A jump rope could be crudely modeled by the equation putt = Tuxx u (0,t) = sin (wt) U (1, t) = sin (wt) where p is the rope density, T is the tension, and w is the forcing frequency. In general there are initial conditions associated with the problem as well, but in this problem we will only be interested in the particular solution. We will assume that damping causes the initial conditions to be forgotten (however, we will *not* include the damping explicitly, as it makes the algebra much messier). (a) Split the problem into two parts u (x, t) = ub (x, t) +u+ (x, t), and solve the boundary problem by direct integration. (b) Identify the interior problem, noting that the time-dependent boundary conditions of the boundary problem produce an interior problem with a non-homogeneous forcing. (c) Solve the interior problem by eigenfunction expansion. [Note: When you have reduced your problem to an ODE by means of the transform, you may ignore the homogeneous solutions (set both integration constants to zero), because we are ignoring the initial conditions.] Now, the goal of a jump rope is to get the whole rope to oscillate more or less as a unit. That is, we want the shape of the solution to be dominated by the first mode sin (TX), without lots of wiggles due to the subsequent terms in the expansion. (d) Note that the coefficients of your eigenfunction expansion depend on {p, T, w}. For a jump rope, p is likely to be fixed. Why? On the other hand, t and w are likely to be adjustable. Why? (e) Find a simple relation between 1 and w that induces resonance in the first mode sin (7x), and thus optimizes the jump rope experience.