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Topic: nonlinear differential equation
Replies: 8   Last Post: Mar 1, 2005 2:14 AM

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Posts: 24
Registered: 12/13/04
Re: Re: nonlinear differential equation
Posted: Feb 28, 2005 3:55 AM
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In article <cvrqae$p3s$>, 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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