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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

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    $\begingroup$ In the second case, we must put $y(t)\delta (x)$. $\endgroup$ – Alex Trounev Jun 28 at 2:39
  • $\begingroup$ 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. $\endgroup$ – Cinaed Simson Jun 28 at 22:43
  • $\begingroup$ @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. $\endgroup$ – Gabriel Sandoval Jun 29 at 4:57

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