage that expression into something that matches the identities.
Any thoughts would be appreciated.