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Solution for the One-dimensional Diffusion Equation

We call this constant -\lambda^2 . This gives us the two differential equations, \begin{equation*} F''+\lambda^2F=0; \hspace{1cm} G'=-\lambda^2G, \end{equation*} with general solutions \begin{equation*} F(x)=A\sin(\lambda x)+B\cos(\lambda x); \hspace{1cm} G(t)=Ce^{-\lambda^2t}. \end{equation*}