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Re: Re: nonlinear differential equation

• To: mathgroup at smc.vnet.net
• Subject: [mg54699] Re: [mg54647] Re: nonlinear differential equation
• From: DrBob <drbob at bigfoot.com>
• Date: Sun, 27 Feb 2005 01:29:34 -0500 (EST)
• References: <cvhequ$qft$1@smc.vnet.net> <200502250618.BAA02402@smc.vnet.net>
• Reply-to: drbob at bigfoot.com
• Sender: owner-wri-mathgroup at wolfram.com

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 -
            48*a*c])])/Sqrt[#1]],
          -((3*b + Sqrt[9*b^2 -
            48*a*c])/(-3*b +
            Sqrt[9*b^2 - 48*a*
            c]))]*Sqrt[
          1 - (12*c)/((-3*b +
            Sqrt[9*b^2 - 48*a*c])*
            #1)]*Sqrt[
          1 + (12*c)/((3*b +
            Sqrt[9*b^2 - 48*a*c])*
            #1)]*#1)/(Sqrt[3]*
         Sqrt[c/(3*b + Sqrt[
            9*b^2 - 48*a*c])]*
         Sqrt[6*c + 3*b*#1 +
           2*a*#1^2]) & ][
     -(t/Sqrt[3]) + C[1]]],
   Function[{t}, InverseFunction[
      (I*EllipticF[I*ArcSinh[
            (2*Sqrt[3]*Sqrt[
            c/(3*b + Sqrt[9*b^2 -
            48*a*c])])/Sqrt[#1]],
          -((3*b + Sqrt[9*b^2 -
            48*a*c])/(-3*b +
            Sqrt[9*b^2 - 48*a*
            c]))]*Sqrt[
          1 - (12*c)/((-3*b +
            Sqrt[9*b^2 - 48*a*c])*
            #1)]*Sqrt[
          1 + (12*c)/((3*b +
            Sqrt[9*b^2 - 48*a*c])*
            #1)]*#1)/(Sqrt[3]*
         Sqrt[c/(3*b + Sqrt[
            9*b^2 - 48*a*c])]*
         Sqrt[6*c + 3*b*#1 +
           2*a*#1^2]) & ][
     t/Sqrt[3] + C[1]]]}

Bobby

On Fri, 25 Feb 2005 01:18:45 -0500 (EST), Jens-Peer Kuska <kuska at informatik.uni-leipzig.de> wrote:

> Hi,
>
> deqn = s''[t] - a*s[t]^2 - b*s[t] - c == 0;
>
> ddeqn=Integrate[#, t] & /@ Expand[s'[t]*#] & /@ deqn
>
> gives you a nonlinear first order equation and DSolve[] can express the
>
> solution in InverseFunction[] of elliptic integrals.
>
> Regards
>
>   Jens
>
> "Umby" <umprisco at unina.it> schrieb im Newsbeitrag
> news:cvhequ$qft$1 at smc.vnet.net...
>> hi group,
>>
>> could anyone help me in solving the following nonlinear differential
>> equation:
>> s''[t] - a1s[t]^2 - b1 s[t] - c1 = 0
>> s[0] = 0, s'[0] = v0
>>
>> is it possible to solve it?
>>
>> thanks
>> -u
>>
>>
>
>
>
>
>



-- 
DrBob at bigfoot.com
www.eclecticdreams.net

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