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MathGroup Archive 2006

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Re: Why doesn't Mathematica solve this simple differential equation?

  • To: mathgroup at smc.vnet.net
  • Subject: [mg69254] Re: Why doesn't Mathematica solve this simple differential equation?
  • From: Peter Pein <petsie at dordos.net>
  • Date: Sun, 3 Sep 2006 23:46:58 -0400 (EDT)
  • References: <eddqq8$3vq$1@smc.vnet.net>
  • Sender: owner-wri-mathgroup at wolfram.com

Joseph Gwinn schrieb:
> Here is the system I'm trying to solve.  It's an electrical circuit 
> consisting of a capacitor C1 (with initial voltage 4.0 volts), a 
> resistor R1, and a diode in series.  
> 
> 
> Approach 1:
> 
> eqns11 = {Q1'[t] == -Iloop[t], Q1[t] == C1*Vc[t],     Vr[t] == 
> R1*Is*Exp[Vd[t]/0.026], Vc[t] == Vr[t] + Vd[t], Vc[0] == 4.0}
> 
> eqns12 = eqns11 /. {C1 -> 1.0*10^-6, R1 -> 16, Is -> 10^-13}
> 
> eqns12soln = NDSolve[eqns12, Q1, {t, 0, 1}]
> 

You want to solve for Q1. Unless there is an dependency in a previous 
definition of Iloop between Q1 and the voltages, the relevant equations 
remaining are:
Q1'[t] == -Iloop[t] and Q1[t] == C1*Vc[t]
with unknown(?) Iloop. Obviously this can not be solved.

> 
> Approach 2:
> 
> eqns21 = {Vc'[t] == -Id[t]/C1, Vc[t] == 0.026*Log[Id[t]/Is] + R1*Id[t],     
> Vc[0] == 4.0}
> 
> eqns22 = eqns11 /. {C1 -> 1.0*10^-6, R1 -> 16, Is -> 10^-13}
> 
> eqns22soln = NDSolve[eqns22, Vc, {t, 0, 1}]

The same happens here more functions (all undefined?) than equations.
> 
> 
> Both approaches fail with Mathematica complaining that "NDSolve::ndode: 
> Input is not an ordinary differential equation".
> 
> Another, simpler, problem (same circuit but without the R1) solves 
> happily, so long as I eliminate all intermediate variables manually.
> 
> eqns1 = {Vd'[t] == -Is*Exp[Vd[t]/0.026]/C, Vd[0] == 4.0}
> 
> eqns2 = eqns1 /. {C -> 1.0*10^-6, Is -> 10^-13}
> 
> eqns2soln = NDSolve[eqns2, Vd, {t, 0, 1}]
> 

One function, one equation - Mathematica is happy

> 
> Any ideas?
> 
> Joe Gwinn
> 

consider this example, where elimination is trivial:

In[4]:=
DSolve[{f'[x] == g[x] - f[x], g[x] == Sin[x]}, f[x], x]
 From In[4]:=
"DSolve::deqx: Supplied equations are not differential equations of the 
given functions."
Out[4]=
DSolve[{f'[x] == g[x] - f[x], g[x] == Sin[x]}, f[x], x]

and

In[5]:=
DSolve[
   Eliminate[{f'[x] == g[x] - f[x], g[x] == Sin[x]}, g[x]],
  f, x]

solves the deq.

HTH
Peter


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