20/99
\begin{frame}
\frametitle{Exponential Growth and Decay}

\begin{exampleblock}{}
Assume that $y(t)$ be a function, and $k$ a constant such that
\begin{talign}
y' = ky
\end{talign}
\end{exampleblock}
\pause\medskip

We have seen functions with this behavior:
\pause
\begin{talign}
y(t) = Ce^{kt} &&\mpause[1]{y'(t) = k(Ce^{kt})} \mpause[2]{= ky(t)}
\end{talign}
\pause\pause\pause
Note that
\begin{talign}
y(0) = Ce^0 = C
\end{talign}
\pause\vspace{-1ex}

\begin{block}{}
The only solutions of the differential equation
\begin{talign}
y' = ky
\end{talign}
are the exponential functions
\begin{talign}
y(t) = Ce^{kt}
\end{talign}
where $C$ is any real number.
\end{block}
\end{frame}