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First time in this forum, so greetings to everyone!

I am currently working with some physical models in the field of natural ventilation and I came across the following 5th order polynomial equation (quintic function):

This is the steady state solution of a physical system of coupled ODEs. And I am interested in finding a explicit relationship for X = f(), (the roots in terms of C), possibly analytically. The values of X (that makes physical sense) needs to be real and between 0 and 1. Additionally, C is a constant (a time scale of my system), which need to be larger or equal than 0 (and to infinity).

I can plot this equation numerically

If anyone could give me some insight about this equations or if it is possible an explicit solution for the root with the constrains C>=0 and X[0 1], I would be extremely grateful!

Thanks in advance,

Daniel

EDIT: improved explanation of one line.

I am currently working with some physical models in the field of natural ventilation and I came across the following 5th order polynomial equation (quintic function):

**$X^{5}+ C X - C =0$**This is the steady state solution of a physical system of coupled ODEs. And I am interested in finding a explicit relationship for X = f(), (the roots in terms of C), possibly analytically. The values of X (that makes physical sense) needs to be real and between 0 and 1. Additionally, C is a constant (a time scale of my system), which need to be larger or equal than 0 (and to infinity).

I can plot this equation numerically

**,**but I was wondering if I can obtain a explicit/analytical solution of this equation for**C>=0 and X[0 1]**. I have been reading a little bit about quintic functions, and how some of them can be solved (this one seems to be in a similar way as Quintics in Bring–Jerrard form).If anyone could give me some insight about this equations or if it is possible an explicit solution for the root with the constrains C>=0 and X[0 1], I would be extremely grateful!

Thanks in advance,

Daniel

EDIT: improved explanation of one line.

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