3. Use the method of separation of variables to solve the problem Ut = xUxx x € [0, 1], t≥0, U (x, 0) = sinx, U (0, t) = 0, U.(t)-0. = Plot the solutions at several instants of time and describe in qualitative terms the behaviour of the solution to as too. What is lim→ U(x, t)? boundary conditions -Dirichlet to the left and (Observe that this problem has mixed Neumann to the right.)

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