# Finding simpler symmetries to differential equations

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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:

$$\frac{dy}{dx}=\frac{12}{19}\frac{y\left(1+\left(\frac{73}{24}x^2\right)+\left(\frac{37}{26}x^4\right)\right)}{x\left(1+\left(\frac{121}{304}x^2\right)\right)}$$

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)

The decay time of stars can be found by solving the following integral:
$$T(a_{0},e_{0})=\frac{12(c_{0}^4)}{19\gamma}\int_{0}^{e_0}{\frac{e^{29/19}[1+(121/304)e^2]^{1181/2299}}{(1-e^2)^{3/2}}}de\tag1$$
Where $$\gamma=\frac{64G^3}{5c^5}m_{1}m_{2}(m_{1}+m_{2})$$
For $e_{0}$ close to $1$ the equation becomes:
$$T(a_{0},e_{0})\approx\frac{768}{425}T_{f}a_{0}(1-e_{0}^2)^{7/2}\tag2$$
Where $$T_{f}=\frac{a_{0}^4}{4\gamma}$$

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?

edited Feb 21, 2017

Powers in the integrand in your equation (1) are very non-trivial, so I don't think that hypergeometric function can be further simplified.

The links don't work. Please try the corrected ones independently of their generation before posting the corrections. Use the chain symbol in the editor to hide the URL under the [here].

@Arnold Neumaier I have edited the links. They work now

The links now do work, but content of the second link is still uninformative.

I'm sorry about that I am unable to pin info via Desmos.

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