2. The telegraph equations for the local voltage V(x, t) and current I (x, t) along a transmission line are Vr = -LIt - RI & Ix =-CVt - GV, (1) where L, R and C denotes the impedance, resistance and capacitance of the line; G is the line conduct tance. The transmission line has length & and there is no activity along it at t = 0 (so that V(x, 0) = I(x, 0) = 0). The voltage at the end x = ( is suddenly increased to V((, t) = Vo in order to send a signal to the other end at x = 0, which is held at zero voltage: V(0, t) = 0. The problem is to establish what signal, S(t) = I(0, t), is received. (a) (3 pts) First, eliminate I(x, t) to write a single PDE for V(x, t). Hint: differentiate in x the first PDE in (1), then use the second PDE in (1) to replace Ir. Show that this equation reduces to either the wave equation or the diffusion equation in certain limits of the parameters { L, R, C, G}. (b) (4 pts) Now (and from hereon) consider the case G = R and C = L. Find the steady state voltage solution Vss (x). (c) (11 pts) Next, attack the initial-value problem using separation of variables. (d) (4 pts) Last, use your solution for V(x, t) and the first PDE in (1), evaluated for x = 0, to write a first-order ODE in time for S(t). Solve this ODE. To save you a little algebra, you may quote the integrals e" sin(yz) dz = 1 - ye" cos(yz) + rez sin yz - e" cos(yz)dz _ re cos( yz) - r + ye sin yz 72 + 72 72 +72