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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: [mg46228] Re: simplifying first-order diff eq solution
  • From: Paul Abbott <paul at physics.uwa.edu.au>
  • Date: Thu, 12 Feb 2004 07:15:54 -0500 (EST)
  • Organization: The University of Western Australia
  • References: <bvq7gc$5ma$1@smc.vnet.net>
  • Sender: owner-wri-mathgroup at wolfram.com

In article <bvq7gc$5ma$1 at smc.vnet.net>,
 "J.S." <childrenoflessergod at yahoo.com> wrote:

> Hi, 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']})

But Mathematica can, and does, compute the definite integral

  Integrate[Exp[u^2/2 + s u], {u, 0, t}]

obtaining

  Sqrt[Pi/2] (Erfi[(s + t)/Sqrt[2]] - Erfi[s/Sqrt[2]]) / E^(s^2/2)

You may find this horrible -- but it is an explicit closed form 
expression for the integral, which is the best you can hope for.

> I am sure Mathematica is intelligent enough to reduce the results to
> this simple form, but how do I do it? 

In general, a closed-form expression in terms of known special functions 
is better than a definited integral.

> For example, why does Mathematica try to express the answer in Erfi[] 
> (instead of erf[]), using complex variables? 

Because Erfi is the simplest way of expressing this result. E.g., try

  Integrate[Exp[u^2/2], {u, 0, t}]

You could express this in terms of Erf using complex variables -- but 
why do you want to. It is not "simpler".

Cheers,
Paul

-- 
Paul Abbott                                   Phone: +61 8 9380 2734
School of Physics, M013                         Fax: +61 8 9380 1014
The University of Western Australia      (CRICOS Provider No 00126G)         
35 Stirling Highway
Crawley WA 6009                      mailto:paul at physics.uwa.edu.au 
AUSTRALIA                            http://physics.uwa.edu.au/~paul


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