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<a name="Functions-and-Variables-for-contrib_005fode"></a>
<div class="header">
<p>
Next: <a href="maxima_208.html#Possible-improvements-to-contrib_005fode" accesskey="n" rel="next">Possible improvements to contrib_ode</a>, Previous: <a href="maxima_206.html#Introduction-to-contrib_005fode" accesskey="p" rel="previous">Introduction to contrib_ode</a>, Up: <a href="maxima_205.html#contrib_005fode_002dpkg" accesskey="u" rel="up">contrib_ode-pkg</a> [<a href="maxima_toc.html#SEC_Contents" title="Table of contents" rel="contents">Contents</a>][<a href="maxima_423.html#Function-and-Variable-Index" title="Index" rel="index">Index</a>]</p>
</div>
<a name="Functions-and-Variables-for-contrib_005fode-1"></a>
<h3 class="section">49.2 Functions and Variables for contrib_ode</h3>
<a name="contrib_005fode"></a><a name="Item_003a-contrib_005fode_002fdeffn_002fcontrib_005fode"></a><dl>
<dt><a name="index-contrib_005fode"></a>Function: <strong>contrib_ode</strong> <em>(<var>eqn</var>, <var>y</var>, <var>x</var>)</em></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>
Categories:<a href="maxima_424.html#Category_003a-Package-contrib_005fode">Package contrib_ode</a>
·</div>
</dd></dl>
<a name="odelin"></a><a name="Item_003a-contrib_005fode_002fdeffn_002fodelin"></a><dl>
<dt><a name="index-odelin"></a>Function: <strong>odelin</strong> <em>(<var>eqn</var>, <var>y</var>, <var>x</var>)</em></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>
<div class="example">
<pre class="example">(%i1) load("contrib_ode")$
</pre><pre class="example">(%i2) odelin(x*(x+1)*'diff(y,x,2)+(x+5)*'diff(y,x,1)+(-4)*y,y,x);
gauss_a(- 6, - 2, - 3, - x) gauss_b(- 6, - 2, - 3, - x)
(%o2) {---------------------------, ---------------------------}
4 4
x x
</pre></div>
<div class=categorybox>
Categories:<a href="maxima_424.html#Category_003a-Package-contrib_005fode">Package contrib_ode</a>
·</div>
</dd></dl>
<a name="ode_005fcheck"></a><a name="Item_003a-contrib_005fode_002fdeffn_002fode_005fcheck"></a><dl>
<dt><a name="index-ode_005fcheck"></a>Function: <strong>ode_check</strong> <em>(<var>eqn</var>, <var>soln</var>)</em></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>
<div class="example">
<pre class="example">(%i1) load("contrib_ode")$
</pre><pre class="example">(%i2) eqn:'diff(y,x,2)+(a*x+b)*y;
2
d y
(%o2) --- + (b + a x) y
2
dx
</pre><pre class="example">(%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 (b + a x)
(%o3) [y = bessel_y(-, --------------) %k2 sqrt(a x + b)
3 3 a
3/2
1 2 (b + a x)
+ bessel_j(-, --------------) %k1 sqrt(a x + b)]
3 3 a
</pre><pre class="example">(%i4) ode_check(eqn,ans[1]);
(%o4) 0
</pre></div>
<div class=categorybox>
Categories:<a href="maxima_424.html#Category_003a-Package-contrib_005fode">Package contrib_ode</a>
·</div>
</dd></dl>
<a name="gauss_005fa"></a><a name="Item_003a-contrib_005fode_002fdeffn_002fgauss_005fa"></a><dl>
<dt><a name="index-gauss_005fa"></a>Function: <strong>gauss_a</strong> <em>(<var>a</var>, <var>b</var>, <var>c</var>, <var>x</var>)</em></dt>
<dd>
<p><code>gauss_a(a,b,c,x)</code> and <code>gauss_b(a,b,c,x)</code> are 2F1
hypergeometric 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) - a*b*y = 0</code> (A&S 15.5.1).
</p>
<p>The only use of these functions is in solutions of ODEs returned by
<code><a href="#odelin">odelin</a></code> and <code><a href="#contrib_005fode">contrib_ode</a></code>. The definition and use of these
functions may change in future releases of Maxima.
</p>
<p>See also <code><a href="#gauss_005fb">gauss_b</a></code>, <code><a href="#dgauss_005fa">dgauss_a</a></code> and <code><a href="#gauss_005fb">gauss_b</a></code>.
</p>
<div class=categorybox>
Categories:<a href="maxima_424.html#Category_003a-Package-contrib_005fode">Package contrib_ode</a>
·</div>
</dd></dl>
<a name="gauss_005fb"></a><a name="Item_003a-contrib_005fode_002fdeffn_002fgauss_005fb"></a><dl>
<dt><a name="index-gauss_005fb"></a>Function: <strong>gauss_b</strong> <em>(<var>a</var>, <var>b</var>, <var>c</var>, <var>x</var>)</em></dt>
<dd><p>See <code><a href="#gauss_005fa">gauss_a</a></code>.
</p>
<div class=categorybox>
Categories:<a href="maxima_424.html#Category_003a-Package-contrib_005fode">Package contrib_ode</a>
·</div>
</dd></dl>
<a name="dgauss_005fa"></a><a name="Item_003a-contrib_005fode_002fdeffn_002fdgauss_005fa"></a><dl>
<dt><a name="index-dgauss_005fa"></a>Function: <strong>dgauss_a</strong> <em>(<var>a</var>, <var>b</var>, <var>c</var>, <var>x</var>)</em></dt>
<dd><p>The derivative with respect to <var>x</var>
of <code><a href="#gauss_005fa">gauss_a</a></code><code>(<var>a</var>, <var>b</var>, <var>c</var>, <var>x</var>)</code>.
</p>
<div class=categorybox>
Categories:<a href="maxima_424.html#Category_003a-Package-contrib_005fode">Package contrib_ode</a>
·</div>
</dd></dl>
<a name="dgauss_005fb"></a><a name="Item_003a-contrib_005fode_002fdeffn_002fdgauss_005fb"></a><dl>
<dt><a name="index-dgauss_005fb"></a>Function: <strong>dgauss_b</strong> <em>(<var>a</var>, <var>b</var>, <var>c</var>, <var>x</var>)</em></dt>
<dd><p>The derivative with respect to <var>x</var>
of <code><a href="#gauss_005fb">gauss_b</a></code><code>(<var>a</var>, <var>b</var>, <var>c</var>, <var>x</var>)</code>.
</p>
<div class=categorybox>
Categories:<a href="maxima_424.html#Category_003a-Package-contrib_005fode">Package contrib_ode</a>
·</div>
</dd></dl>
<a name="kummer_005fm"></a><a name="Item_003a-contrib_005fode_002fdeffn_002fkummer_005fm"></a><dl>
<dt><a name="index-kummer_005fm"></a>Function: <strong>kummer_m</strong> <em>(<var>a</var>, <var>b</var>, <var>x</var>)</em></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><a href="#odelin">odelin</a></code> and <code><a href="#contrib_005fode">contrib_ode</a></code>. The definition and use of this
function may change in future releases of Maxima.
</p>
<p>See also <code><a href="#kummer_005fu">kummer_u</a></code>, <code><a href="#dkummer_005fm">dkummer_m</a></code>, and <code><a href="#dkummer_005fu">dkummer_u</a></code>.
</p>
<div class=categorybox>
Categories:<a href="maxima_424.html#Category_003a-Package-contrib_005fode">Package contrib_ode</a>
·</div>
</dd></dl>
<a name="kummer_005fu"></a><a name="Item_003a-contrib_005fode_002fdeffn_002fkummer_005fu"></a><dl>
<dt><a name="index-kummer_005fu"></a>Function: <strong>kummer_u</strong> <em>(<var>a</var>, <var>b</var>, <var>x</var>)</em></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><a href="#kummer_005fm">kummer_m</a></code>.
</p>
<div class=categorybox>
Categories:<a href="maxima_424.html#Category_003a-Package-contrib_005fode">Package contrib_ode</a>
·</div>
</dd></dl>
<a name="dkummer_005fm"></a><a name="Item_003a-contrib_005fode_002fdeffn_002fdkummer_005fm"></a><dl>
<dt><a name="index-dkummer_005fm"></a>Function: <strong>dkummer_m</strong> <em>(<var>a</var>, <var>b</var>, <var>x</var>)</em></dt>
<dd><p>The derivative with respect to <var>x</var>
of <code><a href="#kummer_005fm">kummer_m</a></code><code>(<var>a</var>, <var>b</var>, <var>x</var>)</code>.
</p>
<div class=categorybox>
Categories:<a href="maxima_424.html#Category_003a-Package-contrib_005fode">Package contrib_ode</a>
·</div>
</dd></dl>
<a name="dkummer_005fu"></a><a name="Item_003a-contrib_005fode_002fdeffn_002fdkummer_005fu"></a><dl>
<dt><a name="index-dkummer_005fu"></a>Function: <strong>dkummer_u</strong> <em>(<var>a</var>, <var>b</var>, <var>x</var>)</em></dt>
<dd><p>The derivative with respect to <var>x</var>
of <code><a href="#kummer_005fu">kummer_u</a></code><code>(<var>a</var>, <var>b</var>, <var>x</var>)</code>.
</p>
<div class=categorybox>
Categories:<a href="maxima_424.html#Category_003a-Package-contrib_005fode">Package contrib_ode</a>
·</div>
</dd></dl>
<a name="bessel_005fsimplify"></a><a name="Item_003a-contrib_005fode_002fdeffn_002fbessel_005fsimplify"></a><dl>
<dt><a name="index-bessel_005fsimplify"></a>Function: <strong>bessel_simplify</strong> <em>(<var>expr</var>)</em></dt>
<dd><p>Simplifies expressions containing Bessel functions <code><a href="maxima_85.html#bessel_005fj">bessel_j</a></code>,
<code><a href="maxima_85.html#bessel_005fy">bessel_y</a></code>, <code><a href="maxima_85.html#bessel_005fi">bessel_i</a></code>, <code><a href="maxima_85.html#bessel_005fk">bessel_k</a></code>,
<code><a href="maxima_85.html#hankel_005f1">hankel_1</a></code>, <code><a href="maxima_85.html#hankel_005f2">hankel_2</a></code>, <code><a href="maxima_90.html#struve_005fh">struve_h</a></code>
and <code><a href="maxima_90.html#struve_005fl">struve_l</a></code>.
Recurrence relations (DLMF §10.6(i))(A&S 9.1.27)
are used to replace functions of highest order n
by functions of order n-1 and n-2.
</p>
<p>This process is repeated until all the orders
differ by less than 2.
</p>
<div class="example">
<pre class="example">(%i1) load("contrib_ode")$
</pre><pre class="example">(%i2) bessel_simplify(4*bessel_j(n,x^2)*(x^2-n^2/x^2)
+x*((bessel_j(n-2,x^2)-bessel_j(n,x^2))*x
-(bessel_j(n,x^2)-bessel_j(n+2,x^2))*x)
-2*bessel_j(n+1,x^2)+2*bessel_j(n-1,x^2));
(%o2) 0
</pre><pre class="example">(%i3) bessel_simplify( -2*bessel_j(1,z)*z^3 - 10*bessel_j(2,z)*z^2
+ 15*%pi*bessel_j(1,z)*struve_h(3,z)*z - 15*%pi*struve_h(1,z)
*bessel_j(3,z)*z - 15*%pi*bessel_j(0,z)*struve_h(2,z)*z
+ 15*%pi*struve_h(0,z)*bessel_j(2,z)*z - 30*%pi*bessel_j(1,z)
*struve_h(2,z) + 30*%pi*struve_h(1,z)*bessel_j(2,z));
(%o3) 0
</pre></div>
<div class=categorybox>
Categories:<a href="maxima_424.html#Category_003a-Package-contrib_005fode">Package contrib_ode</a>
·<a href="maxima_424.html#Category_003a-Bessel-functions">Bessel functions</a>
·<a href="maxima_424.html#Category_003a-Special-functions">Special functions</a>
·</div>
</dd></dl>
<a name="expintegral_005fe_005fsimplify"></a><a name="Item_003a-contrib_005fode_002fdeffn_002fexpintegral_005fe_005fsimplify"></a><dl>
<dt><a name="index-expintegral_005fe_005fsimplify"></a>Function: <strong>expintegral_e_simplify</strong> <em>(<var>expr</var>)</em></dt>
<dd><p>Simplify expressions containing exponential integral <code><a href="maxima_88.html#expintegral_005fe">expintegral_e</a></code>
using the recurrence (A&S 5.1.14).
</p>
<p>expintegral_e(n+1,z)
= (1/n) * (exp(-z)-z*expintegral_e(n,z)) n = 1,2,3 ....
</p>
<div class=categorybox>
Categories:<a href="maxima_424.html#Category_003a-Package-contrib_005fode">Package contrib_ode</a>
·<a href="maxima_424.html#Category_003a-Exponential-Integrals">Exponential Integrals</a>
·<a href="maxima_424.html#Category_003a-Special-functions">Special functions</a>
·</div>
</dd></dl>
<a name="Item_003a-contrib_005fode_002fnode_002fPossible-improvements-to-contrib_005fode"></a><hr>
<div class="header">
<p>
Next: <a href="maxima_208.html#Possible-improvements-to-contrib_005fode" accesskey="n" rel="next">Possible improvements to contrib_ode</a>, Previous: <a href="maxima_206.html#Introduction-to-contrib_005fode" accesskey="p" rel="previous">Introduction to contrib_ode</a>, Up: <a href="maxima_205.html#contrib_005fode_002dpkg" accesskey="u" rel="up">contrib_ode-pkg</a> [<a href="maxima_toc.html#SEC_Contents" title="Table of contents" rel="contents">Contents</a>][<a href="maxima_423.html#Function-and-Variable-Index" title="Index" rel="index">Index</a>]</p>
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