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]>
<html xmlns="http://www.w3.org/1999/xhtml"
xmlns:xlink="http://www.w3.org/1999/xlink" >
<head>
<title>Section1.13</title>
<link rel="stylesheet" type="text/css" href="graphicstyle.css" />
<script type="text/javascript" src="bookax1.js" />
</head>
<body>
<a href="book-contents.xhtml" style="margin-right: 10px;">Book Contents</a><a href="section-1.12.xhtml" style="margin-right: 10px;">Previous Section 1.12 Integration</a><a href="section-1.14.xhtml" style="margin-right: 10px;">Next Section 1.14 Solution of Equations</a>
<a href="book-index.xhtml">Book Index</a><div class="section" id="sec-1.13">
<h2 class="sectiontitle">1.13 Differential Equations</h2>
<a name="ugIntroDiffEqns" class="label"/>
<p>The general approach used in integration also carries over to the
solution of linear differential equations.
</p>
<p>Let's solve some differential equations.
Let <math xmlns="&mathml;" mathsize="big"><mstyle><mi>y</mi></mstyle></math> be the unknown function in terms of <math xmlns="&mathml;" mathsize="big"><mstyle><mi>x</mi></mstyle></math>.
</p>
<div id="spadComm1-173" class="spadComm" >
<form id="formComm1-173" action="javascript:makeRequest('1-173');" >
<input id="comm1-173" type="text" class="command" style="width: 11em;" value="y := operator 'y" />
</form>
<span id="commSav1-173" class="commSav" >y := operator 'y</span>
<div id="mathAns1-173" ></div>
</div>
<div class="math">
<table>
<tr><td>
<math xmlns="&mathml;" mathsize="big" display="block"><mstyle><mi>y</mi></mstyle></math>
</td></tr>
</table>
</div>
<div class="returnType">
Type: BasicOperator
</div>
<p>Here we solve a third order equation with polynomial coefficients.
</p>
<div id="spadComm1-174" class="spadComm" >
<form id="formComm1-174" action="javascript:makeRequest('1-174');" >
<input id="comm1-174" type="text" class="command" style="width: 60em;" value="deq := x**3 * D(y x, x, 3) + x**2 * D(y x, x, 2) - 2 * x * D(y x, x) + 2 * y x = 2 * x**4" />
</form>
<span id="commSav1-174" class="commSav" >deq := x**3 * D(y x, x, 3) + x**2 * D(y x, x, 2) - 2 * x * D(y x, x) + 2 * y x = 2 * x**4</span>
<div id="mathAns1-174" ></div>
</div>
<div class="math">
<table>
<tr><td>
<math xmlns="&mathml;" mathsize="big" display="block"><mstyle><mrow><mrow><mrow><mrow><msup><mi>x</mi><mn>3</mn></msup></mrow><mo></mo><mrow><mrow><msubsup><mi>y</mi><mrow><mo></mo></mrow><mrow><mrow><mo>′</mo><mo>′</mo><mo>′</mo></mrow></mrow></msubsup></mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mrow><mo>+</mo><mrow><mrow><msup><mi>x</mi><mn>2</mn></msup></mrow><mo></mo><mrow><mrow><msubsup><mi>y</mi><mrow><mo></mo></mrow><mrow><mrow><mo>′</mo><mo>′</mo></mrow></mrow></msubsup></mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mrow><mo>-</mo><mrow><mn>2</mn><mo></mo><mi>x</mi><mo></mo><mrow><mrow><msubsup><mi>y</mi><mrow><mo></mo></mrow><mrow><mo>′</mo></mrow></msubsup></mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mrow><mo>+</mo><mrow><mn>2</mn><mo></mo><mrow><mi>y</mi><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mrow></mrow><mo>=</mo><mrow><mn>2</mn><mo></mo><mrow><msup><mi>x</mi><mn>4</mn></msup></mrow></mrow></mrow></mstyle></math>
</td></tr>
</table>
</div>
<div class="returnType">
Type: Equation Expression Integer
</div>
<div id="spadComm1-175" class="spadComm" >
<form id="formComm1-175" action="javascript:makeRequest('1-175');" >
<input id="comm1-175" type="text" class="command" style="width: 11em;" value="solve(deq, y, x)" />
</form>
<span id="commSav1-175" class="commSav" >solve(deq, y, x)</span>
<div id="mathAns1-175" ></div>
</div>
<p><!-- NOTE: the book has a different solution and it appears to be
less complicated than this one. -->
<math xmlns="&mathml;" mathsize="big" display="block"><mstyle><mrow><mtable><mtr><mtd><mo>[</mo><mrow><mi>particular</mi><mo>=</mo><mfrac><mrow><mrow><msup><mi>x</mi><mn>5</mn></msup></mrow><mo>-</mo><mrow><mn>10</mn><mo></mo><mrow><msup><mi>x</mi><mn>3</mn></msup></mrow></mrow><mo>+</mo><mrow><mn>20</mn><mo></mo><mrow><msup><mi>x</mi><mn>2</mn></msup></mrow></mrow><mo>+</mo><mn>4</mn></mrow><mrow><mn>15</mn><mo></mo><mi>x</mi></mrow></mfrac></mrow><mo>,</mo></mtd></mtr><mtr><mtd></mtd></mtr><mtr><mtd><mrow><mi>basis</mi><mo>=</mo><mrow><mo>[</mo><mfrac><mrow><mrow><mn>2</mn><mo></mo><mrow><msup><mi>x</mi><mn>3</mn></msup></mrow></mrow><mo>-</mo><mrow><mn>3</mn><mo></mo><mrow><msup><mi>x</mi><mn>2</mn></msup></mrow></mrow><mo>+</mo><mn>1</mn></mrow><mi>x</mi></mfrac><mo>,</mo><mfrac><mrow><mrow><msup><mi>x</mi><mn>3</mn></msup></mrow><mo>-</mo><mn>1</mn></mrow><mi>x</mi></mfrac><mo>,</mo><mfrac><mrow><mrow><msup><mi>x</mi><mn>3</mn></msup></mrow><mo>-</mo><mrow><mn>3</mn><mo></mo><mrow><msup><mi>x</mi><mn>2</mn></msup></mrow></mrow><mo>-</mo><mn>1</mn></mrow><mi>x</mi></mfrac><mo>]</mo></mrow></mrow><mo>]</mo></mtd></mtr></mtable></mrow></mstyle></math>
</p>
<div class="returnType">
Type: Union(Record(particular: Expression Integer,basis: List Expression Integer),...)
</div>
<p>Here we find all the algebraic function solutions of the equation.
</p>
<div id="spadComm1-176" class="spadComm" >
<form id="formComm1-176" action="javascript:makeRequest('1-176');" >
<input id="comm1-176" type="text" class="command" style="width: 42em;" value="deq := (x**2 + 1) * D(y x, x, 2) + 3 * x * D(y x, x) + y x = 0" />
</form>
<span id="commSav1-176" class="commSav" >deq := (x**2 + 1) * D(y x, x, 2) + 3 * x * D(y x, x) + y x = 0</span>
<div id="mathAns1-176" ></div>
</div>
<div class="math">
<table>
<tr><td>
<math xmlns="&mathml;" mathsize="big" display="block"><mstyle><mrow><mrow><mrow><mrow><mo>(</mo><mrow><msup><mi>x</mi><mn>2</mn></msup></mrow><mo>+</mo><mn>1</mn><mo>)</mo></mrow><mo></mo><mrow><mrow><msubsup><mi>y</mi><mrow><mo></mo></mrow><mrow><mrow><mo>′</mo><mo>′</mo></mrow></mrow></msubsup></mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mrow><mo>+</mo><mrow><mn>3</mn><mo></mo><mi>x</mi><mo></mo><mrow><mrow><msubsup><mi>y</mi><mrow><mo></mo></mrow><mrow><mo>′</mo></mrow></msubsup></mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mrow><mo>+</mo><mrow><mi>y</mi><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mrow><mo>=</mo><mn>0</mn></mrow></mstyle></math>
</td></tr>
</table>
</div>
<div class="returnType">
Type: Equation Expression Integer
</div>
<div id="spadComm1-177" class="spadComm" >
<form id="formComm1-177" action="javascript:makeRequest('1-177');" >
<input id="comm1-177" type="text" class="command" style="width: 11em;" value="solve(deq, y, x)" />
</form>
<span id="commSav1-177" class="commSav" >solve(deq, y, x)</span>
<div id="mathAns1-177" ></div>
</div>
<div class="math">
<table>
<tr><td>
<math xmlns="&mathml;" mathsize="big" display="block"><mstyle><mrow><mo>[</mo><mrow><mi>particular</mi><mo>=</mo><mn>0</mn></mrow><mo>,</mo><mrow><mi>basis</mi><mo>=</mo><mrow><mo>[</mo><mfrac><mn>1</mn><mrow><msqrt><mrow><mrow><msup><mi>x</mi><mn>2</mn></msup></mrow><mo>+</mo><mn>1</mn></mrow></msqrt></mrow></mfrac><mo>,</mo><mfrac><mrow><mo>log</mo><mo>(</mo><mrow><mrow><msqrt><mrow><mrow><msup><mi>x</mi><mn>2</mn></msup></mrow><mo>+</mo><mn>1</mn></mrow></msqrt></mrow><mo>-</mo><mi>x</mi></mrow><mo>)</mo></mrow><mrow><msqrt><mrow><mrow><msup><mi>x</mi><mn>2</mn></msup></mrow><mo>+</mo><mn>1</mn></mrow></msqrt></mrow></mfrac><mo>]</mo></mrow></mrow><mo>]</mo></mrow></mstyle></math>
</td></tr>
</table>
</div>
<div class="returnType">
Type: Union(Record(particular: Expression Integer,basis: List Expression Integer),...)
</div>
<p>Coefficients of differential equations can come from arbitrary
constant fields. For example, coefficients can contain algebraic
numbers.
</p>
<p>This example has solutions whose logarithmic derivative is an
algebraic function of degree two.
</p>
<div id="spadComm1-178" class="spadComm" >
<form id="formComm1-178" action="javascript:makeRequest('1-178');" >
<input id="comm1-178" type="text" class="command" style="width: 37em;" value="eq := 2*x**3 * D(y x,x,2) + 3*x**2 * D(y x,x) - 2 * y x" />
</form>
<span id="commSav1-178" class="commSav" >eq := 2*x**3 * D(y x,x,2) + 3*x**2 * D(y x,x) - 2 * y x</span>
<div id="mathAns1-178" ></div>
</div>
<div class="math">
<table>
<tr><td>
<math xmlns="&mathml;" mathsize="big" display="block"><mstyle><mrow><mrow><mn>2</mn><mo></mo><mrow><msup><mi>x</mi><mn>3</mn></msup></mrow><mo></mo><mrow><mrow><msubsup><mi>y</mi><mrow><mo></mo></mrow><mrow><mrow><mo>′</mo><mo>′</mo></mrow></mrow></msubsup></mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mrow><mo>+</mo><mrow><mn>3</mn><mo></mo><mrow><msup><mi>x</mi><mn>2</mn></msup></mrow><mo></mo><mrow><mrow><msubsup><mi>y</mi><mrow><mo></mo></mrow><mrow><mo>′</mo></mrow></msubsup></mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mrow><mo>-</mo><mrow><mn>2</mn><mo></mo><mrow><mi>y</mi><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mrow></mrow></mstyle></math>
</td></tr>
</table>
</div>
<div class="returnType">
Type: Expression Integer
</div>
<div id="spadComm1-179" class="spadComm" >
<form id="formComm1-179" action="javascript:makeRequest('1-179');" >
<input id="comm1-179" type="text" class="command" style="width: 13em;" value="solve(eq,y,x).basis" />
</form>
<span id="commSav1-179" class="commSav" >solve(eq,y,x).basis</span>
<div id="mathAns1-179" ></div>
</div>
<div class="math">
<table>
<tr><td>
<math xmlns="&mathml;" mathsize="big" display="block"><mstyle><mrow><mo>[</mo><mrow><msup><mi>e</mi><mrow><mo>(</mo><mo>-</mo><mfrac><mn>2</mn><mrow><msqrt><mi>x</mi></msqrt></mrow></mfrac><mo>)</mo></mrow></msup></mrow><mo>,</mo><mrow><msup><mi>e</mi><mfrac><mn>2</mn><mrow><msqrt><mi>x</mi></msqrt></mrow></mfrac></msup></mrow><mo>]</mo></mrow></mstyle></math>
</td></tr>
</table>
</div>
<div class="returnType">
Type: List Expression Integer
</div>
<p>Here's another differential equation to solve.
</p>
<div id="spadComm1-180" class="spadComm" >
<form id="formComm1-180" action="javascript:makeRequest('1-180');" >
<input id="comm1-180" type="text" class="command" style="width: 31em;" value="deq := D(y x, x) = y(x) / (x + y(x) * log y x)" />
</form>
<span id="commSav1-180" class="commSav" >deq := D(y x, x) = y(x) / (x + y(x) * log y x)</span>
<div id="mathAns1-180" ></div>
</div>
<div class="math">
<table>
<tr><td>
<math xmlns="&mathml;" mathsize="big" display="block"><mstyle><mrow><mrow><mrow><msubsup><mi>y</mi><mrow><mo></mo></mrow><mrow><mo>′</mo></mrow></msubsup></mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mfrac><mrow><mi>y</mi><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mrow><mrow><mrow><mi>y</mi><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo></mo><mrow><mo>log</mo><mo>(</mo><mrow><mi>y</mi><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>)</mo></mrow></mrow><mo>+</mo><mi>x</mi></mrow></mfrac></mrow></mstyle></math>
</td></tr>
</table>
</div>
<div class="returnType">
Type: Equation Expression Integer
</div>
<div id="spadComm1-181" class="spadComm" >
<form id="formComm1-181" action="javascript:makeRequest('1-181');" >
<input id="comm1-181" type="text" class="command" style="width: 11em;" value="solve(deq, y, x)" />
</form>
<span id="commSav1-181" class="commSav" >solve(deq, y, x)</span>
<div id="mathAns1-181" ></div>
</div>
<div class="math">
<table>
<tr><td>
<math xmlns="&mathml;" mathsize="big" display="block"><mstyle><mfrac><mrow><mrow><mrow><mi>y</mi><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo></mo><mrow><msup><mrow><mo>log</mo><mo>(</mo><mrow><mi>y</mi><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>)</mo></mrow><mn>2</mn></msup></mrow></mrow><mo>-</mo><mrow><mn>2</mn><mo></mo><mi>x</mi></mrow></mrow><mrow><mn>2</mn><mo></mo><mrow><mi>y</mi><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mrow></mfrac></mstyle></math>
</td></tr>
</table>
</div>
<div class="returnType">
Type: Union(Expression Integer,...)
</div>
<p>Rather than attempting to get a closed form solution of
a differential equation, you instead might want to find an
approximate solution in the form of a series.
</p>
<p>Let's solve a system of nonlinear first order equations and get a
solution in power series. Tell Axiom that <math xmlns="&mathml;" mathsize="big"><mstyle><mi>x</mi></mstyle></math> is also an
operator.
</p>
<div id="spadComm1-182" class="spadComm" >
<form id="formComm1-182" action="javascript:makeRequest('1-182');" >
<input id="comm1-182" type="text" class="command" style="width: 11em;" value="x := operator 'x" />
</form>
<span id="commSav1-182" class="commSav" >x := operator 'x</span>
<div id="mathAns1-182" ></div>
</div>
<div class="math">
<table>
<tr><td>
<math xmlns="&mathml;" mathsize="big" display="block"><mstyle><mi>x</mi></mstyle></math>
</td></tr>
</table>
</div>
<div class="returnType">
Type: BasicOperator
</div>
<p>Here are the two equations forming our system.
</p>
<div id="spadComm1-183" class="spadComm" >
<form id="formComm1-183" action="javascript:makeRequest('1-183');" >
<input id="comm1-183" type="text" class="command" style="width: 21em;" value="eq1 := D(x(t), t) = 1 + x(t)**2" />
</form>
<span id="commSav1-183" class="commSav" >eq1 := D(x(t), t) = 1 + x(t)**2</span>
<div id="mathAns1-183" ></div>
</div>
<div class="math">
<table>
<tr><td>
<math xmlns="&mathml;" mathsize="big" display="block"><mstyle><mrow><mrow><mrow><msubsup><mi>x</mi><mrow><mo></mo></mrow><mrow><mo>′</mo></mrow></msubsup></mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow><mo>=</mo><mrow><mrow><msup><mrow><mi>x</mi><mo>(</mo><mi>t</mi><mo>)</mo></mrow><mn>2</mn></msup></mrow><mo>+</mo><mn>1</mn></mrow></mrow></mstyle></math>
</td></tr>
</table>
</div>
<div class="returnType">
Type: Equation Expression Integer
</div>
<div id="spadComm1-184" class="spadComm" >
<form id="formComm1-184" action="javascript:makeRequest('1-184');" >
<input id="comm1-184" type="text" class="command" style="width: 21em;" value="eq2 := D(y(t), t) = x(t) * y(t)" />
</form>
<span id="commSav1-184" class="commSav" >eq2 := D(y(t), t) = x(t) * y(t)</span>
<div id="mathAns1-184" ></div>
</div>
<div class="math">
<table>
<tr><td>
<math xmlns="&mathml;" mathsize="big" display="block"><mstyle><mrow><mrow><mrow><msubsup><mi>y</mi><mrow><mo></mo></mrow><mrow><mo>′</mo></mrow></msubsup></mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow><mo>=</mo><mrow><mrow><mi>x</mi><mo>(</mo><mi>t</mi><mo>)</mo></mrow><mo></mo><mrow><mi>y</mi><mo>(</mo><mi>t</mi><mo>)</mo></mrow></mrow></mrow></mstyle></math>
</td></tr>
</table>
</div>
<div class="returnType">
Type: Equation Expression Integer
</div>
<p>We can solve the system around <math xmlns="&mathml;" mathsize="big"><mstyle><mrow><mi>t</mi><mo>=</mo><mn>0</mn></mrow></mstyle></math> with the initial
conditions <math xmlns="&mathml;" mathsize="big"><mstyle><mrow><mi>x</mi><mo>(</mo><mn>0</mn><mo>)</mo><mo>=</mo><mn>0</mn></mrow></mstyle></math> and <math xmlns="&mathml;" mathsize="big"><mstyle><mrow><mi>y</mi><mo>(</mo><mn>0</mn><mo>)</mo><mo>=</mo><mn>1</mn></mrow></mstyle></math>. Notice that since
we give the unknowns in the order <math xmlns="&mathml;" mathsize="big"><mstyle><mrow><mo>[</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>]</mo></mrow></mstyle></math>, the answer is a list
of two series in the order
<math xmlns="&mathml;" mathsize="big"><mstyle><mrow><mo>[</mo><mrow><mtext>series for </mtext></mrow><mi>x</mi><mo>(</mo><mi>t</mi><mo>)</mo><mo>,</mo><mrow><mtext>series for </mtext></mrow><mi>y</mi><mo>(</mo><mi>t</mi><mo>)</mo><mo>]</mo></mrow></mstyle></math>.
</p>
<div id="spadComm1-185" class="spadComm" >
<form id="formComm1-185" action="javascript:makeRequest('1-185');" >
<input id="comm1-185" type="text" class="command" style="width: 40em;" value="seriesSolve([eq2, eq1], [x, y], t = 0, [y(0) = 1, x(0) = 0])" />
</form>
<span id="commSav1-185" class="commSav" >seriesSolve([eq2, eq1], [x, y], t = 0, [y(0) = 1, x(0) = 0])</span>
<div id="mathAns1-185" ></div>
</div>
<div class="math">
<table>
<tr><td>
<math xmlns="&mathml;" mathsize="big" display="block"><mstyle><mrow><mo>[</mo><mrow><mo></mo><mi>t</mi><mo>+</mo><mrow><mfrac><mn>1</mn><mn>3</mn></mfrac><mo></mo><mrow><msup><mi>t</mi><mn>3</mn></msup></mrow></mrow><mo>+</mo><mrow><mfrac><mn>2</mn><mn>15</mn></mfrac><mo></mo><mrow><msup><mi>t</mi><mn>5</mn></msup></mrow></mrow><mo>+</mo><mrow><mfrac><mn>17</mn><mn>315</mn></mfrac><mo></mo><mrow><msup><mi>t</mi><mn>7</mn></msup></mrow></mrow><mo>+</mo><mrow><mfrac><mn>62</mn><mn>2835</mn></mfrac><mo></mo><mrow><msup><mi>t</mi><mn>9</mn></msup></mrow></mrow><mo>+</mo><mrow><mi>O</mi><mo>(</mo><mrow><msup><mi>t</mi><mn>11</mn></msup></mrow><mo>)</mo></mrow></mrow><mo>,</mo><mrow><mn>1</mn><mo>+</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mrow><msup><mi>t</mi><mn>2</mn></msup></mrow></mrow><mo>+</mo><mrow><mfrac><mn>5</mn><mn>24</mn></mfrac><mo></mo><mrow><msup><mi>t</mi><mn>4</mn></msup></mrow></mrow><mo>+</mo><mrow><mfrac><mn>61</mn><mn>720</mn></mfrac><mo></mo><mrow><msup><mi>t</mi><mn>6</mn></msup></mrow></mrow><mo>+</mo><mrow><mfrac><mn>277</mn><mn>8064</mn></mfrac><mo></mo><mrow><msup><mi>t</mi><mn>8</mn></msup></mrow></mrow><mo>+</mo><mrow><mfrac><mn>50521</mn><mn>3628800</mn></mfrac><mo></mo><mrow><msup><mi>t</mi><mn>10</mn></msup></mrow></mrow><mo>+</mo><mrow><mi>O</mi><mo>(</mo><mrow><msup><mi>t</mi><mn>11</mn></msup></mrow><mo>)</mo></mrow></mrow><mo>]</mo></mrow></mstyle></math>
</td></tr>
</table>
</div>
<div class="returnType">
Type: List UnivariateTaylorSeries(Expression Integer,t,0)
</div>
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