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Re: Solving a nasty rational differential equation
*To*: mathgroup at smc.vnet.net
*Subject*: [mg74633] Re: Solving a nasty rational differential equation
*From*: Roger Bagula <rlbagula at sbcglobal.net>
*Date*: Thu, 29 Mar 2007 02:30:03 -0500 (EST)
*References*: <eud32p$k50$1@smc.vnet.net>
Roger Bagula wrote:
> I have this nasty differential equation: ( Lorentz invariant elliptical
> invariant Klein-Gordon)
> -(hbar2/(2*m))*Sum[D[Phi[x[i]],{x(i),2}],{i,1,4}]+2*m*c2*J{Sqrt[m/m0]]=0
> The substition I'm working with is the Kerr mass one of:
> m/m0->(r/r0)^2
> and J as an elliptical invariant like:
> J[x_]=(-x20+228*x15-494*x10-228*x5-1)3/(1728*x5*(x10+11*x5-1)5)
> I isolate the radial part of the second differential ( in a polar four
> sopace the angular part isn't importyant mostly to mass radial solutions)
> The Phi part is straight forfowd so I'm left with a double interagation
> of a nasty rational function: ( m^2->r^4)
> f[x_]=(-x20+228*x15-494*x10-228*x5-1)3/(1728*x*(x10+11*x5-1)5)
> or if
> hbar^2/(m^2*c^2)-->r^2
> g[x_]=(-x20+228*x15-494*x10-228*x5-1)3/(1728*x^7 *(x10+11*x5-1)5)
> What I tried after the Integration wouldn't stop in my Mathematica
> was doing a term wise integration ( 64 terms, every 5th one non-zero).
>
>
For some reason my software strips out the "^" powers characters in the
post.
I'll try again : ( various versions of this elliptic invariant are well
known in Mathematics)
The Point group dodecahedron/ icosahedron symmetry is a dual
on exchange of vertices with sides in a Euler topology.
J[x_]=(-x^20+228*x^15-494*x^10-228*x^5-1)^3/(1728*x^5*(x^10+11*x^5-1)^5)
f[x_]=x^4*J[x]
g[x]=x^2*J[x]
Using an r0=1 relative radius I get an answer.
I just tried it again with r0=1 and I did get an answer ( for r0 as a
variable it won't solve),
but that won't solve for a radius.
The Log[r] and r Log[r] seem to prevent solution in Mathematica.
It is necessary to get relative r radial levels as energy levels of the
particle states.
I attached my working file. The maximum power seems to be x^22 as with the
upper polynomial integration ( x11 and Sqrt[x] relatively).
The model involves the E8 dodecahedron/ icosahedron symmetry of the
early universe.
r0=1 assumes a scale of one relative to some singularity radius in the
Unification field
at alpha =1. The symmetry breaking to produce charge and gravity is
E8-> D4*A4 ( A4 is SU(5) like or standard model like : not su(5) as the
wave function that comes out is Exp[solution] ;
this symmetry breaking would seem to cause the inflationary era.
I have already integrated the D4 elliptic Invariant Klein -Gordon.
E8->22 energy levels (?) ( unified field like)
D4->10 enegry levels (Gravity like?)
A4-> 12 energy levels (?) ( electromagnetic like)
It is a model for pre -standard model symmetry breaking in the early
era big bang.
It agrees with a Dynkin diagram / graph breaking of the group/ lie
algebras.
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