Re: 2 coupled diff. eqns

• To: mathgroup at smc.vnet.net
• Subject: [mg21169] Re: 2 coupled diff. eqns
• From: Pasquale Nardone <Pasquale.Nardone at ulb.ac.be>
• Date: Fri, 17 Dec 1999 01:22:50 -0500 (EST)
• Organization: Brussels Free Universities VUB/ULB
• References: <831vap\$g58@smc.vnet.net>
• Sender: owner-wri-mathgroup at wolfram.com

```just define

z'=q-c*(z/|z|)
where
q=a+I*b is a complex constant
Now you go to a polar representation
z=rho*Exp[I*theta]
you also define
q=mu*Exp[I*phi] (mu=Sqrt[a^2+b^2], and Tan[phi]=b/a, as usual)
rho'+I*rho*theta'=mu*Exp[I*(phi-theta)]-c
which you decompose in real and imaginary part leading to

rho'=mu*Cos[(phi-theta)]-c
rho*theta'=mu*Sin[(phi-theta)]

you go then to rho versus theta (instead of rho(t) and
theta(t)->rho(theta))
and the eq is:
(1/rho)*(drho/dtheta)=Cotan[phi-theta]-(c/mu)*(1/Sin[phi-theta])
which is integrable:
Log[rho]=-Log[Sin[phi-theta]]+(c/mu)*Log[Tan[(phi-theta)/2]]+Constant

now you put rho[theta] in rho*theta'=mu*Sin[(phi-theta)] to find
the dependance of "t" versus theta to finish the problem
--
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Pasquale Nardone                          *
Universite Libre de Bruxelles             *
CP 231, Sciences-Physique-Agrigation      *
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1050 Bruxelles, Belgium                   *
tel: 650,55,15 fax: 650,57,67 (+32,2)     *
http://homepages.ulb.ac.be/~pnardon/       *
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```

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