/usr/share/doc/axiom-doc/hypertex/equdifferentialpowerseries.xhtml

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  <div align="center"><img align="middle" src="doctitle.png"/></div>
  <hr/>
  <div align="center">
   Power Series Solutions of Differential Equations
  </div>
  <hr/>
The command to solve differential equations in power series around a
particular initial point with specific initial conditions is called
<a href="dbopseriessolve.xhtml">seriesSolve</a>. It can take a variety of
parameters, so we illustrate its use with some examples.

Since the coefficients of some solutions are quite large, we reset the
default to compute only seven terms.
<ul>
 <li>
  <input type="submit" id="p1" class="noresult" 
    onclick="makeRequest('p1');"
    value=")set streams calculate 7" />
  <div id="ansp1"><div></div></div>
 </li>
</ul>
You can solve a single nonlinear equation of any order. For example, we 
solve
<pre>
  y''' = sin(y'') * exp(y) + cos(x)
</pre>
subject to y(0)=1, y'(0)=0, y''(0)=0

We first tell Axiom that the symbol 'y denotes a new operator.
<ul>
 <li>
  <input type="submit" id="p2" class="subbut" 
    onclick="makeRequest('p2');"
    value="y:=operator 'y" />
  <div id="ansp2"><div></div></div>
 </li>
</ul>
Enter the differential equation using y like any system function.
<ul>
 <li>
  <input type="submit" id="p3" class="subbut" 
    onclick="handleFree(['p1','p2','p3']);"
    value="eq:=D(y(x),x,3)-sin(D(y(x),x,2))*exp(y(x))=cos(x)" />
  <div id="ansp3"><div></div></div>
 </li>
</ul>
Solve it around x=0 with the initial conditions y(0)=1, y'(0)=y''(0)=0.
<ul>
 <li>
  <input type="submit" id="p4" class="subbut" 
    onclick="handleFree(['p1','p2','p3','p4']);"
    value="seriesSolve(eq,y,x=0,[1,0,0])" />
  <div id="ansp4"><div></div></div>
 </li>
</ul>
You can also solve a system of nonlinear first order equations. For 
example, we solve a system that has tan(t) and sec(t) as solutions.

We tell Axiom that x is also an operator.
<ul>
 <li>
  <input type="submit" id="p5" class="subbut" 
    onclick="makeRequest('p5');"
    value="x:=operator 'x" />
  <div id="ansp5"><div></div></div>
 </li>
</ul>
Enter the two equations forming our system.
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 <li> <input type="submit" id="p6" class="subbut" 
    onclick="handleFree(['p5','p6']);"
    value="eq1:=D(x(t),t)=1+x(t)^2" />
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 </li>
 <li>
  <input type="submit" id="p7" class="subbut" 
    onclick="handleFree(['p2','p5','p7']);"
    value="eq2:=D(y(t),t)=x(t)*y(t)" />
  <div id="ansp7"><div></div></div>
 </li>
</ul>
Solve the system around t=0 with the initial conditions x(0)=0 and y(0)=1.
Notice that since we give the unknowns in the order [x,y], the answer is a
list of two series in the order [series for x(t), series for y(t)].
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 <li>
  <input type="submit" id="p8" class="subbut" 
    onclick="handleFree(['p1','p2','p5','p6','p7','p8']);"
    value="seriesSolve([eq2,eq1],[x,y],t=0,[y(0)=1,x(0)=0])" />
  <div id="ansp8"><div></div></div>
 </li>
</ul>
The order in which we give the equations and the initial conditions has no
effect on the order of the solution.
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axiom-hypertex-data 20120501-8 / usr / share / doc / axiom-doc / hypertex / equdifferentialpowerseries.xhtml
