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Re: Mathematica won't solve simple diff. eqn. system

  • To: mathgroup at smc.vnet.net
  • Subject: [mg24727] Re: Mathematica won't solve simple diff. eqn. system
  • From: changpoohbear at my-deja.com
  • Date: Wed, 9 Aug 2000 02:31:26 -0400 (EDT)
  • References: <8mdlt6$5jt@smc.vnet.net>
  • Sender: owner-wri-mathgroup at wolfram.com


Christopher,

It appears that the general linear system that you are trying to solve
is in the form of

a i1'[t] + b i2'[t] + c i1[t] + d i2[t] ==0
e i1'[t] + f i2'[t] + g i1[t] + h i2[t] ==0
i1[0] == alpha
i2[0] == beta

Since this is an autonomous system, it also can be represented in
matrix form as

A i'[t] = B i[t]

where

A={{a,b},{e,f}}; B=-{{c,d},{g,h}}; i[t]={{i1[t]},{i2[t]}};

*IF* A is invertible, the solution can be given as

i[t] = MatrixExp[Inverse[A].B].i[0]

where

i[0] = {{i1[0]},{i2[0]}}

In your example given below, I don't believe the solutions given are
correct ({{-4,8},{-4,8}} also is not invertible!).  For your specific
example, observe that

Eliminate[{-4 i1'[t] + 8i2'[t] - 25i1[t] + 20 i2[t] ==
      0,-4i1'[t] + 8 i2'[t] - 10 i1[t] + 40 i2[t] == 0},
      {i1'[t],i2'[t]}]

yields

3 i1[t]==-4 i2[t]

Using this fact, together with the fact that i1[0]=i2[0]=0, one should
obtain

i1[t]=i2[t]=0

I don't know why my version of Mathematica 3.0.x cannot realise this,
and just returns its input as its output when using DSolve.  <Shrug>

I hope this helps!

Mike

In article <8mdlt6$5jt at smc.vnet.net>,
  "Christopher R. Carlen" <crcarle at sandia.gov> wrote:
> Mathematica 4.0 and linear constant coefficient differential
equations:
>
> I have the following system:
>
> -4 i1'[t] + 8 i2'[t] - 25 i1[t] + 20 i2[t] == 0
> -4 i1'[t] + 8 i2'[t] - 10 i1[t] + 40 i2[t] == 0
> i1[0]==0
> i2[0]==0
>
> Which when I try to solve with DSolve, it fails.
>
> It seems any system of the form:
>
> y1'[t] + y2'[t] + C1 y1[t] + C2 y2[t] == 0
> y1'[t] + y2'[t] + C3 y1[t] + C4 y2[t] == 0
>
> can't be solved.  If the coefficients on y1' and y2' are not the same
> between the two equations, then it can be solved.
>
> The problem is that there is a solution to the above system, which I
> have verified.  That solution is:
>
> i1[t_] = 4 + 64 E^(-5 t) - 68 E^(-4 t)
> i2[t_] = 1 - 52 E^(-5 t) + 51 E^(-4 t)
>
> So the question is:  If there is a solution (and not a very difficult
> one) why can't Mathematica find it???  Is there some way to coerce
Mathematica to
> produce the equation, in both the symbolic and numerical situations?
>
> These types of systems arise frequently in the study of electronic
> circuits.  Numerical solvers like SPICE solve them without any
> difficulty.  I have struggled with getting Mathematica to solve them
for a long
> time.  Sometimes I force a numerical solution by perturbing the
> coefficients a bit, as long as the error is acceptible.
>
> But I would like to understand better what the hangup is.  I have had
a
> diff. eqns. course, but haven't gone into systems yet.
>
> Thanks.
> --
> _______________________
> Christopher R. Carlen
> Sr. Laser/Optical Tech.
> Sandia National Labs
>
>


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