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<html>
<head>
  <title>Differential Equations</title>
  <link rel="stylesheet" href="yacas.css" TYPE="text/css" MEDIA="screen">
</head>
<body>
<a name="c15">

</a>
(directly go to documentation on : <a href="refchapter15.html#OdeSolve" target='Chapters' title="general ODE solver">OdeSolve</a>, <a href="refchapter15.html#OdeTest" target='Chapters' title="test the solution of an ODE">OdeTest</a>, <a href="refchapter15.html#OdeOrder" target='Chapters' title="return order of an ODE">OdeOrder</a>.
)<h1>
15. Differential Equations
</h1>
In this chapter, some facilities for solving differential
equations are described. Currently only simple equations without
auxiliary conditions are supported.

<p> </p>
<center><table>
<tr BGCOLOR=#E0E0E0>
<td><a href="refchapter15.html#OdeSolve" target='Chapters' title="general ODE solver">OdeSolve</a></td>
<td>general ODE solver</td>
</tr>
<tr BGCOLOR=#E0E0E0>
<td><a href="refchapter15.html#OdeTest" target='Chapters' title="test the solution of an ODE">OdeTest</a></td>
<td>test the solution of an ODE</td>
</tr>
<tr BGCOLOR=#E0E0E0>
<td><a href="refchapter15.html#OdeOrder" target='Chapters' title="return order of an ODE">OdeOrder</a></td>
<td>return order of an ODE</td>
</tr>
</table></center>

<p>

<a name="OdeSolve">

</a>
<a name="odesolve">

</a>
<h3>
<hr>OdeSolve -- general ODE solver
</h3>
<h5 align=right>Standard library</h5><h5>
Calling format:
</h5>
<table cellpadding="0" width="100%">
<tr><td width=100% bgcolor="#DDDDEE"><pre>
OdeSolve(expr1==expr2)
</pre></tr>
</table>

<h5>
Parameters:
</h5>
<b><tt>expr1,expr2</tt></b> -- expressions containing a function to solve for


<p>

<h5>
Description:
</h5>
This function currently can solve second order homogeneous linear real constant
coefficient equations. The solution is returned with unique constants
generated by <b><tt>UniqueConstant</tt></b>. The roots of the auxiliary equation are 
used as the arguments of exponentials. If the roots are complex conjugate
pairs, then the solution returned is in the form of exponentials, sines
and cosines.


<p>
First and second derivatives are entered as <b><tt>y',y''</tt></b>. Higher order derivatives
may be entered as <b><tt>y(n)</tt></b>, where <b><tt>n</tt></b> is any integer. 


<p>

<h5>
Examples:
</h5>
<table cellpadding="0" width="100%">
<tr><td width=100% bgcolor="#DDDDEE"><pre>
In&gt; OdeSolve( y'' + y == 0 )
Out&gt; C42*Sin(x)+C43*Cos(x);
In&gt; OdeSolve( 2*y'' + 3*y' + 5*y == 0 )
Out&gt; Exp(((-3)*x)/4)*(C78*Sin(Sqrt(31/16)*x)+C79*Cos(Sqrt(31/16)*x));
In&gt; OdeSolve( y'' - 4*y == 0 )
Out&gt; C132*Exp((-2)*x)+C136*Exp(2*x);
In&gt; OdeSolve( y'' +2*y' + y == 0 )
Out&gt; (C183+C184*x)*Exp(-x);
</pre></tr>
</table>


<p>

<h5>
See also:
</h5>
<a href="ref.html?Solve" target="Chapters">
Solve
</a>
, <a href="ref.html?RootsWithMultiples" target="Chapters">
RootsWithMultiples
</a>
.<a name="OdeTest">

</a>
<a name="odetest">

</a>
<h3>
<hr>OdeTest -- test the solution of an ODE
</h3>
<h5 align=right>Standard library</h5><h5>
Calling format:
</h5>
<table cellpadding="0" width="100%">
<tr><td width=100% bgcolor="#DDDDEE"><pre>
OdeTest(eqn,testsol)
</pre></tr>
</table>

<h5>
Parameters:
</h5>
<b><tt>eqn</tt></b> -- equation to test


<p>
<b><tt>testsol</tt></b> -- test solution


<p>

<h5>
Description:
</h5>
This function automates the verification of the solution of an ODE.
It can also be used to quickly see how a particular equation operates
on a function.


<p>

<h5>
Examples:
</h5>
<table cellpadding="0" width="100%">
<tr><td width=100% bgcolor="#DDDDEE"><pre>
In&gt; OdeTest(y''+y,Sin(x)+Cos(x))
Out&gt; 0;
In&gt; OdeTest(y''+2*y,Sin(x)+Cos(x))
Out&gt; Sin(x)+Cos(x);
</pre></tr>
</table>


<p>

<h5>
See also:
</h5>
<a href="ref.html?OdeSolve" target="Chapters">
OdeSolve
</a>
.<a name="OdeOrder">

</a>
<a name="odeorder">

</a>
<h3>
<hr>OdeOrder -- return order of an ODE
</h3>
<h5 align=right>Standard library</h5><h5>
Calling format:
</h5>
<table cellpadding="0" width="100%">
<tr><td width=100% bgcolor="#DDDDEE"><pre>
OdeOrder(eqn)
</pre></tr>
</table>

<h5>
Parameters:
</h5>
<b><tt>eqn</tt></b> -- equation 


<p>

<h5>
Description:
</h5>
This function returns the order of the differential equation, which is
order of the highest derivative. If no derivatives appear, zero is returned.


<p>

<h5>
Examples:
</h5>
<table cellpadding="0" width="100%">
<tr><td width=100% bgcolor="#DDDDEE"><pre>
In&gt; OdeOrder(y'' + 2*y' == 0)
Out&gt; 2;
In&gt; OdeOrder(Sin(x)*y(5) + 2*y' == 0)
Out&gt; 5;
In&gt; OdeOrder(2*y + Sin(y) == 0)
Out&gt; 0;
</pre></tr>
</table>


<p>

<h5>
See also:
</h5>
<a href="ref.html?OdeSolve" target="Chapters">
OdeSolve
</a>
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