Gconsider the following b/ivp for the wave equation, defined for 0 ≤ x ≤ 1 and t ≥ 0: utt = c 2uxx, u(0, t) = u(1, t) = 0, u(x, 0) = x(1 − x), ut(x, 0) = 0. (a) briefly describe, and sketch, a physical situation which this models. be sure to explain the effect of both boundary conditions and both initial conditions. (b) assume that u(x, t) = f(x)g(t). find utt and uxx. also, determine the boundary conditions for f(x) from the boundary conditions for u(x, t), and one initial condition for g(t). (the three bc/ics equal to zero will give you these.) (c) plug u = fg back into the pde and divide both sides by c 2 fg, i. e., “separate variables.” briefly justify why this quantity must be a constant. call this constant −λ. write down two odes: one for g(t) with one initial condition, and a sturmlioville problem for f(x). (d) determine f(x), λ, and g(t). (e) using your solution to part (d) and the principle of superposition, find the general solution to this pde. (f) use the last initial condition, u(x, 0) = x(1 − x), to find the particular solution that satisfies the initial and boundary conditions. (g) what is the long-term behavior of this system? give both a mathematical and physical justification.