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This is a separable PDE which came up in my exam and I couldn't derive the $X$ solution from deductions on the boundary conditions. The boundary condition $ u(0, t) = \frac{\partial u}{\partial x}(0, t) $ doesn't give the usual form where $C_1$ or $C_2$ may be deduced as zero. Can someone show me how to use the initial condition properly? Any help is greatly appreciated!

\[ u_t = u_{xx} \] subject to the boundary conditions \[ u(0, t) = \frac{\partial u}{\partial x}(0, t) \] and \[ u(\pi, t) = 0 \] for (t > 0), along with the initial condition \[ u(x, 0) = \sin(\pi x). \]

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