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
$$
\begin{equation*}
F(x)=A\sin(\lambda x)+B\cos(\lambda x); \hspace{1cm} G(t)=Ce^{-\lambda^2t}.
\end{equation*}
$$