Help understanding a vibrating string question

[John004]

Homework Statement

So I don't really understand what the professor means by "show why the displacements y(x,t) should satisfy this boundary value problem" in problem 1. Doesn't that basically boil down to deriving the wave equation? At least in problem 2 he says what he wants us to show.

Homework Equations

u_tt = c²u_xx

The Attempt at a Solution

The problem set is in the attachment

 

[Dr.D]

I would interpret that problem to mean
1) develop the equation of motion for the system, ending up with the wave equation
2) show that, for this situation, the boundary conditions are those suggested in the problem statement.
In the process, you will need to show how c is related to the given information, so don't just write "c" but put it in terms of the given data.