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\begin{document}

{\bf Question}

For each of the following one-parameter families of functions,
find a first-order differential equation ({\bf not} containing the
constant $c$) which is satisfied by all members of the family. In
each case sketch graphs of a few typical functions in the family.

\begin{enumerate}

\item $y=ce^{-2x}\qquad (*)$

\item $y=cx^2-x$

\item $y=(c+x)e^{3x}\qquad (*)$

\end{enumerate}





\vspace{0.25in}

{\bf Answer}

\begin{enumerate}

\item

\begin{eqnarray*}
y=ce^{-2x} & \Rightarrow & \dy = -2ce^{-2x}\\
\textrm{eliminate }c & & \\
\Rightarrow & & \\
ye^{2x} = c & \Rightarrow & \dy = -2(ye^{2x})e^{-2x}\\
& & \Rightarrow \dy = -2y
\end{eqnarray*}

\item

$$y = cx^2 - x \Rightarrow \dy = 2cx - 1$$
$$\Rightarrow c = \frac{y+x}{x^2}$$
\begin{eqnarray*}
\Rightarrow \dy & = & 2 \left ( \frac{y+x}{x^2} \right ) x - 1\\
& = & 2 \left ( \frac{y+x}{x} \right ) -1 \\
& = & 2 \left ( \frac{y}{x} + 1 \right ) -1\\
& = & s \frac{y}{x} + 1
\end{eqnarray*}

\item

$$y = (c+x)e^{3x}$$
\begin{eqnarray*}
\Rightarrow \dy & = & 3(c+x)e^{3x} + e^{3x}\\
& = & 3ce^{3x} + (3x+1)e^{3x}
\end{eqnarray*}
$$\Rightarrow c = e^{-3x} y - x$$
\begin{eqnarray*}
\Rightarrow \dy & = & 3 \left ( e^{-3x} y - x \right ) +
(3x+1)e^{3x}\\
& = & 3y + e^{3x} 
\end{eqnarray*}

\end{enumerate}

\end{document}