Re: How solve a system of ODE with periodic coeffs.

• To: mathgroup at smc.vnet.net
• Subject: [mg30721] Re: How solve a system of ODE with periodic coeffs.
• From: "W. H. Harker" <wharke at sprynet.com>
• Date: Sat, 8 Sep 2001 02:56:00 -0400 (EDT)
• References: <9nal83\$np0\$1@smc.vnet.net>
• Sender: owner-wri-mathgroup at wolfram.com

```In article <9nal83\$np0\$1 at smc.vnet.net>, guillerm at aida.usal.es says...
> I would like solved System of ODE with periodic  coeffs. as this:
>
> DSolve[{Derivative[1][x1][t] == Cos[t]*x1[t] + 1.5*x2[t],
>    Derivative[1][x2][t] == 2.1*x1[t] + 2.3*x2[t], x1[0] == 1, x2[0] ==
> 0}, {x1[t], x2[t]}, t]
>
> I wish obtain the analitycal solution. I have tested with DSolve but
> doesn`t work (Even using Rationalize, giving numeric values of initial
> conditions,i.e.: a =1).

This is Hill's equation (it's close to a Mathieu equation but a little
more complicated). It can have exponentially growing oscillatory
solutions, quasiperiodic solutions and, a small subset, periodic
solutions. Unless you are extremely lucky there are no closed form
solutions to this ODE. Use NDSolve.

Reference: "Hill's Equation"
W. Magnus & S. Winkler
Interscience 1966

Hope this is of some help

Wes Harker

```

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