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MathGroup Archive 2004

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Re: "Rubbish" integration outputs

  • To: mathgroup at smc.vnet.net
  • Subject: [mg45749] Re: "Rubbish" integration outputs
  • From: Roland Franzius <roland.franzius at uos.de>
  • Date: Fri, 23 Jan 2004 03:15:33 -0500 (EST)
  • Organization: Universitaet Hannover
  • References: <bulomk$8ir$1@smc.vnet.net>
  • Sender: owner-wri-mathgroup at wolfram.com

Iwonder wrote:
> Hi,
> 
> I always find me in troubles when integrating ODE that I think Mathematica
> should be able to do. It's all right for the very simplest, but consider for
> instance the following:
> 
> \!\(DSolve[\((x + 2  x\^2 + x\^3)\) \(y'\)[x] + \(x\^2\) \(y''\)[x] == A, y,
>     x]\)
> 
> 'A' a constant. Well, the problem is that I don't even the clue that
> Mathematica *cannot* integrate it, there is some output with gibberish, like
> K$14646 and the like (those are intermediary variables I guess). So what is
> wrong? Is Mathematica expected to return such outputs?
> 
> Cheers.

Use the general formula youself
with y' = z, P=x^2/(x+2x^2+x^3), Q=-A/(x+2x^2+x^3)

z' + P(x) z + Q(x)==0
is solved by

z = exp(int^x P(s) ds) ( int^x  ( exp(- int^r P(s) ds)) Q(r)  dr )

as one verifies by elementary calculus.

F[x_] := Integrate[P[s1], {s1, a, x}]
z[x_] := Exp[-F[x]]Integrate[Exp[F[s2]] Q[s2], {s2, a, x}]
z'[x] + P[x]z[x] // FullSimplify


You can not expect Mathematica or anyone else to integrate  exponentials 
of a rational functions two times. So expect something ugly and be 
pleased if the formal expression is printable. In the best case some 
special functions may do the job or the series expansion may be simple. 
NIntegrate gives you every number.

-- 

Roland Franzius


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