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<B>NUM_ODESOLVE</B> _ _ _  _ _ _  _ _ _  _ _ _ <B>operator</B><P>
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The <em>Runge-Kutta</em> method of order 3 finds an approximate graph for 
the solution of real <em>ODE initial value problem</em>. 
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syntax: </H3>
<em>num_odesolve</em>(&lt;exp&gt;,&lt;depvar&gt;=&lt;start&gt;, 
 &lt;indep&gt;=(&lt;from&gt; .. &lt;to&gt;) 
 [,accuracy=&lt;a&gt;][,iterations=&lt;i&gt;]) 
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or 
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<em>num_odesolve</em>({&lt;exp&gt;,&lt;exp&gt;,...}, 
 { &lt;depvar&gt;=&lt;start&gt;,&lt;depvar&gt;=&lt;start&gt;,...} 
 &lt;indep&gt;=(&lt;from&gt; .. &lt;to&gt;) 
 [,accuracy=&lt;a&gt;][,iterations=&lt;i&gt;]) 
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where 
&lt;depvar&gt; and &lt;start&gt; specify the dependent variable(s) 
and the starting point value (vector), 
&lt;indep&gt;, &lt;from&gt; and &lt;to&gt; specify the independent variable 
and the integration interval (starting point and end point), 
&lt;exp&gt; are equations or expressions which 
contain the first derivative of the independent variable 
with respect to the dependent variable. 
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The ODEs are converted to an explicit form, which then is 
used for a Runge Kutta iteration over the given range. The 
number of steps is controlled by the value of &lt;i&gt; 
(default: 20). If the steps are too coarse to reach the desired 
accuracy in the neighborhood of the starting point, the number is 
increased automatically. 
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Result is a list of pairs, each representing a point of the 
approximate solution of the ODE problem. 
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examples: </H3>
<P><PRE><TT>
depend(y,x);

num_odesolve(df(y,x)=y,y=1,x=(0 .. 1), iterations=5);


  ,{0.2,1.2214},{0.4,1.49181796},{0.6,1.8221064563},
    {0.8,2.2255208258},{1.0,2.7182511366}}

</TT></PRE><P>In most cases you must declare the dependency relation 
between the variables explicitly using 
<A HREF=r37_0192.html>depend</A>; 
otherwise the formal derivative might be converted to zero. 
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The operator 
<A HREF=r37_0179.html>solve</A> is used to convert the form into 
an explicit ODE. If that process fails or if it has no unique result, 
the evaluation is stopped with an error message. 
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