Consider the following partial differential equations. (A) k + cx = 212 (B) k hu = (C) a - (D) k - hu || (E) a2: +c(L x)= au - = at (F) a. +cx = 8u (G) a au (H) k = In each part, determine which differential equation is the correct model for the given function. (a) A string of length Z coincides with the interval [0, 1] on the x-axis. An external vertical force proportional to the horizontal distance from the left end acts on the string for t>0. u(x, t) is the displacement of the string. (b) A string of length I coincides with the interval [0, Z] on the x-axis. u(x, t) is the displacement of the string. (c) A string of length Z coincides with the interval [0, Z] on the x-axis. Transverse vibrations are damped with a force proportional to the instantaneous velocity. u(x, t) is the displacement of the string. (d) A rod of length I coincides with the interval [0, Z] on the x-axis. There is heat transfer from the lateral surface into the surrounding medium which is held at a constant temperature of 0. u(x, t) is the temparature of the rod.