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.

**References**:**Re: nonlinear differential equation***From:*"Jens-Peer Kuska" <kuska@informatik.uni-leipzig.de>