Question 1. Consider the second order partial differential equation 20 u at2 (1) for an unknown real-valued function u = u(t, r), where t represents time, r represents a point in space, and c > 0 is a constant. 1. For any twice differentiable functions F = F(x) and G = G(x), show that u(t. 2) = F(x + ct) + G(x - ct) satisfies (1) Partial differential equations such as (1) are often solved as initial value problems, where the initial description of the unknown function and its time derivative are provided at each point in space. Along these lines, suppose we are given that u(0, 2) = g(x) and 54 (0, z) = h(r), for some given functions g and h. 2. Assume that u(t, r) = F(x + ct) + G(x - ct) for some functions F and G, as described in problem 1.1. If u = u(t, z) solves the initial value problem described above, show that g(x) = F(x) + G(z) and h(z) = cF'(z) - cG'(r). 3. By integrating the last equation for h(r), show that for any constant a E R, [ h(s) ds = cF(x) - cG(z) -cF(a) + cG(a), and from here solve a linear system to show that F (2) = = (9(x ) + ! [ h(=) as + F(@) - G(a ) ) and G(x) - 5 (9(1) - 2 { h(s) as - F(@) + G(a) ) 4. Lastly, given that u(t, r) = F(x + ct) + G(x - ct), arrive at an explicit formula for the solution to the initial value problem for (1): u(t, 4) = 5 Lo(z + ct) + g(x - ct)] + 2 / h(s)ds. Question 2. Consider the function f(r.y, 2) = aye ("-tufts"). Find all points (x, y, 2) such that Vf (x, y, 2) = (0, 0, 0)