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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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