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Unsolvable ODE

• To: mathgroup at smc.vnet.net
• Subject: [mg113074] Unsolvable ODE
• From: István Zachar <zac at freemail.hu>
• Date: Tue, 12 Oct 2010 04:30:29 -0400 (EDT)

Dear All,

Consider the following set of ODEs which Mathematica can solve easily:

In[2]:= Solve[{
  0 == f - 2 p m + q a + q b,
  0 == p m - q a - kA a y2,
  0 == p m - q b - kB b (x1 + x2 + y1),
  0 == -kB b x1 + 2 kB b x2,
  0 == kB b x1 - dtemp x2 - kB b x2,
  0 == -kB b y1 + 2 kA a y2,
  0 == kB b y1 - dtemp y2 - kA a y2},
 {m, a, b, x1, x2, y1, y2}]

Out[2]= {{m -> (f kB + dtemp q)/(kB p), x1 -> (2 f)/(3 dtemp),
  x2 -> f/(3 dtemp), y1 -> 0, y2 -> 0, a -> (f kB + dtemp q)/(kB q),
  b -> dtemp/kB}, {m -> (f kA + 3 dtemp q)/(3 kA p), x1 -> 0, x2 -> 0,
   y1 -> -((2 f kA q)/(kB (f kA - 3 dtemp q))), y2 -> f/(3 dtemp),
  a -> dtemp/kA,
  b -> (-f kA + 3 dtemp q)/(3 kA q)}, {m -> (
   f + (dtemp q)/kA + (dtemp q)/kB)/(2 p),
  x1 -> (-f + (3 dtemp q)/kA - (3 dtemp q)/kB)/(3 dtemp),
  x2 -> (-f + (3 dtemp q)/kA - (3 dtemp q)/kB)/(6 dtemp),
  y1 -> (f kA kB + dtemp kA q - dtemp kB q)/(dtemp kA kB),
  y2 -> (f - (dtemp q)/kA + (dtemp q)/kB)/(2 dtemp), a -> dtemp/kA,
  b -> dtemp/kB}}

Now rearrange some terms in the equations, but keep the complexity and
order at the same level. Strangely, Mathematica 7.0.1.0 (and 6.0 as well)
seems to have some problem solving it for all the variables.

In[1]:= Solve[{
  0 == f - 2 m p + a q + b q,
  0 == m p - a q - a kA (x1 + x2 + y2),
  0 == m p - b q - b kB y1,
  0 == -a kA x1 + 2 a kA x2,
  0 == a kA x1 - dtemp x2 - a kA x2,
  0 == -b kB y1 + 2 a kA y2,
  0 == b kB y1 - dtemp y2 - a kA y2
  }, {m, a, b, x1, x2, y1, y2}]

During evaluation of In[1]:= Solve::svars: Equations may not give
solutions for all "solve" variables. >>

Out[1]= {{m -> (f + b q + (dtemp q)/kA)/(2 p),
  x1 -> (f + 3 b q - (3 dtemp q)/kA)/(6 dtemp),
  x2 -> (f + 3 b q - (3 dtemp q)/kA)/(12 dtemp),
  y1 -> (f - b q + (dtemp q)/kA)/(2 b kB),
  y2 -> (f - b q + (dtemp q)/kA)/(4 dtemp), a -> dtemp/kA}}

Note that the solution contains 'b', thus the whole ODE set is not
solved for every variable, just as the error message states. Now I
have a very strong feeling about this ODE set being equivalent to the
first one, and being soluble for all variables. Does anyone have any
idea what could go wrong here? Is it the internal algorithm of Solve?
Can anyone prove analytically that this latter ODE system cannot be
solved for all variables WITHOUT using Mathematica?

Istvan

• Follow-Ups:
  • Re: Unsolvable ODE
    • From: Daniel Lichtblau <danl@wolfram.com>