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

• To: mathgroup at smc.vnet.net
• Subject: [mg24742] Re: Mathematica won't solve simple diff. eqn. system
• From: Jens-Peer Kuska <kuska at informatik.uni-leipzig.de>
• Date: Wed, 9 Aug 2000 02:31:47 -0400 (EDT)
• Organization: Universitaet Leipzig
• References: <8mdlt6\$5jt@smc.vnet.net>
• Sender: owner-wri-mathgroup at wolfram.com

```"Christopher R. Carlen" 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
>
Hi,

to solve the system Mathematica must be able to bring the system
into an explict form like

y1'[t]==someExpr1
y2'[t]==someExpr2

That means to solve your system for y1'[] and y2'[].
But the equations you supply  can't solved in this way
because

y1[t]+y2[t]->z[t]

gives

z'[t]+C1 y1[t]+C2 y2[t]==0
z'[t]+C3 y1[t]+C4 y2[t]==0

so, either C1==C3 && C2==C4 or the system has no solution.

Can you you supply the correct equations (without C1,C2,C3,C4)
that solve you equations ? Because

In[]:=deqn = {y1'[t] + y2'[t] + C1 y1[t] + C2 y2[t] == 0,
y1'[t] + y2'[t] + C3 y1[t] + C4 y2[t] == 0};

In[]:= sol = {y1[t] -> 4 + 64 E^(-5 t) - 68 E^(-4 t),
y2[t] -> 1 - 52 E^(-5 t) + 51 E^(-4 t)};

In[]:= deqn /. Flatten[{#, D[#, t]} &  /@ sol] // FullSimplify
Out[]=

{4*C1 + C2 + (4*(-15 + 16*C1 - 13*C2))/E^(5*t) ==
(17*(-4 + 4*C1 - 3*C2))/E^(4*t),
4*C3 + C4 + (4*(-15 + 16*C3 - 13*C4))/E^(5*t) ==
(17*(-4 + 4*C3 - 3*C4))/E^(4*t)}

does not show that this is a solution. It gives a overdetermined
system when I try to solve it for C3 and C4.

Regards
Jens

> 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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