Exercise 1. If u is a function of x and t with continuous second-order partial derivatives, we set @ = aI + bt and = Mr + nt. Given that wibl an bm = 0, use the chain rule to show that Utt = uxx + 2bnuxt + n^2u.
Exercise 2. Show that if u(x,t) = F(x+c) + G(x-c) satisfies u(x,0) = f(c) and u(x,0) = g(x) for all x ∈ ℝ, then f(c) = g(2) = 2(F(1) - F(-1)).
Exercise 3. Continuing the notation of the preceding exercise, if G1(x) and G2(x) are both antiderivatives of g(x), show that "(x,t) does not depend on which one we choose to represent g(x). [Suggestion: Write down an equation relating G1 and G2.]
Exercise 4. Use exercises 1 and 3 to help you solve the 1-D wave equation on the domain [0,0) subject to the given initial data:
u(x,0) = 2x, u(x,0) = 1+x,
u(0,t) = f(t), u(0,t) = 0,
u(1,t) = c-t, u(1,t) = c.
Exercise 5. Suppose that u(x,t) and v(x,t) have continuous second-order partial derivatives and are related through the equations du/dt = A^2 * d^2u/dx^2 and dv/dt = B^2 * d^2v/dx^2 for some positive constants A and B. Show that u and v are both solutions of the 1-D wave equation.
