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%TCIDATA{<META NAME="Title" CONTENT="Differential Equations">}
%TCIDATA{Created=Mon Aug 19 14:52:24 1996}
%TCIDATA{LastRevised=Wed May 14 11:53:56 1997}
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\input{tcilatex}
\begin{document}


\subsection{Differential Equations\label{Differential Equations}\hrulefill}

Many differential equations have answers in terms of familiar functions. You
can use the Solve ODE command to find those functions. If you specify
initial conditions for an equation, you can solve the equation numerically.

\subparagraph{\FRAME{itbpF}{50.75pt}{21.75pt}{6pt}{}{}{ex3.wmf}{\special%
{language "Scientific Word";type "GRAPHIC";display "PICT";valid_file
"F";width 50.75pt;height 21.75pt;depth 6pt;original-width
48.875pt;original-height 19.5625pt;cropleft "0";croptop "1";cropright
"1";cropbottom "0";filename 'ex3.WMF';file-properties "XNPEU";}}Compute $%
\dfrac{dx}{dy}=y$}

\begin{enumerate}
\item  Click\FRAME{itbpF}{19.5625pt}{18.8125pt}{2pt}{}{}{fraction.wmf}{%
\special{language "Scientific Word";type "GRAPHIC";display "PICT";valid_file
"F";width 19.5625pt;height 18.8125pt;depth 2pt;original-width
17.8125pt;original-height 16.9375pt;cropleft "0";croptop "1";cropright
"1";cropbottom "0";filename 'fraction.wmf';file-properties "XNPEU";}}.

\item  Type \textbf{dx}, press Tab, type \textbf{dy},\ and press spacebar.

\item  Type \textbf{=y}.

\item  From the Maple menu, choose Solve ODE and then choose Exact. Maple
computes the equation and gives this result: \medskip 
\[
\fbox{%
\begin{tabular}{lll}
\rule[-0.18in]{0in}{0.4in} & $\dfrac{dx}{dy}=y$, Exact solution is : $%
x\left( y\right) =\frac{1}{2}y^{2}+C_{1}$ & \rule[-0.18in]{0in}{0.4in}%
\end{tabular}
}
\]
\medskip 
\end{enumerate}

\subparagraph{\FRAME{itbpF}{50.75pt}{21.75pt}{6pt}{}{}{ex3.wmf}{\special%
{language "Scientific Word";type "GRAPHIC";display "PICT";valid-file
"F";width 50.75pt;height 21.75pt;depth 6pt;original-width
48.875pt;original-height 19.5625pt;cropleft "0";croptop "1";cropright
"1";cropbottom "0";filename 'ex3.WMF';file-properties "XNPEU";}}Compute $%
\left[ 
\begin{array}{l}
y^{\prime }=y \\ 
y\left( 0\right) =1
\end{array}
\right] $}

\begin{enumerate}
\item  Click\FRAME{itbpF}{21.125pt}{19.75pt}{2pt}{}{}{sqbrack.wmf}{\special%
{language "Scientific Word";type "GRAPHIC";maintain-aspect-ratio
TRUE;display "PICT";valid_file "F";width 21.125pt;height 19.75pt;depth
2pt;original-width 20.3125pt;original-height 18.8125pt;cropleft "0";croptop
"1";cropright "1";cropbottom "0";filename 'Sqbrack.wmf';file-properties
"XNPEU";}}, click\FRAME{itbpF}{19.5625pt}{18.8125pt}{2pt}{}{}{matrix.wmf}{%
\special{language "Scientific Word";type "GRAPHIC";display "PICT";valid_file
"F";width 19.5625pt;height 18.8125pt;depth 2pt;original-width
17.8125pt;original-height 16.9375pt;cropleft "0";croptop "1";cropright
"1";cropbottom "0";filename 'Matrix.wmf';file-properties "XNPEU";}}on the
Math2 toolbar, and specify 2 rows and 1 column.

\item  Type \textbf{y'=y }and press Tab.

\item  Type \textbf{y}, click\FRAME{itbpF}{19.5625pt}{18.8125pt}{2pt}{}{}{%
parens.wmf}{\special{language "Scientific Word";type "GRAPHIC";display
"PICT";valid_file "F";width 19.5625pt;height 18.8125pt;depth
2pt;original-width 20.3125pt;original-height 20.3125pt;cropleft "0";croptop
"1";cropright "1";cropbottom "0";filename 'parens.wmf';file-properties
"XNPEU";}}, type \textbf{0},\textbf{\ }and press spacebar.

\item  Type \textbf{=1 }and press spacebar twice.

\item  From the Maple menu, choose Solve ODE and then choose Exact.

\item  Type \textbf{x }as the independent variable you want Maple to use and
choose OK. Maple computes the equation and returns this result:\medskip 
\[
\fbox{%
\begin{tabular}{lll}
\rule[-0.18in]{0in}{0.45in} & $\left[ 
\begin{array}{l}
y^{\prime }=y \\ 
y\left( 0\right) =1
\end{array}
\right] $, Exact solution is : $y\left( x\right) =e^{x}$ & 
\rule[-0.18in]{0in}{0.45in}%
\end{tabular}
}
\]
\medskip 
\end{enumerate}

\subparagraph{Compute these differential equations\textit{\label{exercises}}}

\begin{quotation}
\[
\dfrac{dx}{dy}=2xy\text{\hspace{0.5in}}\ y^{\prime \prime }=y\text{\hspace{%
0.5in}}\ y^{\prime \prime }+y=0
\]
\end{quotation}

\end{document}
