Consider a string of length L that is held fixed at each end. We want to solve the wave equation for y(x,t), the vertical displacement of the string as a function of position and time. The wave equation is any(I, t) = 202 dray (1, t), where c is the speed of the wave. You should assume that the ends of the string are at r = 0 and x = L. (a) Write the boundary conditions for the string at r = 0 and r = L. (b) Assume that the solution is separable so that y(x, t) = X(x)T(t). Substitute that solution in and determine the equations satisfied by X(x) and T(t). Use -A for the separation constant. (c) Solve the obtained equation for X(x). (d) Use the obtained boundary conditions to determine the allowed values of 1. (e) Solve the obtained equation for T(t). (f) Now let's assume that we release the string from rest. Write out the initial condition this implies and use it to simplify your expression for T(t). (g) Write the general solution for the displacement of the string. What does this solution look like at t = 0? (h) To continue any further, we need to know the initial shape of the string. Assume that you pull up on the string at > = a and release it from rest when y(a, 0) = h. In other words, the initial shape of the string looks like an inverted letter v. Mathematically, the initial displacement of the string is: y(x, 0) = (L -I), IZa. What type of Fourier series should you use to describe this initial displacement? Ex- plain