5. (11 points) Problem for the wave equation asks for a function u(x, t) that satisfies, for -00 0, u(x, 0) = f(x), ut(r, 0) = 9(I). Let U, F, and G be Fourier transforms of u, f, and g, respectively. Prove that U(X, t) = F(X) + 2iAc 1 G ( X ) exact + F(X) 1 G ( X) e-iAct 2iAc4. (30 points) Use the method of separation of variables u(x,y) = X(1)Y(y) to solve the following eqaution, with initial condition Hint: You may use the following steps: Step 1: Using u(x,y) = X(X)Y(y), nd u: and y . Step 2: Substitute III and y into the equation uI+ 2 uy= o , I ! Step 3: Apply required calculation to reach l2% =- % - Step 4: Explain why two fractions are equal to a constant number K . Step 5: The equation in step 4, gives us 2 ODEs. State and solve the ODEs. Step 6: Find the final solution for the equation \"1+ 2 uy= 0 _ Apply the initial condition to nd the unique solution