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MathGroup Archive 1997

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Re: DSolve & Cosh

  • To: mathgroup at smc.vnet.net
  • Subject: [mg6236] Re: [mg6194] DSolve & Cosh
  • From: seanross at worldnet.att.net
  • Date: Fri, 28 Feb 1997 03:22:10 -0500
  • Sender: owner-wri-mathgroup at wolfram.com

Dwarf wrote:
> 
> Hello all,
> 
> I am trying to solve the hanging chain problem using Mathematica
> (Student Ver 2.2 -- I don't have Win 95 or a Mac to be able to use 3.0)
> and getting Mma to solve the resulting ODE is giving me fits!  I know the
> solution is neatly expressed in terms of Cosh, but instead I get some
> monstrous (but equivalent) expression in terms of exponentials.
> 
> Is anyone aware of a function package which will convert exponentials to
> hyperbolic sines & cosines?  Something like ComplexToTrig, perhaps?  I can
> demonstrate thet the two answers are equivalent, but I had hoped for
> something more direct.
> 
> Here is the ODE, for those who care:
> 
> C y'[x]==Sqrt[(lambda+rho g y[x])^2-C^2]
> 
> and the answer (in terms of Cosh)
> 
> Y[x_]:=(C Cosh[rho g x/C+C2]-lambda)/(rho g)
> 
> and the answer I get from Mma (which requires the use of Calculus`DSolve`)
> 
> {{y[x] -> (C^2 + E^((2 g rho (x + C[1]))/C) -2 E^((g rho (x +
> C[1]))/C) lambda)/ (2 E^((g rho (x + C[1]))/C) g rho)}}
> 
> Any help would be appreciated. . .
> 
> +----------------------------+
> |                            |    Good things come in small packages,
> |   Greg Anderson            |    but big things don't, unless they're
> |   dwarf at wam.umd.edu        |    inflatable or require some assembly.
> |   timbwolf at eng.umd.edu     |
> |                            |   -- The Tick
> +----------------------------+

I have had limited success using manual transformation rules like
expr/.{1/Exp[whatever]+Exp[whatever]->2 Cosh[whatever]}.

The results are not totally satisfactory.  Mathematica has some 
fixations on certain expressions.  It thinks that exponentials are 
"simpler" than hyperbolic functions and 1/x is "simpler" than x^-1.  
Somewhere along the line, someone had to decide what was a better 
expression.  There is still no substitute for pencil and paper.  Why not 
take the result mathematica gives you and convert it into hyperbolic 
functions.  The algebra shouldn't be too hard.


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