Understanding boundary conditions and forced system - Wave Equation

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

• In the second case, we must put $y(t)\delta (x)$. – Alex Trounev Jun 28 at 2:39
• The problem is ill poised. You need to expand the d'Alembert operator - where did $y(t)$ come from? Is $u(x,y,t)$? Or by $y(t)$ did you mean $u(0,t)=A\cos(\omega t)$? Where are the first derivatives of $u(x,t)$? And $\Box u(0,t)=y(t)$ doesn't make sense. I recommend moving this to math.stackexchange.com. – Cinaed Simson Jun 28 at 22:43
• @CinaedSimson $y(t)$ is the way in which the mechanism moves. So, $u(0,t)=A\cos(\omega t)$. Pleas read carefully because you're completely misunderstanding what I asked. I'm not moving to math.stackexchange because my problem is understanding the difference between a forced system and the boundary conditions, which is the physical part of the mathematics involved. Regards. – Gabriel Sandoval Jun 29 at 4:57