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Re: Finding branches where general solution is possible

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  • Subject: [mg131883] Re: Finding branches where general solution is possible
  • From: Narasimham <mathma18 at gmail.com>
  • Date: Wed, 23 Oct 2013 23:45:14 -0400 (EDT)
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> Am Mittwoch, 16. Oktober 2013 10:46:06 UTC+2 schrieb
> Narasimham:
> > For following function with period 2 Pi in which
> branches is it possible to get a general solution?
> > Narasimham 
> > 
> > DSolve[{si''[th] Tan[si[th]]==(1+si'[th])
> (1+2si'[th]),si[0]==Pi/4 },si,th]
> > 
> > NDSolve[{si''[th] Tan[si[th]]==(1+si'[th])
> (1+2si'[th]),si'[0]==0,si[0]==Pi/4},si,{th,0,6Pi}];
> > 
> > SI[u_]=si[u]/.First[%];Plot[SI[th],{th,0,6Pi}]
> > DSolve::bvnul: For some branches of the general
> solution, the given boundary conditions lead to an
> empty solution.>>
> 
> Version 8 of Mathematica has no difficulty in
> DSolve-ing the equation generally.
> The solution is given in terms of InverseFunction as
> follows:
> 
> In[1]:= eq = 
>  Derivative[2][s][t] == 
> Cot[s[t]]*(1 + Derivative[1][s][t])*(1 +
> + 2*Derivative[1][s][t])
> 
> Out[1]= Derivative[2][s][t] == 
> Cot[s[t]]*(1 + Derivative[1][s][t])*(1 +
> + 2*Derivative[1][s][t])
> 
> In[2]:= DSolve[eq, s[t], t]
> 
> Out[2]= {
>    {s[t] -> 
>    InverseFunction[
>      2*(-((I*E^C[1]*Cos[#1/2]^3*
>               Log[2*I*E^C[1]*Cos[#1] - 
>                 Sqrt[2]*Cos[#1/2]^2*
> Sqrt[(-2 + E^(2*C[1]) -
> -2 + E^(2*C[1]) - E^(2*C[1])*Cos[2*#1])*
>                    Sec[#1/2]^4]]*
>                           
> Sqrt[(-(2 - E^(2*C[1]) +
> - E^(2*C[1]) + E^(2*C[1])*Cos[2*#1]))*
>                 Sec[#1/2]^4]*Sin[#1/2])/
>             Sqrt[(-E^(2*C[1]))*(2 - E^(2*C[1]) + 
> E^(2*C[1])*Cos[2*#1])*Sin[#1]^2]) -
> ])*Sin[#1]^2]) - #1/2) & ][
>     t + C[2]]}, 
>    {s[t] -> 
>    InverseFunction[-2*(-((I*E^C[1]*Cos[#1/2]^3*
>               Log[2*I*E^C[1]*Cos[#1] - 
>                 Sqrt[2]*Cos[#1/2]^2*
> Sqrt[(-2 + E^(2*C[1]) -
> -2 + E^(2*C[1]) - E^(2*C[1])*Cos[2*#1])*
>                    Sec[#1/2]^4]]*
>                           
> Sqrt[(-(2 - E^(2*C[1]) +
> - E^(2*C[1]) + E^(2*C[1])*Cos[2*#1]))*
>                 Sec[#1/2]^4]*Sin[#1/2])/
>             Sqrt[(-E^(2*C[1]))*(2 - E^(2*C[1]) + 
> E^(2*C[1])*Cos[2*#1])*Sin[#1]^2]) +
> ])*Sin[#1]^2]) + #1/2) & ][
>     t + C[2]]}}
> 
> Best regards,
> Wolfgang


Thanks Wolfgang, I am having difficulty in simplifying this Inverse periodic function for a given boundary condition at t = 0 for #1. E.g., can I get a closed form/general expression for s as a function of t, at least for boundary condition s[0] == Pi/2 ? ... so as to be able to plot the numerical case given from that expression ?

Regards
Narasimham



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