I have developed a differential equation for the variation of a star's semi-major axis with respect to its eccentricity.
It is as follows:
dydx=1219y(1+(7324x2)+(3726x4))x(1+(121304x2))
Where y is the semi-major axis and x is the eccentricity.
Where y is the semi-major axis and x is the eccentricity. The 3-D plots of this equation can be found [here](http://www.wolframalpha.com/input/?i=3D+plot++%5Cfrac%7B12%7D%7B19%7D%5Cfrac%7By%5Cleft(1%2B%5Cleft(%5Cfrac%7B73%7D%7B24%7Dx%5E2%5Cright)%2B%5Cleft(%5Cfrac%7B37%7D%7B26%7Dx%5E4%5Cright)%5Cright)%7D%7Bx%5Cleft(1%2B%5Cleft(%5Cfrac%7B121%7D%7B304%7Dx%5E2%5Cright)%5Cright)%7D)
And this is the solution to the above DE [here](http://www.wolframalpha.com/input/?i=solve+y'%3D+%5Cfrac%7B12%7D%7B19%7D%5Cfrac%7By%5Cleft(1%2B%5Cleft(%5Cfrac%7B73%7D%7B24%7Dx%5E2%5Cright)%2B%5Cleft(%5Cfrac%7B37%7D%7B26%7Dx%5E4%5Cright)%5Cright)%7D%7Bx%5Cleft(1%2B%5Cleft(%5Cfrac%7B121%7D%7B304%7Dx%5E2%5Cright)%5Cright)%7D)
The decay time of stars can be found by solving the following integral:
T(a0,e0)=12(c40)19γ∫e00e29/19[1+(121/304)e2]1181/2299(1−e2)3/2de
Where
γ=64G35c5m1m2(m1+m2)
For
e0 close to
1 the equation becomes:
T(a0,e0)≈768425Tfa0(1−e20)7/2
Where
Tf=a404γ
I used Appell's hypergeometric functions to solve integral (1), but is there any way in which I can express the solutions in terms of few special functions with simpler symmetries, so that the analysis becomes easier.There is a well defined symmetry for the above equation from the plot. Hence, is it possible to express this in terms of other special function (which have different symmetries).
EDIT: I was suggested that since the powers in the integrand in equation (1) are very non-trivial, probably the hypergeometric function can't be further simplified. But I fail to understand why this might seem to pose a problem. Can't this D.E. be solved by Lie symmetry methods? Or can this solution's field be treated using Frobenius' theorem and the dimensions of it analysed?