> "Like intuitively for describing forced oscillations I would have rather started with an ansatz where the periodic forcing is in the constant term"

That's what I did in comment #7. I set the forcing terms as annual and biannual sinusoids. That's the starting point for any of this discussion. Without a forcing, it becomes an initial conditions problem. With the forcing, it becomes more of a boundary value problem, where the continuous forcing drives the solution.