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<div class="subsection-level-extent" id="Integrating-Differential-Equations">
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<h4 class="subsection" id="Integrating-Differential-Equations-1"><span>1.2.5 Integrating Differential Equations<a class="copiable-link" href="#Integrating-Differential-Equations-1"> ¶</a></span></h4>
<p>Octave has built-in functions for solving nonlinear differential
equations of the form
</p>
<div class="example">
<div class="group"><pre class="example-preformatted">dx
-- = f (x, t)
dt
</pre></div></div>
<p>with the initial condition
</p>
<div class="example">
<pre class="example-preformatted">x(t = t0) = x0
</pre></div>
<p>For Octave to integrate equations of this form, you must first provide a
definition of the function
<code class="code">f(x,t)</code>.
This is straightforward, and may be accomplished by entering the
function body directly on the command line. For example, the following
commands define the right-hand side function for an interesting pair of
nonlinear differential equations. Note that while you are entering a
function, Octave responds with a different prompt, to indicate that it
is waiting for you to complete your input.
</p>
<div class="example">
<div class="group"><pre class="example-preformatted">octave:1> function xdot = f (x, t)
>
> r = 0.25;
> k = 1.4;
> a = 1.5;
> b = 0.16;
> c = 0.9;
> d = 0.8;
>
> xdot(1) = r*x(1)*(1 - x(1)/k) - a*x(1)*x(2)/(1 + b*x(1));
> xdot(2) = c*a*x(1)*x(2)/(1 + b*x(1)) - d*x(2);
>
> endfunction
</pre></div></div>
<p>Given the initial condition
</p>
<div class="example">
<pre class="example-preformatted">octave:2> x0 = [1; 2];
</pre></div>
<p>and the set of output times as a column vector (note that the first
output time corresponds to the initial condition given above)
</p>
<div class="example">
<pre class="example-preformatted">octave:3> t = linspace (0, 50, 200)';
</pre></div>
<p>it is easy to integrate the set of differential equations:
</p>
<div class="example">
<pre class="example-preformatted">octave:4> x = lsode ("f", x0, t);
</pre></div>
<p>The function <code class="code">lsode</code> uses the Livermore Solver for Ordinary
Differential Equations, described in A. C. Hindmarsh,
<cite class="cite">ODEPACK, a Systematized Collection of ODE Solvers</cite>, in: Scientific
Computing, R. S. Stepleman et al. (Eds.), North-Holland, Amsterdam,
1983, pages 55–64.
</p>
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