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MathGroup Archive 2004

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Re: simplifying first-order diff eq solution

  • To: mathgroup at smc.vnet.net
  • Subject: [mg46134] Re: simplifying first-order diff eq solution
  • From: ghrabovsky at tds.net (George Hrabovsky)
  • Date: Sat, 7 Feb 2004 04:02:27 -0500 (EST)
  • References: <bvq7gc$5ma$1@smc.vnet.net> <bvt1lo$oln$1@smc.vnet.net>
  • Sender: owner-wri-mathgroup at wolfram.com

I go this result with:

DSolve[{G'[t] == -( s + t) G[t] + 
        1 + G[t]^2, G[0] == 0}, G[t], t] // FullSimplify

Out[203]=
                                 2
                      1/2 (s + t)
{{G[t] -> s + t + (2 E             s)/
 
            2
           s /2
      (-2 E     + Sqrt[2 Pi] s 
 
                  s             s + t
         (Erfi[-------] - Erfi[-------]))}}
               Sqrt[2]         Sqrt[2]


bobhanlon at aol.com (Bob Hanlon) wrote in message news:<bvt1lo$oln$1 at smc.vnet.net>...
> $Version
> 
> 4.2 for Mac OS X (August 22, 2002)
> 
> sol=FullSimplify[
>     DSolve[{G'[t]==-(s+t) G[t]+1+G[t]^2,G[0]==0},G[t],t][[1]]]
> 
> {G[t] -> (2*E^((1/2)*(s + t)^2)*s)/
>      (Sqrt[2*Pi]*s*(Erfi[s/Sqrt[2]] - 
>         Erfi[(s + t)/Sqrt[2]]) - 2*E^(s^2/2)) + s + t}
> 
> However, Version 5.0.1 for Mac OS X fails to solve this.
> 
> 
> Bob Hanlon
> 
> In article <bvq7gc$5ma$1 at smc.vnet.net>, "J.S." <childrenoflessergod at yahoo.com>
> wrote:
> 
> << I want to solve a first-order simple non-linear differential
> equation. Incidentally, I even know the solution. Now try to solve this
> using Mathematica:
> 
> DSolve[{G'[t] == -( s + t) G[t] + 1 + G[t]^2, G[0] == 0}, G[t], t]
> 
> You will get a horrible series of Erfi[], while the answer is simply
> 
> s+t - s Exp[t^2 + st]/(1+s Int_{0}^{t} {dt' Exp[t'^2 / 2 + s t']})
> 
> I am sure Mathematica is intelligent enough to reduce the results to
> this simple form, but how do I do it? For example, why does Mathematica
> try to express the answer in Erfi[] (instead of erf[]), using complex
> variables? Can I tell Mathematica that all my variables are real
> numbers?


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