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wes
Posts:
24
Registered:
12/13/04


Re: Re: nonlinear differential equation
Posted:
Feb 28, 2005 3:55 AM


In article <cvrqae$p3s$1@smc.vnet.net>, drbob@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 tm]
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, SpringerVerlag 1954.



