Question 1. Consider the second order partial differential equation (1) for an unknown real-valued function u = u(t, x), where t represents time, x represents a point in space, and c> 0 is a constant. 1. For any twice differentiable functions F = satisfies (1). G(x), show that u(t, x) = F(x + ct)+ G(x – ct) F(x) and G = Partial differential equations such as (1) the unknown function and its time derivative are provided at each point in space. Along these lines, suppose we are given that u(0, x) = g(x) and (0, x) = h(x), for some given functions g and h. are often solved as initial value problems, where the initial description of 2. Assume that u(t, x) = F(x + ct) + G(x – ct) for some functions F and G, as described in problem 1.1. If u = u(t, x) solves the initial value problem described above, show that g(x) = F(x)+ G(x) and h(x) = cF'(x) – cGʻ (x). 3. By integrating the last equation for h(x), show that for any constant a ER, h(s) ds = cF(x) – cG(x) – cF(a) + cG(a), and from here solve a linear system to show that F(x) g(x) + h(s) ds + F(a) – G(a) ), and G(») = (ole) - 1 1 h(s) ds – F(a) + G(a) 4. Lastly, given that u(t, x) = F(x + ct) + G(x - value problem for (1): ct), arrive at an explicit formula for the solution to the initial 1 rx+ct u(t, x) = [g(x + ct) + g(x – ct)] + h(s)ds. 2c x-ct

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