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MathGroup Archive 1999

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DSolve algorithm

  • To: mathgroup at smc.vnet.net
  • Subject: [mg18987] DSolve algorithm
  • From: "Atul Sharma" <atulksharma at yahoo.com>
  • Date: Tue, 3 Aug 1999 13:44:34 -0400
  • Sender: owner-wri-mathgroup at wolfram.com

I most certainly do not want to come across as a Mathematica basher. The
truth is that I used Mathematica as a student and continue to use it despite
annoying pressure from Information Services to join the site license program
which they have with another program. Having said that, I am a bit perplexed by the
difference noted below and wonder if DSolve can be augmented to allow it to
deal with the problem:

I have a system of differential equations representing a multicompartment
pharmacokinetic model of dialysis and impaired kidney function. By tedious
algebra, I can approximate this system with a second order differential
equation with time varying coefficients, describing what happens in the
blood compartment. If I make some inspired variable substitutions, I can get
the equation in the following reduced form:

-a y[x] -qf (-g +x) y'[x] +qf^2 x vi y''[x] ==gu

In this form, Mathematica can generate a solution in terms of Kummer
functions. No solution is forthcoming with either the original variables,
the original system, or earlier steps in the road to the Kummer differential
equation. This itself doesn't bother me, though I do have a number of
variants of this basic problem that I also would like to solve. What
provoked this question was what happened when I asked the other system dsolve
command to solve the same problem. The reduced equation above returns a
solution in terms of Whittaker functions equivalent to Mathematica's
solution. However, I had never bothered to try the unreduced equation and
only realized yesterday that another system gives me a solution with the unreduced
equation and even with the original system of equations.

I obviously realize that generalizing from n=1 is  dangerous at the best of
times, and this certainly isn't enough to make me want to switch from a
platfrom with which I'm very comfortable. However, I do have a number of
variants of this system that I'm working with and need to solve, and I
wonder if Mathematica DSolve can be made smarter or if v 4.0 has a 'new, improved'
DSolve functionality. I also confess to some curiousity as to why this would
be the case.

Any comments would be appreciated, as I would like to understand this
difference.

Atul

--------------------------------------------

Atul Sharma MD, FRCP(C)
Pediatric Nephrologist,
McGill University/Montreal Children's Hospital

email: mdsa at musica.mcgill.ca




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