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MathGroup Archive 2005

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Re: Re: nonlinear differential equation

  • To: mathgroup at smc.vnet.net
  • Subject: [mg54743] Re: Re: nonlinear differential equation
  • From: wes <wesh at acm.org>
  • Date: Mon, 28 Feb 2005 03:28:10 -0500 (EST)
  • References: <cvhequ$qft$1@smc.vnet.net> <200502250618.BAA02402@smc.vnet.net> <cvrqae$p3s$1@smc.vnet.net>
  • Sender: owner-wri-mathgroup at wolfram.com

In article <cvrqae$p3s$1 at smc.vnet.net>, drbob at bigfoot.com says...
> Yikes!!! Good luck inverting the functions involved.
> 
> Off[Solve::verif, Solve::tdep]
> deqn = Derivative[2][s][t] -
>       a*s[t]^2 - b*s[t] - c == 0;
> ddeqn =
>    ((Integrate[#1, t] & ) /@
>       Expand[Derivative[1][s][t]*
>         #1] & ) /@ deqn
> s /. DSolve[{%}, s, t]
> (-c)*s[t] - (1/2)*b*s[t]^2 -
>     (1/3)*a*s[t]^3 +
>     (1/2)*Derivative[1][s][t]^
>       2 == 0
> 
> {Function[{t}, InverseFunction[
>       (I*EllipticF[I*ArcSinh[
>             (2*Sqrt[3]*Sqrt[
>             c/(3*b + Sqrt[9*b^2 -

The solution is formed from the integral wrt s of the reciprical square 
root of (1/3)a s^3 + (1/2)b s^2 + c s + v0^2. The result depends
strongly on the roots of the polynomial in s. If they are real the
result will be an inverse Jacobi elliptic function equal to d t, d a 
constant. Inverting the result is usually something like

	sin[y[s]] = sn[d t|m]

with y[s] not being too complicated. Mathematica doesn't handle this 
type of problem gracefully yet. See "Handbook of Elliptic Integrals For
Engineers And Physicists", Bryd & Friedman, Springer-Verlag 1954.


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