Consider the following initial boundary value problem for the wave equation under homogeneous Neumann boundary conditions. Utt (x,t) = Uxx(x, t) for (x,t) = (0,π) × (0,∞) ux (0,t) = 0 = ux(π,t) when t> 0 u(x, 0) = (x) ut(x, 0) = v(x) x = [0, π]. We will assume that and are differentiable on [0, 7] and also satisfy the homogenous Neumann boundary condition themselves. i) Using separation of variables derive an expression for a solution u to the above as an infinite sum. You will need to show that u takes the form u(x,t) = A0 +tBo + Σ (An cos(nt) + B₂ sin(nt)) cos(nx). n=1 Describe the steps you carry out and give explicit expressions for An and Bn in terms of and . Hint: You can use the following without proof: If a differentiable function g : [0,7] → R satisfies the homogeneous Neumann condition 02 (0) = 0 = 02 (7) then it can be expressed via: ½ П მ მ S₁g(s) cos (ns) ds if n ≥ 1 g(x) = = Σ In cos (nx) where In = n=0 חי √₁₂ g(s) ds if n = = 0

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