1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
|
<!DOCTYPE html PUBLIC "-//W3C//DTD HTML 4.01 Transitional//EN" "http://www.w3.org/TR/html401/loose.dtd">
<html>
<!-- Created on September, 20 2006 by texi2html 1.76 -->
<!--
Written by: Lionel Cons <Lionel.Cons@cern.ch> (original author)
Karl Berry <karl@freefriends.org>
Olaf Bachmann <obachman@mathematik.uni-kl.de>
and many others.
Maintained by: Many creative people <dev@texi2html.cvshome.org>
Send bugs and suggestions to <users@texi2html.cvshome.org>
-->
<head>
<title>Maxima Manual: 22. Differential Equations</title>
<meta name="description" content="Maxima Manual: 22. Differential Equations">
<meta name="keywords" content="Maxima Manual: 22. Differential Equations">
<meta name="resource-type" content="document">
<meta name="distribution" content="global">
<meta name="Generator" content="texi2html 1.76">
<meta http-equiv="Content-Type" content="text/html; charset=us-ascii">
<style type="text/css">
<!--
a.summary-letter {text-decoration: none}
pre.display {font-family: serif}
pre.format {font-family: serif}
pre.menu-comment {font-family: serif}
pre.menu-preformatted {font-family: serif}
pre.smalldisplay {font-family: serif; font-size: smaller}
pre.smallexample {font-size: smaller}
pre.smallformat {font-family: serif; font-size: smaller}
pre.smalllisp {font-size: smaller}
span.sansserif {font-family:sans-serif; font-weight:normal;}
ul.toc {list-style: none}
body
{
color: black;
background: white;
margin-left: 8%;
margin-right: 13%;
}
h1
{
margin-left: +8%;
font-size: 150%;
font-family: sans-serif
}
h2
{
font-size: 125%;
font-family: sans-serif
}
h3
{
font-size: 100%;
font-family: sans-serif
}
h2,h3,h4,h5,h6 { margin-left: +4%; }
div.textbox
{
border: solid;
border-width: thin;
/* width: 100%; */
padding-top: 1em;
padding-bottom: 1em;
padding-left: 2em;
padding-right: 2em
}
div.titlebox
{
border: none;
padding-top: 1em;
padding-bottom: 1em;
padding-left: 2em;
padding-right: 2em;
background: rgb(200,255,255);
font-family: sans-serif
}
div.synopsisbox
{
border: none;
padding-top: 1em;
padding-bottom: 1em;
padding-left: 2em;
padding-right: 2em;
background: rgb(255,220,255);
/*background: rgb(200,255,255); */
/* font-family: fixed */
}
pre.example
{
border: none;
padding-top: 1em;
padding-bottom: 1em;
padding-left: 1em;
padding-right: 1em;
background: rgb(247,242,180); /* kind of sandy */
/* background: rgb(200,255,255); */ /* sky blue */
font-family: "Lucida Console", monospace
}
div.spacerbox
{
border: none;
padding-top: 2em;
padding-bottom: 2em
}
div.image
{
margin: 0;
padding: 1em;
text-align: center;
}
-->
</style>
<link rel="icon" href="http://maxima.sourceforge.net/favicon.ico"/>
</head>
<body lang="en" bgcolor="#FFFFFF" text="#000000" link="#0000FF" vlink="#800080" alink="#FF0000">
<a name="Differential-Equations"></a>
<a name="SEC74"></a>
<table cellpadding="1" cellspacing="1" border="0">
<tr><td valign="middle" align="left">[<a href="maxima_21.html#SEC73" title="Previous section in reading order"> < </a>]</td>
<td valign="middle" align="left">[<a href="#SEC75" title="Next section in reading order"> > </a>]</td>
<td valign="middle" align="left"> </td>
<td valign="middle" align="left">[<a href="maxima_21.html#SEC72" title="Beginning of this chapter or previous chapter"> << </a>]</td>
<td valign="middle" align="left">[<a href="maxima.html#SEC_Top" title="Up section"> Up </a>]</td>
<td valign="middle" align="left">[<a href="maxima_23.html#SEC76" title="Next chapter"> >> </a>]</td>
<td valign="middle" align="left"> </td>
<td valign="middle" align="left"> </td>
<td valign="middle" align="left"> </td>
<td valign="middle" align="left"> </td>
<td valign="middle" align="left">[<a href="maxima.html#SEC_Top" title="Cover (top) of document">Top</a>]</td>
<td valign="middle" align="left">[<a href="maxima_toc.html#SEC_Contents" title="Table of contents">Contents</a>]</td>
<td valign="middle" align="left">[<a href="maxima_72.html#SEC264" title="Index">Index</a>]</td>
<td valign="middle" align="left">[<a href="maxima_abt.html#SEC_About" title="About (help)"> ? </a>]</td>
</tr></table>
<h1 class="chapter"> 22. Differential Equations </h1>
<table class="menu" border="0" cellspacing="0">
<tr><td align="left" valign="top"><a href="#SEC75">22.1 Definitions for Differential Equations</a></td><td> </td><td align="left" valign="top">
</td></tr>
</table>
<hr size="6">
<a name="Definitions-for-Differential-Equations"></a>
<a name="SEC75"></a>
<table cellpadding="1" cellspacing="1" border="0">
<tr><td valign="middle" align="left">[<a href="#SEC74" title="Previous section in reading order"> < </a>]</td>
<td valign="middle" align="left">[<a href="maxima_23.html#SEC76" title="Next section in reading order"> > </a>]</td>
<td valign="middle" align="left"> </td>
<td valign="middle" align="left">[<a href="#SEC74" title="Beginning of this chapter or previous chapter"> << </a>]</td>
<td valign="middle" align="left">[<a href="#SEC74" title="Up section"> Up </a>]</td>
<td valign="middle" align="left">[<a href="maxima_23.html#SEC76" title="Next chapter"> >> </a>]</td>
<td valign="middle" align="left"> </td>
<td valign="middle" align="left"> </td>
<td valign="middle" align="left"> </td>
<td valign="middle" align="left"> </td>
<td valign="middle" align="left">[<a href="maxima.html#SEC_Top" title="Cover (top) of document">Top</a>]</td>
<td valign="middle" align="left">[<a href="maxima_toc.html#SEC_Contents" title="Table of contents">Contents</a>]</td>
<td valign="middle" align="left">[<a href="maxima_72.html#SEC264" title="Index">Index</a>]</td>
<td valign="middle" align="left">[<a href="maxima_abt.html#SEC_About" title="About (help)"> ? </a>]</td>
</tr></table>
<h2 class="section"> 22.1 Definitions for Differential Equations </h2>
<dl>
<dt><u>Function:</u> <b>bc2</b><i> (<var>solution</var>, <var>xval1</var>, <var>yval1</var>, <var>xval2</var>, <var>yval2</var>)</i>
<a name="IDX692"></a>
</dt>
<dd><p>Solves boundary value problem for second order differential equation.
Here: <var>solution</var> is a general solution to the equation, as
found by <code>ode2</code>, <var>xval1</var> is an equation for the independent
variable in the form <code><var>x</var> = <var>x0</var></code>, and <var>yval1</var> is
an equation for the dependent variable in the form
<code><var>y</var> = <var>y0</var></code>. The <var>xval2</var> and <var>yval2</var> are
equations for these variables at another point.
See <code>ode2</code> for example of usage.
</p>
</dd></dl>
<dl>
<dt><u>Function:</u> <b>desolve</b><i> (<var>eqn</var>, <var>x</var>)</i>
<a name="IDX693"></a>
</dt>
<dt><u>Function:</u> <b>desolve</b><i> ([<var>eqn_1</var>, ..., <var>eqn_n</var>], [<var>x_1</var>, ..., <var>x_n</var>])</i>
<a name="IDX694"></a>
</dt>
<dd><p>The function <code>dsolve</code> solves systems of linear
ordinary differential equations using Laplace transform.
Here the <var>eqn</var>'s are differential equations in
the dependent variables <var>x_1</var>, ..., <var>x_n</var>.
The functional relationships must be explicitly
indicated in both the equations and the variables. For example
</p>
<table><tr><td> </td><td><pre class="example">'diff(f,x,2)=sin(x)+'diff(g,x);
'diff(f,x)+x^2-f=2*'diff(g,x,2);
</pre></td></tr></table>
<p>is not the proper format. The correct way is:
</p>
<table><tr><td> </td><td><pre class="example">'diff(f(x),x,2)=sin(x)+'diff(g(x),x);
'diff(f(x),x)+x^2-f=2*'diff(g(x),x,2);
</pre></td></tr></table>
<p>The call is then <code>desolve([%o3,%o4],[f(x),g(x)]);</code> .
</p>
<p>If initial conditions at 0 are known, they should be supplied before
calling <code>desolve</code> by using <code>atvalue</code>.
</p>
<table><tr><td> </td><td><pre class="example">(%i1) <b><tt>'diff(f(x),x)='diff(g(x),x)+sin(x);</tt></b>
d d
(%o1) -- (f(x)) = -- (g(x)) + sin(x)
dx dx
(%i2) <b><tt>'diff(g(x),x,2)='diff(f(x),x)-cos(x);</tt></b>
2
d d
(%o2) --- (g(x)) = -- (f(x)) - cos(x)
2 dx
dx
(%i3) <b><tt>atvalue('diff(g(x),x),x=0,a);</tt></b>
(%o3) a
(%i4) <b><tt>atvalue(f(x),x=0,1);</tt></b>
(%o4) 1
(%i5) <b><tt>desolve([%o1,%o2],[f(x),g(x)]);</tt></b>
x
(%o5) [f(x) = a %e - a + 1, g(x) =
x
cos(x) + a %e - a + g(0) - 1]
(%i6) <b><tt>[%o1,%o2],%o5,diff;</tt></b>
x x x x
(%o6) [a %e = a %e , a %e - cos(x) = a %e - cos(x)]
</pre></td></tr></table>
<p>If <code>desolve</code> cannot obtain a solution, it returns <code>false</code>.
</p>
</dd></dl>
<dl>
<dt><u>Function:</u> <b>ic1</b><i> (<var>solution</var>, <var>xval</var>, <var>yval</var>)</i>
<a name="IDX695"></a>
</dt>
<dd><p>Solves initial value problem for first order differential equation.
Here: <var>solution</var> is a general solution to the equation, as
found by <code>ode2</code>, <var>xval</var> is an equation for the independent
variable in the form <code><var>x</var> = <var>x0</var></code>, and <var>yval</var> is
an equation for the dependent variable in the form
<code><var>y</var> = <var>y0</var></code>. See <code>ode2</code> for example of usage.
</p>
</dd></dl>
<dl>
<dt><u>Function:</u> <b>ic2</b><i> (<var>solution</var>, <var>xval</var>, <var>yval</var>, <var>dval</var>)</i>
<a name="IDX696"></a>
</dt>
<dd><p>Solves initial value problem for second order differential equation.
Here: <var>solution</var> is a general solution to the equation, as
found by <code>ode2</code>, <var>xval</var> is an equation for the independent
variable in the form <code><var>x</var> = <var>x0</var></code>, <var>yval</var> is
an equation for the dependent variable in the form
<code><var>y</var> = <var>y0</var></code>, and <var>dval</var> is an equation for
the derivative of the dependent variable with respect to
independent variable evaluated at the point <var>xval</var>.
See <code>ode2</code> for example of usage.
</p>
</dd></dl>
<dl>
<dt><u>Function:</u> <b>ode2</b><i> (<var>eqn</var>, <var>dvar</var>, <var>ivar</var>)</i>
<a name="IDX697"></a>
</dt>
<dd><p>The function <code>ode2</code> solves ordinary differential equations of first or second order.
It takes three arguments: an ODE <var>eqn</var>, the dependent variable
<var>dvar</var>, and the independent variable <var>ivar</var>.
When successful, it returns either an explicit or implicit solution for the
dependent variable. <code>%c</code> is used to represent the constant in the case
of first order equations, and <code>%k1</code> and <code>%k2</code> the constants for second
order equations. If <code>ode2</code> cannot obtain a solution for whatever
reason, it returns <code>false</code>, after perhaps printing out an error message.
The methods implemented for first order equations in the order in
which they are tested are: linear, separable, exact - perhaps
requiring an integrating factor, homogeneous, Bernoulli's equation,
and a generalized homogeneous method.
For second order: constant coefficient, exact, linear homogeneous with
non-constant coefficients which can be transformed to constant
coefficient, the Euler or equidimensional equation, the method of
variation of parameters, and equations which are free of either the
independent or of the dependent variable so that they can be reduced
to two first order linear equations to be solved sequentially.
In the course of solving ODEs, several variables are set purely for
informational purposes: <code>method</code> denotes the method of solution used
e.g. <code>linear</code>, <code>intfactor</code> denotes any integrating factor
used, <code>odeindex</code> denotes the index for Bernoulli's method or for the generalized
homogeneous method, and <code>yp</code> denotes the particular solution for the
variation of parameters technique.
</p>
<p>In order to solve initial value problems (IVPs) and
boundary value problems (BVPs), the routine <code>ic1</code> is available
for first order equations, and <code>ic2</code> and <code>bc2</code> for second
order IVPs and BVPs, respectively.
</p>
<p>Example:
</p>
<table><tr><td> </td><td><pre class="example">(%i1) <b><tt>x^2*'diff(y,x) + 3*y*x = sin(x)/x;</tt></b>
2 dy sin(x)
(%o1) x -- + 3 x y = ------
dx x
(%i2) <b><tt>ode2(%,y,x);</tt></b>
%c - cos(x)
(%o2) y = -----------
3
x
(%i3) <b><tt>ic1(%o2,x=%pi,y=0);</tt></b>
cos(x) + 1
(%o3) y = - ----------
3
x
(%i4) <b><tt>'diff(y,x,2) + y*'diff(y,x)^3 = 0;</tt></b>
2
d y dy 3
(%o4) --- + y (--) = 0
2 dx
dx
(%i5) <b><tt>ode2(%,y,x);</tt></b>
3
y + 6 %k1 y
(%o5) ------------ = x + %k2
6
(%i6) <b><tt>ratsimp(ic2(%o5,x=0,y=0,'diff(y,x)=2));</tt></b>
3
2 y - 3 y
(%o6) - ---------- = x
6
(%i7) <b><tt>bc2(%o5,x=0,y=1,x=1,y=3);</tt></b>
3
y - 10 y 3
(%o7) --------- = x - -
6 2
</pre></td></tr></table>
</dd></dl>
<hr size="6">
<table cellpadding="1" cellspacing="1" border="0">
<tr><td valign="middle" align="left">[<a href="#SEC74" title="Beginning of this chapter or previous chapter"> << </a>]</td>
<td valign="middle" align="left">[<a href="maxima_23.html#SEC76" title="Next chapter"> >> </a>]</td>
<td valign="middle" align="left"> </td>
<td valign="middle" align="left"> </td>
<td valign="middle" align="left"> </td>
<td valign="middle" align="left"> </td>
<td valign="middle" align="left"> </td>
<td valign="middle" align="left">[<a href="maxima.html#SEC_Top" title="Cover (top) of document">Top</a>]</td>
<td valign="middle" align="left">[<a href="maxima_toc.html#SEC_Contents" title="Table of contents">Contents</a>]</td>
<td valign="middle" align="left">[<a href="maxima_72.html#SEC264" title="Index">Index</a>]</td>
<td valign="middle" align="left">[<a href="maxima_abt.html#SEC_About" title="About (help)"> ? </a>]</td>
</tr></table>
<p>
<font size="-1">
This document was generated by <em>Robert Dodier</em> on <em>September, 20 2006</em> using <a href="http://texi2html.cvshome.org/"><em>texi2html 1.76</em></a>.
</font>
<br>
</p>
</body>
</html>
|
