Re: DSolve
- To: mathgroup at smc.vnet.net
- Subject: [mg18775] Re: DSolve
- From: Paul Abbott <paul at physics.uwa.edu.au>
- Date: Tue, 20 Jul 1999 01:33:31 -0400
- Organization: University of Western Australia
- References: <7mp2bh$l7j@smc.vnet.net>
- Sender: owner-wri-mathgroup at wolfram.com
"N. Shamsundar" wrote:
> Here is a pair of differential equations that Version 3 could not solve
> (actually, it was still trying after 15 minutes on a 400 MHz PII!)
>
> w:=Exp[-a x];
> DSolve[{u'[x]==-u[x]+v[x]+w (1-a)+Cos[Pi x]-(1+Pi) Sin[Pi x],
> v''''[x]==u[x]+v[x]+(Pi^4-1) Sin[Pi x]-Cos[Pi x]-w, u[0]==2,
> v[0]==0,v''[0]==0,v[1]==0,v''[1]==0},
> {u[x],v[x]},x]
>
> These are linear equations with constant coefficients, and the exact solution
> is
> u=w+cos(Pi x), v=sin(Pi x)
>
> I would appreciate someone running this calculation in Version 4.
The problem is not version specific. If you try solving your pair of equations
without the boundary conditions you will get a rather long answer involving Root
objects. If you eliminate u[x] you obtain a 5th order equation in v leading to
a fifth order auxilliary equation with non-trivial roots (and this is where
Mathematica gets stuck).
The following Notebook suggests a general approach for solving this type of
problem.
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"x"], \(\((RootSum[#1\^5 + #1\^4 - #1 -
2 &, \(\[ExponentialE]\^\(\(-DSolve`t\)\ \
#1\)\ #1\^3 - \[ExponentialE]\^\(\(-DSolve`t\)\ #1\)\ #1\^2 + \[ExponentialE]\
\^\(\(-DSolve`t\)\ #1\)\ #1 - \[ExponentialE]\^\(\(-DSolve`t\)\ #1\)\)\/\(5\ \
#1\^3 - #1\^2 + #1 - 1\) &]\ \((\(-\[ExponentialE]\^\(\(-a\)\ DSolve`t\)\)\ a \
+ \[ExponentialE]\^\(\(-a\)\ DSolve`t\) +
cos(\[Pi]\ DSolve`t) - \[Pi]\ \(sin(\[Pi]\ \
DSolve`t)\) - sin(\[Pi]\ DSolve`t))\) +
RootSum[#1\^5 + #1\^4 - #1 -
2 &, \[ExponentialE]\^\(\(-DSolve`t\)\ #1\
\)\/\(5\ #1\^4 + 4\ #1\^3 - 1\) &]\ \((\(-\(cos(\[Pi]\ DSolve`t)\)\) - \
\[ExponentialE]\^\(\(-a\)\ DSolve`t\) + \[Pi]\^4\ \(sin(\[Pi]\ DSolve`t)\) -
sin(\[Pi]\ DSolve`t))\))\) \
\[DifferentialD]DSolve`t\)}], ")"}],
" ", \(RootSum[#1\^5 + #1\^4 - #1 -
2 &, \[ExponentialE]\^\(x\ #1\)\/\(5\ #1\^4 + 4\ \
#1\^3 - 1\) &]\)}]}]}]}], "}"}], "}"}], TraditionalForm]], "Output"]
}, Closed]]
}, Closed]]
}
]
____________________________________________________________________
Paul Abbott Phone: +61-8-9380-2734
Department of Physics Fax: +61-8-9380-1014
The University of Western Australia
Nedlands WA 6907 mailto:paul at physics.uwa.edu.au
AUSTRALIA http://physics.uwa.edu.au/~paul
God IS a weakly left-handed dice player
____________________________________________________________________