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8. Suppose that a pipe of length L with closed ends has an initial dye concentration of mass M0 centered at one-quarter of the pipe's

8. Suppose that a pipe of length L with closed ends has an initial dye concentration of mass M0 centered at one-quarter of the pipe's length. The long-time solution for the concentration is given by u(x,t)=LM0u(x,t)=L2M0cos(L2x)u(x,t)=L2M0cos(L2x)exp(L242Dt)u(x,t)=L2M0sin(Lx)exp(L22Dt)

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