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

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  <hr/>
  <div align="center">
   Closed-Form Solutions of Non-Linear Differential Equations
  </div>
  <hr/>
This is an example that shows how to solve a non-linear first order 
ordinary differential equation manually when an integrating factor can
be found just by integration. At the end, we show you how to solve it
directly.

Let's solve the differential equation
<pre>
  y' = y/(x + y log y)
</pre>
Using the notation
<pre>
  m(x,y)+n(x,y)y' = 0
</pre>
we have m=-y and n=x+y*log y
<ul>
 <li>
  <input type="submit" id="p1" class="subbut" 
    onclick="makeRequest('p1');"
    value="m:=-y" />
  <div id="ansp1"><div></div></div>
 </li>
</ul>
<ul>
 <li>
  <input type="submit" id="p2" class="subbut" 
    onclick="handleFree(['p1','p2']);"
    value="n:=x+y*log y" />
  <div id="ansp2"><div></div></div>
 </li>
</ul>
We first check for exactness, that is, does dm/dy=dn/dx?
<ul>
 <li>
  <input type="submit" id="p3" class="subbut" 
    onclick="handleFree(['p1','p2','p3']);"
    value="D(m,y)-D(n,x)" />
  <div id="ansp3"><div></div></div>
 </li>
</ul>
This is not zero, so the equation is not exact. Therefore we must look
for an integrating factor, that is, a function mu(x,y) such that 
d(mu m)/dy=d(mu n)/dx. Normally, we first search for mu(x,y) depending only
on x or only on y. Let's search for such a mu(x) first.
<ul>
 <li>
  <input type="submit" id="p4" class="subbut" 
    onclick="makeRequest('p4');"
    value="mu:=operator 'mu" />
  <div id="ansp4"><div></div></div>
 </li>
 <li>
  <input type="submit" id="p5" class="subbut" 
    onclick="handleFree(['p1','p2','p4','p5']);"
    value="a:=D(mu(x)*m,y)-D(mu(x)*n,x)" />
  <div id="ansp5"><div></div></div>
 </li>
</ul>
If the above is zero for a function mu that does not depend on y, then
mu(x) is an integrating factor.
<ul>
 <li>
  <input type="submit" id="p6" class="subbut" 
    onclick="handleFree(['p1','p2','p4','p5','p6']);"
    value="solve(a=0,mu,x)" />
  <div id="ansp6"><div></div></div>
 </li>
</ul>
The solution depends on y, so there is no integrating factor that depends
on x only. Let's look for on that depends on y only.
<ul>
 <li>
  <input type="submit" id="p7" class="subbut" 
    onclick="handleFree(['p1','p2','p4','p7']);"
    value="b:=D(mu(y)*m,y)-D(mu(y)*n,x)" />
  <div id="ansp7"><div></div></div>
 </li>
</ul>
<ul>
 <li>
  <input type="submit" id="p8" class="subbut" 
    onclick="handleFree(['p1','p2','p4','p7','p8']);"
    value="sb:=solve(b=0,mu,y)" />
  <div id="ansp8"><div></div></div>
 </li>
</ul>
We've found one. The above mu(y) is an integrating factor. We must multiply
our initial equation (that is, m and n) by the integrating factor.
<ul>
 <li>
  <input type="submit" id="p9" class="subbut" 
    onclick="handleFree(['p1','p2','p4','p7','p8','p9']);"
    value="intFactor:=sb.basis.1" />
  <div id="ansp9"><div></div></div>
 </li>
 <li>
  <input type="submit" id="p10" class="subbut" 
    onclick="handleFree(['p1','p2','p4','p7','p8','p9','p10']);"
    value="m:=intFactor*m" />
  <div id="ansp10"><div></div></div>
 </li>
</ul>
<ul>
 <li>
  <input type="submit" id="p11" class="subbut" 
    onclick="handleFree(['p1','p2','p4','p7','p8','p9','p11']);"
    value="n:=intFactor*n" />
  <div id="ansp11"><div></div></div>
 </li>
</ul>
Let's check for exactness.
<ul>
 <li>
  <input type="submit" id="p12" class="subbut" 
    onclick="handleFree(['p1','p2','p4','p7','p8','p9','p10','p11','p12']);"
    value="D(m,y)-D(n,x)" />
  <div id="ansp12"><div></div></div>
 </li>
</ul>
We must solve the exact equation, that is, find a function s(x,y) such that
ds/dx=m and ds/dy=n. We start by writing 
<pre>
  s(x,y) = h(y) + integrate(m,x)
</pre>
where h(y) is an unknown function of y. This guarantees that ds/dx=m.
<ul>
 <li>
  <input type="submit" id="p13" class="subbut" 
    onclick="makeRequest('p13');"
    value="h:=operator 'h" />
  <div id="ansp13"><div></div></div>
 </li>
 <li>
  <input type="submit" id="p14" class="subbut" 
    onclick="handleFree(['p1','p2','p4','p7','p8','p9','p10','p13','p14']);"
    value="sol:=h y+integrate(m,x)" />
  <div id="ansp14"><div></div></div>
 </li>
</ul>
All we want is to find h(y) such that ds/dy=n.
<ul>
 <li>
  <input type="submit" id="p15" class="subbut" 
    onclick=
     "handleFree(['p1','p2','p4','p7','p8','p9','p10','p13','p14','p15']);"
    value="dsol:=D(sol,y)" />
  <div id="ansp15"><div></div></div>
 </li>
 <li>
  <input type="submit" id="p16" class="subbut" 
    onclick=
   "handleFree(['p1','p2','p4','p7','p8','p9','p10','p13','p14','p15','p16']);"
    value="nsol:=solve(dsol=n,h,y)" />
  <div id="ansp16"><div></div></div>
 </li>
</ul>
The above particular solution is the h(y) we want, so we just replace h(y)
by it in the implicit solution.
<ul>
 <li>
  <input type="submit" id="p17" class="subbut" 
    onclick=
"handleFree(['p1','p2','p4','p7','p8','p9','p10','p13','p14','p15','p16','p17']);"
    value="eval(sol,h y=nsol.particular)" />
  <div id="ansp17"><div></div></div>
 </li>
</ul>
A first integral of the initial equation is obtained by setting this result
equal to an arbitrary constant.

Now that we've seen how to solve the equation "by hand" we show you how to 
do it with the <a href="dbopsolve.xhtml">solve</a> operation. First define
y to be an operator.
<ul>
 <li>
  <input type="submit" id="p18" class="subbut" 
    onclick="makeRequest('p18');"
    value="y:=operator 'y" />
  <div id="ansp18"><div></div></div>
 </li>
</ul>
Next we create the differential equation.
<ul>
 <li>
  <input type="submit" id="p19" class="subbut" 
    onclick="handleFree(['p18','p19']);"
    value="deq:=D(y x,x)=y(x)/(x+y(x)*log y x)" />
  <div id="ansp19"><div></div></div>
 </li>
</ul>
Finally, we solve it.
<ul>
 <li>
  <input type="submit" id="p20" class="subbut" 
    onclick="handleFree(['p18','p19','p20']);"
    value="solve(deq,y,x)" />
  <div id="ansp20"><div></div></div>
 </li>
</ul>
 </body>
</html>
axiom-hypertex-data 20120501-8 / usr / share / doc / axiom-doc / hypertex / equdifferentialnonlinear.xhtml
