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<a name="contrib_005fode"></a>
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<h1 class="chapter"> 44. contrib_ode </h1>

<table class="menu" border="0" cellspacing="0">
<tr><td align="left" valign="top"><a href="#SEC192">44.1 Introduction to contrib_ode</a></td><td>&nbsp;&nbsp;</td><td align="left" valign="top">
</td></tr>
<tr><td align="left" valign="top"><a href="#SEC193">44.2 Functions and Variables for contrib_ode</a></td><td>&nbsp;&nbsp;</td><td align="left" valign="top">
</td></tr>
<tr><td align="left" valign="top"><a href="#SEC194">44.3 Possible improvements to contrib_ode</a></td><td>&nbsp;&nbsp;</td><td align="left" valign="top">
</td></tr>
<tr><td align="left" valign="top"><a href="#SEC195">44.4 Test cases for contrib_ode</a></td><td>&nbsp;&nbsp;</td><td align="left" valign="top">
</td></tr>
<tr><td align="left" valign="top"><a href="#SEC196">44.5 References for contrib_ode</a></td><td>&nbsp;&nbsp;</td><td align="left" valign="top">
</td></tr>
</table>

<p><a name="Item_003a-Introduction-to-contrib_005fode"></a>
</p><hr size="6">
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<a name="SEC192"></a>
<h2 class="section"> 44.1 Introduction to contrib_ode </h2>

<p>Maxima's ordinary differential equation (ODE) solver <code>ode2</code> solves
elementary linear ODEs of first and second order.  The function
<code>contrib_ode</code> extends <code>ode2</code> with additional methods for linear
and non-linear first order ODEs and linear homogeneous second order ODEs.  
The code is still under development and the calling sequence may change
in future releases.  Once the code has stabilized it may be
moved from the contrib directory and integrated into Maxima.
</p>
<p>This package must be loaded with the command <code>load('contrib_ode)</code>
before use.
</p>
<p>The calling convention for <code>contrib_ode</code> is identical to <code>ode2</code>.  
It takes
three arguments: an ODE (only the left hand side need be given if the
right hand side is 0), the dependent variable, and the independent
variable.  When successful, it returns a list of solutions.
</p>
<p>The form of the solution differs from <code>ode2</code>.
As non-linear equations can have multiple solutions, 
<code>contrib_ode</code> returns a list of solutions.  Each  solution can 
have a number of forms:
</p><ul>
<li>
an explicit solution for the dependent variable,

</li><li>
an implicit solution for the dependent variable,

</li><li>
a parametric solution in terms of variable <code>%t</code>, or

</li><li>
a tranformation into another ODE in variable <code>%u</code>.

</li></ul>

<p><code>%c</code> is used to represent the constant of integration for first order equations.
<code>%k1</code> and <code>%k2</code> are the constants for second order equations.  
If <code>contrib_ode</code>
cannot obtain a solution for whatever reason, it returns <code>false</code>, after
perhaps printing out an error message.
</p>
<p>It is necessary to return a list of solutions, as even first order
non-linear ODEs can have multiple solutions.  For example:
</p>
<pre class="example">(%i1) load('contrib_ode)$

(%i2) eqn:x*'diff(y,x)^2-(1+x*y)*'diff(y,x)+y=0;

                    dy 2             dy
(%o2)            x (--)  - (x y + 1) -- + y = 0
                    dx               dx
(%i3) contrib_ode(eqn,y,x);

                                             x
(%o3)             [y = log(x) + %c, y = %c %e ]
(%i4) method;

(%o4)                        factor
</pre>
<p>Nonlinear ODEs can have singular solutions without constants of
integration, as in the second solution of the following example:
</p>
<pre class="example">(%i1) load('contrib_ode)$

(%i2) eqn:'diff(y,x)^2+x*'diff(y,x)-y=0;

                       dy 2     dy
(%o2)                 (--)  + x -- - y = 0
                       dx       dx
(%i3) contrib_ode(eqn,y,x);

                                           2
                                 2        x
(%o3)              [y = %c x + %c , y = - --]
                                          4
(%i4) method;

(%o4)                       clairault
</pre>

<p>The following ODE has two parametric solutions in terms of the dummy
variable <code>%t</code>.  In this case the parametric solutions can be manipulated
to give explicit solutions.
</p>
<pre class="example">(%i1) load('contrib_ode)$

(%i2) eqn:'diff(y,x)=(x+y)^2;

                          dy          2
(%o2)                     -- = (y + x)
                          dx
(%i3) contrib_ode(eqn,y,x);

(%o3) [[x = %c - atan(sqrt(%t)), y = - x - sqrt(%t)], 
                     [x = atan(sqrt(%t)) + %c, y = sqrt(%t) - x]]
(%i4) method;

(%o4)                       lagrange
</pre>
<p>The following example (Kamke 1.112) demonstrates an implicit solution.
</p>
<pre class="example">(%i1) load('contrib_ode)$

(%i2) assume(x&gt;0,y&gt;0);

(%o2)                    [x &gt; 0, y &gt; 0]
(%i3) eqn:x*'diff(y,x)-x*sqrt(y^2+x^2)-y;

                     dy           2    2
(%o3)              x -- - x sqrt(y  + x ) - y
                     dx
(%i4) contrib_ode(eqn,y,x);

                                  y
(%o4)                  [x - asinh(-) = %c]
                                  x
(%i5) method;

(%o5)                          lie
</pre>
 

<p>The following Riccati equation is transformed into a linear
second order ODE in the variable <code>%u</code>.  Maxima is unable to
solve the new ODE, so it is returned unevaluated.
</p><pre class="example">(%i1) load('contrib_ode)$

(%i2) eqn:x^2*'diff(y,x)=a+b*x^n+c*x^2*y^2;

                    2 dy      2  2      n
(%o2)              x  -- = c x  y  + b x  + a
                      dx
(%i3) contrib_ode(eqn,y,x);

               d%u
               ---                            2
               dx        2     n - 2   a     d %u
(%o3)  [[y = - ----, %u c  (b x      + --) + ---- c = 0]]
               %u c                     2      2
                                       x     dx
(%i4) method;

(%o4)                        riccati
</pre>

<p>For first order ODEs <code>contrib_ode</code> calls <code>ode2</code>.  It then tries the
following methods: factorization, Clairault, Lagrange, Riccati,
Abel and Lie symmetry methods.  The Lie method is not attempted
on Abel equations if the Abel method fails, but it is tried
if the Riccati method returns an unsolved second order ODE.
</p>
<p>For second order ODEs <code>contrib_ode</code> calls <code>ode2</code> then <code>odelin</code>.
</p>
<p>Extensive debugging traces and messages are displayed if the command
<code>put('contrib_ode,true,'verbose)</code> is executed.
</p>
<div class=categorybox>


<p>Categories:&nbsp;&nbsp;<a href="maxima_95.html#Category_003a-Differential-equations">Differential equations</a>
 &middot;
<a href="maxima_95.html#Category_003a-Share-packages">Share packages</a>
 &middot;
<a href="maxima_95.html#Category_003a-Package-contrib_005fode">Package contrib_ode</a>
</p>
</div>



<p><a name="Item_003a-Functions-and-Variables-for-contrib_005fode"></a>
</p><hr size="6">
<a name="Functions-and-Variables-for-contrib_005fode"></a>
<a name="SEC193"></a>
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</tr></table>
<h2 class="section"> 44.2 Functions and Variables for contrib_ode </h2>


<dl>
<dt><u>Function:</u> <b>contrib_ode</b><i> (<var>eqn</var>, <var>y</var>, <var>x</var>)</i>
<a name="IDX1507"></a>
</dt>
<dd><p>Returns a list of solutions of the ODE <var>eqn</var> with 
independent variable <var>x</var> and dependent variable <var>y</var>.
</p>
<div class=categorybox>


<p>Categories:&nbsp;&nbsp;<a href="maxima_95.html#Category_003a-Package-contrib_005fode">Package contrib_ode</a>
</p>
</div>


</dd></dl>

<p><a name="Item_003a-odelin"></a>
</p><dl>
<dt><u>Function:</u> <b>odelin</b><i> (<var>eqn</var>, <var>y</var>, <var>x</var>)</i>
<a name="IDX1508"></a>
</dt>
<dd><p><code>odelin</code> solves linear homogeneous ODEs of first and 
second order with 
independent variable <var>x</var> and dependent variable <var>y</var>.  
It returns a fundamental solution set of the ODE.
</p>
<p>For second order ODEs, <code>odelin</code> uses a method, due to Bronstein
and Lafaille, that searches for solutions in terms of given 
special functions. 
</p>
<pre class="example">(%i1) load('contrib_ode);

(%i2) odelin(x*(x+1)*'diff(y,x,2)+(x+5)*'diff(y,x,1)+(-4)*y,y,x);
...trying factor method
...solving 7 equations in 4 variables
...trying the Bessel solver
...solving 1 equations in 2 variables
...trying the F01 solver
...solving 1 equations in 3 variables
...trying the spherodial wave solver
...solving 1 equations in 4 variables
...trying the square root Bessel solver
...solving 1 equations in 2 variables
...trying the 2F1 solver
...solving 9 equations in 5 variables
       gauss_a(- 6, - 2, - 3, - x)  gauss_b(- 6, - 2, - 3, - x)
(%o2) {---------------------------, ---------------------------}
                    4                            4
                   x                            x

</pre>
<div class=categorybox>


<p>Categories:&nbsp;&nbsp;<a href="maxima_95.html#Category_003a-Package-contrib_005fode">Package contrib_ode</a>
</p>
</div>


</dd></dl>

<p><a name="Item_003a-ode_005fcheck"></a>
</p><dl>
<dt><u>Function:</u> <b>ode_check</b><i> (<var>eqn</var>, <var>soln</var>)</i>
<a name="IDX1509"></a>
</dt>
<dd><p>Returns the value of ODE <var>eqn</var> after substituting a
possible solution <var>soln</var>.  The value is equivalent to 
zero if <var>soln</var> is a solution of <var>eqn</var>.
</p>
<pre class="example">(%i1) load('contrib_ode)$

(%i2) eqn:'diff(y,x,2)+(a*x+b)*y;

                         2
                        d y
(%o2)                   --- + (a x + b) y
                          2
                        dx
(%i3) ans:[y = bessel_y(1/3,2*(a*x+b)^(3/2)/(3*a))*%k2*sqrt(a*x+b)
         +bessel_j(1/3,2*(a*x+b)^(3/2)/(3*a))*%k1*sqrt(a*x+b)];

                                  3/2
                    1  2 (a x + b)
(%o3) [y = bessel_y(-, --------------) %k2 sqrt(a x + b)
                    3       3 a
                                          3/2
                            1  2 (a x + b)
                 + bessel_j(-, --------------) %k1 sqrt(a x + b)]
                            3       3 a
(%i4) ode_check(eqn,ans[1]);

(%o4)                           0
</pre>
<div class=categorybox>


<p>Categories:&nbsp;&nbsp;<a href="maxima_95.html#Category_003a-Package-contrib_005fode">Package contrib_ode</a>
</p>
</div>


</dd></dl>

<p><a name="Item_003a-method"></a>
</p><dl>
<dt><u>System variable:</u> <b>method</b>
<a name="IDX1510"></a>
</dt>
<dd><p>The variable <code>method</code> is set to the successful solution
method. 
</p>
<div class=categorybox>


<p>Categories:&nbsp;&nbsp;<a href="maxima_95.html#Category_003a-Package-contrib_005fode">Package contrib_ode</a>
</p>
</div>


</dd></dl>

<p><a name="Item_003a-_0025c"></a>
</p><dl>
<dt><u>Variable:</u> <b>%c</b>
<a name="IDX1511"></a>
</dt>
<dd><p><code>%c</code> is the integration constant for first order ODEs.
</p>
<div class=categorybox>


<p>Categories:&nbsp;&nbsp;<a href="maxima_95.html#Category_003a-Package-contrib_005fode">Package contrib_ode</a>
</p>
</div>


</dd></dl>

<p><a name="Item_003a-_0025k1"></a>
</p><dl>
<dt><u>Variable:</u> <b>%k1</b>
<a name="IDX1512"></a>
</dt>
<dd><p><code>%k1</code> is the first integration constant for second order ODEs.
</p>
<div class=categorybox>


<p>Categories:&nbsp;&nbsp;<a href="maxima_95.html#Category_003a-Package-contrib_005fode">Package contrib_ode</a>
</p>
</div>


</dd></dl>

<p><a name="Item_003a-_0025k2"></a>
</p><dl>
<dt><u>Variable:</u> <b>%k2</b>
<a name="IDX1513"></a>
</dt>
<dd><p><code>%k2</code> is the second integration constant for second order ODEs.
</p>
<div class=categorybox>


<p>Categories:&nbsp;&nbsp;<a href="maxima_95.html#Category_003a-Package-contrib_005fode">Package contrib_ode</a>
</p>
</div>


</dd></dl>

<p><a name="Item_003a-gauss_005fa"></a>
</p><dl>
<dt><u>Function:</u> <b>gauss_a</b><i> (<var>a</var>, <var>b</var>, <var>c</var>, <var>x</var>)</i>
<a name="IDX1514"></a>
</dt>
<dd><p><code>gauss_a(a,b,c,x)</code> and <code>gauss_b(a,b,c,x)</code> are 2F1 
geometric functions.  They represent any two independent
solutions of the hypergeometric differential equation 
<code>x(1-x) diff(y,x,2) + [c-(a+b+1)x diff(y,x) - aby = 0</code> (A&amp;S 15.5.1).  
</p>
<p>The only use of these functions is in solutions of ODEs returned by 
<code>odelin</code> and <code>contrib_ode</code>.  The definition and use of these 
functions may change in future releases of Maxima.
</p>
<p>See also <code>gauss_b</code>, <code>dgauss_a</code> and <code>gauss_b</code>.
</p>
<div class=categorybox>


<p>Categories:&nbsp;&nbsp;<a href="maxima_95.html#Category_003a-Package-contrib_005fode">Package contrib_ode</a>
</p>
</div>


</dd></dl>

<p><a name="Item_003a-gauss_005fb"></a>
</p><dl>
<dt><u>Function:</u> <b>gauss_b</b><i> (<var>a</var>, <var>b</var>, <var>c</var>, <var>x</var>)</i>
<a name="IDX1515"></a>
</dt>
<dd><p>See <code>gauss_a</code>.
</p>
<div class=categorybox>


<p>Categories:&nbsp;&nbsp;<a href="maxima_95.html#Category_003a-Package-contrib_005fode">Package contrib_ode</a>
</p>
</div>


</dd></dl>

<p><a name="Item_003a-dgauss_005fa"></a>
</p><dl>
<dt><u>Function:</u> <b>dgauss_a</b><i> (<var>a</var>, <var>b</var>, <var>c</var>, <var>x</var>)</i>
<a name="IDX1516"></a>
</dt>
<dd><p>The derivative with respect to <var>x</var> of <code>gauss_a(<var>a</var>, <var>b</var>, <var>c</var>, <var>x</var>)</code>.
</p>
<div class=categorybox>


<p>Categories:&nbsp;&nbsp;<a href="maxima_95.html#Category_003a-Package-contrib_005fode">Package contrib_ode</a>
</p>
</div>


</dd></dl>

<p><a name="Item_003a-dgauss_005fb"></a>
</p><dl>
<dt><u>Function:</u> <b>dgauss_b</b><i> (<var>a</var>, <var>b</var>, <var>c</var>, <var>x</var>)</i>
<a name="IDX1517"></a>
</dt>
<dd><p>The derivative with respect to <var>x</var> of <code>gauss_b(<var>a</var>, <var>b</var>, <var>c</var>, <var>x</var>)</code>.
</p>
<div class=categorybox>


<p>Categories:&nbsp;&nbsp;<a href="maxima_95.html#Category_003a-Package-contrib_005fode">Package contrib_ode</a>
</p>
</div>


</dd></dl>


<p><a name="Item_003a-kummer_005fm"></a>
</p><dl>
<dt><u>Function:</u> <b>kummer_m</b><i> (<var>a</var>, <var>b</var>, <var>x</var>)</i>
<a name="IDX1518"></a>
</dt>
<dd><p>Kummer's M function, as defined in Abramowitz and Stegun,
<i>Handbook of Mathematical Functions</i>, Section 13.1.2.
</p>
<p>The only use of this function is in solutions of ODEs returned by 
<code>odelin</code> and <code>contrib_ode</code>.  The definition and use of this 
function may change in future releases of Maxima.
</p>
<p>See also <code>kummer_u</code>, <code>dkummer_m</code> and <code>dkummer_u</code>.
</p>
<div class=categorybox>


<p>Categories:&nbsp;&nbsp;<a href="maxima_95.html#Category_003a-Package-contrib_005fode">Package contrib_ode</a>
</p>
</div>


</dd></dl>

<p><a name="Item_003a-kummer_005fu"></a>
</p><dl>
<dt><u>Function:</u> <b>kummer_u</b><i> (<var>a</var>, <var>b</var>, <var>x</var>)</i>
<a name="IDX1519"></a>
</dt>
<dd><p>Kummer's U function, as defined in Abramowitz and Stegun,
<i>Handbook of Mathematical Functions</i>, Section 13.1.3.
</p>
<p>See <code>kummer_m</code>.
</p>
<div class=categorybox>


<p>Categories:&nbsp;&nbsp;<a href="maxima_95.html#Category_003a-Package-contrib_005fode">Package contrib_ode</a>
</p>
</div>


</dd></dl>

<p><a name="Item_003a-dkummer_005fm"></a>
</p><dl>
<dt><u>Function:</u> <b>dkummer_m</b><i> (<var>a</var>, <var>b</var>, <var>x</var>)</i>
<a name="IDX1520"></a>
</dt>
<dd><p>The derivative with respect to <var>x</var> of <code>kummer_m(<var>a</var>, <var>b</var>, <var>x</var>)</code>.
</p>
<div class=categorybox>


<p>Categories:&nbsp;&nbsp;<a href="maxima_95.html#Category_003a-Package-contrib_005fode">Package contrib_ode</a>
</p>
</div>


</dd></dl>

<p><a name="Item_003a-dkummer_005fu"></a>
</p><dl>
<dt><u>Function:</u> <b>dkummer_u</b><i> (<var>a</var>, <var>b</var>, <var>x</var>)</i>
<a name="IDX1521"></a>
</dt>
<dd><p>The derivative with respect to <var>x</var> of <code>kummer_u(<var>a</var>, <var>b</var>, <var>x</var>)</code>.
</p>
<div class=categorybox>


<p>Categories:&nbsp;&nbsp;<a href="maxima_95.html#Category_003a-Package-contrib_005fode">Package contrib_ode</a>
</p>
</div>


</dd></dl>



<p><a name="Item_003a-Possible-improvements-to-contrib_005fode"></a>
</p><hr size="6">
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<h2 class="section"> 44.3 Possible improvements to contrib_ode </h2>


<p>These routines are work in progress.  I still need to:
</p>
<ul>
<li>
Extend the FACTOR method <code>ode1_factor</code> to work for multiple roots.

</li><li>
Extend the FACTOR method <code>ode1_factor</code> to attempt to solve higher
  order factors.  At present it only attemps to solve linear factors.

</li><li>
Fix the LAGRANGE routine <code>ode1_lagrange</code> to prefer real roots over
  complex roots.

</li><li>
Add additional methods for Riccati equations.

</li><li>
Improve the detection of Abel equations of second kind.  The exisiting
  pattern matching is weak.

</li><li>
Work on the Lie symmetry group routine <code>ode1_lie</code>.  There are quite a
  few problems with it: some parts are unimplemented; some test cases
  seem to run forever; other test cases crash; yet others return very
  complex &quot;solutions&quot;.  I wonder if it really ready for release yet.

</li><li>
Add more test cases.

</li></ul>

<p><a name="Item_003a-Test-cases-for-contrib_005fode"></a>
</p><hr size="6">
<a name="Test-cases-for-contrib_005fode"></a>
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<h2 class="section"> 44.4 Test cases for contrib_ode </h2>


<p>The routines have been tested on a approximately one thousand  test cases 
from Murphy,
Kamke, Zwillinger and elsewhere.  These are included in the tests subdirectory.
</p>
<ul>
<li>
The Clairault routine <code>ode1_clairault</code> finds all known solutions,
  including singular solutions, of the Clairault equations in Murphy and
  Kamke.

</li><li>
The other routines often return a single solution when multiple
  solutions exist.

</li><li>
Some of the &quot;solutions&quot; from <code>ode1_lie</code> are overly complex and
  impossible to check.

</li><li>
There are some crashes.

</li></ul>

<p><a name="Item_003a-References-for-contrib_005fode"></a>
</p><hr size="6">
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<h2 class="section"> 44.5 References for contrib_ode </h2>


<ol>
<li>
E. Kamke, Differentialgleichungen Losungsmethoden und Losungen, Vol 1,
    Geest &amp; Portig, Leipzig, 1961

</li><li>
G. M. Murphy, Ordinary Differential Equations and Their Solutions,
    Van Nostrand, New York, 1960

</li><li>
D. Zwillinger, Handbook of Differential Equations, 3rd edition,
    Academic Press, 1998

</li><li>
F. Schwarz, Symmetry Analysis of Abel's Equation, Studies in
    Applied Mathematics, 100:269-294 (1998)

</li><li>
F. Schwarz, Algorithmic Solution of Abel's Equation,
    Computing 61, 39-49 (1998)

</li><li>
E. S. Cheb-Terrab, A. D. Roche, Symmetries and First Order
    ODE Patterns, Computer Physics Communications 113 (1998), p 239.
    (<a href="http://lie.uwaterloo.ca/papers/ode_vii.pdf">http://lie.uwaterloo.ca/papers/ode_vii.pdf</a>)

</li><li>
E. S. Cheb-Terrab, T. Kolokolnikov,  First Order ODEs,
    Symmetries and Linear Transformations, European Journal of
    Applied Mathematics, Vol. 14, No. 2, pp. 231-246 (2003).
    (<a href="http://arxiv.org/abs/math-ph/0007023">http://arxiv.org/abs/math-ph/0007023</a>, 
    <a href="http://lie.uwaterloo.ca/papers/ode_iv.pdf">http://lie.uwaterloo.ca/papers/ode_iv.pdf</a>)

</li><li>
G. W. Bluman, S. C. Anco, Symmetry and Integration Methods for
    Differential Equations, Springer, (2002)

</li><li> 
M. Bronstein, S. Lafaille,
Solutions of linear ordinary differential equations in terms
of special functions,
Proceedings of ISSAC 2002, Lille, ACM Press, 23-28. 
(<a href="http://www-sop.inria.fr/cafe/Manuel.Bronstein/publications/issac2002.pdf">http://www-sop.inria.fr/cafe/Manuel.Bronstein/publications/issac2002.pdf</a>)


</li></ol>

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