I'm trying to solve the wave equation for a infinite string which is attached to a mechanism that moves as $y(t)=A\cos(\omega t)$ at $x=0$.
The doubt I have is:
can I see this system as an homogeneous one and set the displacement $y(t)$ as a boundary condition at $x=0$ ?
or, from the beginning, should I solve the non-homogeneous equation $\Box u(x,t)=y(t)$ ?
I can't see clearly what the difference between those situations is