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Re: 2 coupled diff. eqns

  • To: mathgroup at smc.vnet.net
  • Subject: [mg21175] Re: 2 coupled diff. eqns
  • From: Jens-Peer Kuska <kuska at informatik.uni-leipzig.de>
  • Date: Fri, 17 Dec 1999 01:22:59 -0500 (EST)
  • Organization: Universitaet Leipzig
  • References: <831vap$g58@smc.vnet.net>
  • Sender: owner-wri-mathgroup at wolfram.com

Hi Henk,


the system appears to be not solvable. You can try it in polar
coordinates with

deqn = {D[f[t], t] == a - (w f[t])/(Sqrt[f[t]^2 + g[t]^2]), 
    D[g[t], t] == b - (w g[t])/(Sqrt[f[t]^2 + g[t]^2])}


deqn1 = FullSimplify[
    Solve[Simplify[
        deqn /. Flatten[{#, D[#, t]} & /@ {f[t] -> r[t]*Sin[phi[t]], 
                g[t] -> r[t]*Cos[phi[t]]}], 
        Element[r[t], Reals] && r[t] >= 0], {r'[t], phi'[t]}]]

and

DSolve[Flatten[deqn1 /. Rule -> Equal] , {r[t], phi[t]}, t]

But the best seems to make some assuptions on r[t] and phi[t] to get
some limit cases.

Hope that helps
  Jens

Henk Jansen wrote:
> 
> I have the following set of two coupled differential equations:
> 
>                     c f
>    f' = a -  ----------------
>                  ___________
>              \  /  2    2
>               \/  f  + g
> 
>                     c g
>    g' = b -  ----------------
>                  ___________
>              \  /  2    2
>               \/  f  + g
> 
> where a, b and c are constants. Trying to solve this system (if
> possible), after typing
> 
> DSolve[
>     {D[f[t], t] == a - (w f[t])/(Sqrt[f[t]^2 + g[t]^2])\),
>      D[g[t], t] == b - (w g[t])/(Sqrt[f[t]^2 + g[t]^2])\)
>     },
>     {f[t], g[t]},
>     t]
> 
> Mathematica returns with the following message:
> 
>    "Part::partw: Part 2 of g'[f] does not exist."
> 
> without solution. I have two questions:
> 
> 1. Does anyone know how to interprete this message?
> 
> 2. If the system is not solvable, is there a clever coordinate
> transformation for which the system can be solved?
> 
> Thanks,
> 
> Henk Jansen


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