It is given that the vibration of a uniform string is modelled by the wave equation ∂²u/∂t₂ = c₂∂²u/∂x₂, where u(x, t) is the displacement of the string and c is a non-zeroconstant,c = 3. Assume that both ends of the string are fixed. The initial position of the string is set by:

           2x 0 < x ≤ 1/2
u(x,0)= ∃
           -2x + 2 1/2 < x ≤1

The string is also set into motion from its initial position with an initial velocity:

                 0    0 < x ≤ 1/2

∂u(x,0)/∂t = ∃

                  ε    1/2 < x ≤ 1

where ε is any non-zero constant. Follow the steps below and use the method ofseparation of variables u(x, t) = F(x)G(t) to find the displacement u (x, t).

(a) Sketch the initial position of the string.

(b) Write down the fixed-end boundary conditions at x = 0 and x = 1 for the displacement u(x,t), and then deduce the boundary conditions for the functionF(x).

(c) Explain in your own words what is the physical process behind the mathematical relation given by ∂u(x, 0)/∂t.

(d) Using the method of separation of variables, find two constant-coefficient ordinary differential equations.

(e) Given a separation constant k = 0, discuss the general solution of the problem.

(f) Given a separation constant k > 0, discuss the general solution of the problem.

(g) Given a separation constant k = -p₂ < 0

i. Show all steps to find the eigen-function F_n (x).

ii. Given the eigen-values kn , find G_n(t).

Hint: Use the notation Cn and Dn forthe constants involved in the definition of G_n(t).

iii. Write a general form of the u_n(x, t) solution of theproblem.

iv. Use the superposition principle to obtain a general solution of the problem.

v. Find C_n.

vi. Find D_n.

(h) Find the series form for the subsequent time-dependent displacement u(x, t) of the uniform string and write down the first six non-zero terms of the solution.

(i) Using the results of (h) and also a mathematical software of your choice, represent graphically u(x, t) for x ∈ [0,1] , for ε=1 and at five different times t =(1/6, 1/3, 1/2, 2, 3) or any other five interesting times. Comment on the physical behaviour of your string at your chosen times and on long term.

All little steps must be included with clear and not overly complicated explanations as to why each step was completed