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.