Question on Partial Differential Equation.

Please show step by step.

Thank you so much!!!!!!!!!!

(Page 69, Q4.2)The wave equation utt C211,\" 2 0 is an appropriate model for the longitudinal vibrations of a spring. In this application u(t, m) represents displacement parallel to the spring. Suppose that spring has length l and is free at ends. This corresponds to the Neumann boundary conditions, Bu 6U E(t, 0) = a(t, l) = 0, for all t 2 0. Assume the initial conditions 9 and h also satisfy Neumann boundary conditions on [0, l], 611 \"(07 3:) = 901:), a (07 :1?) : h($)' Determine the appropriate extensions of g and h from [0, l] to R so that the solution u(t, :c) given by, u(t, :5) = %[g(:n + at) + g(:c ct)] + i [at h(T)d1-. 20 fat will satisfy the Neumann boundary problem for all t