1. Let x lie in the interval (0, 1). Find u (x, t) so that (a) u satisfies the wave equation utt - c²uxx = 0. (b) u satisfies the boundary condition u (0, t) = u (1, t) = 0. (c) u (x,0) = 0 and u₁(x,0) = x². 2. Now do Neumann boundary conditions: Let x lie in the interval (0,1). Find u (x, t) so that (a) u satisfies the wave equation utt – c²Uxx = 0. (b) u satisfies the boundary condition ux (0, t) = Ux (1, t) = 0. (c) u (x,0) = 0 and u₁ (x,0) = x². 3. Let x lie in the interval (0,1). Find u (x, t) so that (a) u satisfies the diffusion equation ut - kuxx = 0. (b) u satisfies the boundary condition u (0, t) = u (1,t) = 0. (c) u (x,0) = et. 4. Now do Neumann boundary conditions: Let x lie in the interval (0,1). Find u (x, t) so that (a) u satisfies the diffusion equation ut — kuxx = 0. - (b) u satisfies the boundary condition ux (0, t) = Ux (1,t) = 0. (c) u (x,0) = e².

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