Consider the partial differential equation ди 18²u Ət 40x²¹ together with the boundary conditions u(0, t) = 0 and u(,t) = 0 for t≥0 and the initial condition u(x, 0) = x(x) for 0 < x (a) If n is a positive integer, show that the function un (x, t) = e-n²t sin(2nx), satisfies the given partial differential equation and boundary conditions. (b) The general solution of the partial differential equation that satisfies the boundary conditions is u(x, t) =B₁e-n²t sin(2nx). Σ n=1 Write down (but do not evaluate) an integral that can be used to determine the constants Bn. -

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