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*}