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%TCIDATA{<META NAME="Title" CONTENT="Systems of Equations">}
%TCIDATA{Created=Mon Aug 19 14:52:24 1996}
%TCIDATA{LastRevised=Thu Feb 13 16:44:14 1997}
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\section{Systems of Equations\label{Exact solutions} 
\index{Systems of ODEs@Systems of ODEs}\label{Systems of ODEs}}

Systems \label{ODE, solve systems}consisting of more than one equation are
handled in a consistent manner. Such problems include initial-value problems 
\index{Differential equations@Differential equations!initial value problem@initial value problem}%
and systems of differential equations.

\subsection{\ Exact Solutions\label{Boundary value problems}}

The statement of some problems requires more than one equation. You enter
systems with initial conditions, systems of differential equations,
boundary-value problems, or a mixture of these problems using $n\times 1$
matrices, where $n$ is the number of equations and conditions
involved.\medskip You can also enter such systems in a multiline display
after clicking 
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\begin{quote}
$\blacktriangleright $ To enter and solve a system of differential equations
\end{quote}

\begin{enumerate}
\item  Click \FRAME{itbpF}{0.3009in}{0.3009in}{0.0701in}{}{}{matrix.wmf}{%
\special{language "Scientific Word";type "GRAPHIC";display "PICT";valid_file
"F";width 0.3009in;height 0.3009in;depth 0.0701in;original-width
17.8125pt;original-height 16.9375pt;cropleft "0";croptop "1";cropright
"1";cropbottom "0";filename 'Matrix.wmf';file-properties "XNPEU";}}.

\item[2.]  Select 1 column, and set the number of rows equal to the number
of equations.

\item  Choose \textsf{OK}.

\item  From the \textsf{View} menu, choose \textsf{Matrix Lines} and \textsf{%
Input Boxes} (unless \textsf{Matrix Lines} and \textsf{Input Boxes} are
already visible) to show where to enter the required equations, and enter
the equations.

\item  Leave the insertion point in the matrix.

\item  From the \textsf{Solve ODE} submenu, choose \textsf{Exact} or \textsf{%
Laplace.\medskip }
\end{enumerate}

\begin{quote}
$\blacktriangleright $ \textsf{Solve ODE + Exact, Expand}
\end{quote}

\begin{quotation}
$
\begin{array}{r}
y^{\prime }+y=x \\ 
y(0)=1
\end{array}
$ (Specify $x$), Exact solution is : $y\left( x\right) =\frac{xe^{x}-e^{x}+2%
}{e^{x}}=\allowbreak x-1+\frac{2}{e^{x}}$
\end{quotation}

To solve the second-order initial-value problem $y^{\prime \prime }+y=x^{2}$%
, $y(0)=1$, $y^{\prime }(0)=1$, enter these three equations into a $3\times
1 $ matrix and choose \textsf{Laplace} from the \textsf{Solve ODE}
submenu.\medskip

\begin{quote}
$\blacktriangleright $ \textsf{Solve ODE +} \textsf{Laplace}
\end{quote}

\begin{quotation}
$
\begin{array}{r}
y^{\prime \prime }+y=x^{2} \\ 
y(0)=1\;\, \\ 
y^{\prime }(0)=1\;\,
\end{array}
$ (Specify $x$), Laplace solution is: $y\left( x\right) =x^{2}-2+\sin
x+3\cos x\medskip $
\end{quotation}

The following examples illustrate some of the variety of acceptable notation
for entering and solving systems of differential equations in \textsl{%
Scientific Notebook}.\medskip

\begin{quote}
$\blacktriangleright $ \textsf{Solve ODE +} \textsf{Laplace}
\end{quote}

\begin{quotation}
$
\begin{array}{r}
\dfrac{dy}{dx}=\sin x\, \\ 
y(0)=1\;\quad
\end{array}
\vspace{6pt}$, Laplace solution is: $y\left( x\right) =2-\cos x$

$
\begin{array}{r}
D_{xx}y-y=0 \\ 
y(0)=1 \\ 
y^{\prime }\left( 0\right) =0
\end{array}
\vspace{6pt}$, Laplace solution is: $y\left( x\right) =\dfrac{1}{2}e^{x}+%
\dfrac{1}{2}e^{-x}\;$

$
\begin{array}{r}
D_{xxx}y-y=0 \\ 
y(0)=1 \\ 
y^{\prime }\left( 0\right) =0 \\ 
y^{\prime \prime }(0)=0
\end{array}
$, Laplace solution is: $y\left( x\right) =\dfrac{1}{3}e^{x}+\dfrac{2}{3}e^{-%
\frac{1}{2}x}\cos \dfrac{1}{2}\sqrt{3}x\medskip $
\end{quotation}

\textsl{Scientific Notebook} introduces a new independent variable in
certain instances where none is provided.\medskip

\begin{quote}
$\blacktriangleright $ \textsf{Solve ODE + Exact}
\end{quote}

\begin{quotation}
$
\begin{array}{l}
y^{\prime }=x \\ 
x^{\prime }=-y
\end{array}
$ (Specify $t$), Exact solution is: $
\begin{array}{c}
x\left( t\right) =C_{1}\sin t+C_{2}\cos t \\ 
y\left( t\right) =C_{2}\sin t-C_{1}\cos t
\end{array}
\medskip $
\end{quotation}

Notice that an exact solution to this problem involves a two-parameter
family of solutions..

\begin{quote}
$\blacktriangleright $ \textsf{Solve ODE + Laplace}
\end{quote}

\begin{quotation}
$
\begin{array}{r}
y^{\prime }=x \\ 
x^{\prime }=-y \\ 
x(0)=0 \\ 
y(0)=1
\end{array}
$ (Specify $t$), Laplace solution is: $
\begin{array}{c}
x\left( t\right) =-\sin t \\ 
y\left( t\right) =\cos t
\end{array}
$\ \medskip
\end{quotation}

Subscripted dependent variables are allowed.\medskip

\begin{quote}
$\blacktriangleright $ \textsf{Solve ODE + Laplace}
\end{quote}

\begin{quotation}
$
\begin{array}{r}
D_{x}y_{1}+y_{1}=e^{2x} \\ 
y_{1}\left( 0\right) =1
\end{array}
$, Laplace solution is: $y_{1}\left( x\right) =\dfrac{2}{3}e^{-x}+\dfrac{1}{3%
}e^{2x}$

$\;$

$
\begin{array}{r}
D_{xx}y_{1}-y_{1}=0 \\ 
y_{1}(0)=1 \\ 
y_{1}^{\prime }\left( 0\right) =0
\end{array}
$, Laplace solution is: $y_{1}\left( x\right) =\dfrac{1}{2}e^{x}+\dfrac{1}{2}%
e^{-x}\medskip $
\end{quotation}

The next two examples show solutions using \textsf{Exact} for nonlinear
equations. \textsf{Laplace} produces no result for these equations, as
Laplace transforms are appropriate for linear equations only. \textsf{Series}
would also fail in the second example, because $\ln x$ does not have a
series expansion about $x=0$ in powers of $x$.\medskip

\begin{quote}
$\blacktriangleright $ \textsf{Solve ODE + Exact}
\end{quote}

\begin{quotation}
$
\begin{array}{r}
y^{\prime }=y^{2}+4 \\ 
y(0)=-2
\end{array}
$ (Specify $t$), Exact solution is: $y\left( t\right) =2\tan \left( 2t-%
\dfrac{1}{4}\pi \right) $

$\;$

$
\begin{array}{r}
(x+1)y^{\prime }+y=\ln x \\ 
y(1)=10
\end{array}
$ (Specify $x$), Exact solution is: 
\[
y\left( x\right) =\dfrac{1}{x+1}x\ln x-\dfrac{1}{x+1}x+\dfrac{21}{x+1}=\frac{%
x\ln x-x+21}{x+1} 
\]
\bigskip
\end{quotation}

\subsection{\ Series Solutions\label{Series solutions}}

The following examples illustrate series solutions to various types of
systems of differential equations. 
\index{Series@Series!Taylor@Taylor}You can control the number of terms that
appear in the solution by changing \textsf{Series Order}\label%
{Series, Taylor}\label{Taylor series, (2)}\textsf{\ for ODE Solutions} in
the \textsf{Maple Settings} 
\index{Settings@Settings!series order@series order}dialog. For most of the
following examples, the series order has been set at 6.\medskip

\begin{quote}
$\blacktriangleright $ \textsf{Solve ODE + Series}
\end{quote}

\begin{quotation}
$
\begin{array}{r}
y^{\prime }=x \\ 
x^{\prime }=-y \\ 
x(0)=0 \\ 
y(0)=1
\end{array}
$ (Specify $t$), Series solution is : $
\begin{array}{c}
y\left( t\right) =1-%
\frac{1}{2}t^{2}+\frac{1}{24}t^{4}+O\left( t^{6}\right) \\ 
x\left( t\right) =-t+\frac{1}{6}t^{3}-\frac{1}{120}t^{5}+O\left( t^{6}\right)
\end{array}
$

$\;\medskip $

$
\begin{array}{r}
y^{\prime }=y^{2}+4 \\ 
y(0)=-2
\end{array}
$ (Specify $t$), Series solution is: $
\begin{array}{c}
y\left( t\right) =-2+8t-16t^{2}+\dfrac{128}{3}t^{3} \\ 
\text{\quad \quad \quad }-\dfrac{320}{3}t^{4}+\dfrac{4096}{15}t^{5}+O\left(
t^{6}\right)
\end{array}
$

$\;$\medskip

$
\begin{array}{r}
D_{xx}y_{1}-y_{1}=0 \\ 
y_{1}(0)=1 \\ 
y_{1}^{\prime }\left( 0\right) =0
\end{array}
$, Series solution is: $y_{1}\left( x\right) =1+\dfrac{1}{2}x^{2}+\dfrac{1}{%
24}x^{4}+O\left( x^{6}\right) $

$\;\medskip $

$
\begin{array}{r}
y^{\prime }=x \\ 
x^{\prime }=y \\ 
x(0)=0 \\ 
y(0)=1
\end{array}
$ (Specify $t$), Series solution is: $
\begin{array}{l}
x\left( t\right) =t+\dfrac{1}{6}t^{3}+\dfrac{1}{120}t^{5}+O\left(
t^{6}\right) \\ 
y\left( t\right) =1+\dfrac{1}{2}t^{2}+\dfrac{1}{24}t^{4}+O\left( t^{6}\right)
\end{array}
\bigskip $
\end{quotation}

\subsection{\ Related topics}

\begin{itemize}
\item  \hyperref{Ordinary Differential Equations}{}{}{%
DM10-1.tex#Differential equation}

\item  \hyperref{Numerical Methods for ODEs}{}{}{DM10-3.tex#Solve ODE,
numeric}

\item  \hyperref{Partial Differential Equations}{}{}{DM10-4.tex#PDE}

\item  \hyperref{Exercises and Solutions}{}{}{DM10-5.tex#Exercises}
\end{itemize}

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DMIndex.tex#Math Index}\quad {\small Index entries: \hyperref{Differential
equations}{}{}{DMIndex.tex#Differential equations}, \hyperref{Maple settings%
}{}{}{DMIndex.tex#Maple settings}, \hyperref{Solve ODE}{}{}{%
DMIndex.tex#Solve ODE}, \hyperref{Taylor series}{}{}{DMIndex.tex#Taylor
series}}

\end{document}
