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I would like to know whether there is a closed-form solution in $x$ for the equation:

\begin{align*} &e^{-(\alpha\delta+(\gamma-\phi)\frac{\alpha}{\alpha - 1})x} -\frac{\alpha(r+\delta+\phi-\gamma)}{r+\alpha\delta}e^{-(r+\alpha\delta)x} - \frac{r+\alpha(\gamma-\phi-r)}{r+\alpha\delta} = 0,& \end{align*} with following requirements for parameter values: \begin{align*} 0<\alpha<1\\ 0<\delta<<1\\ -1<<\gamma<0\\ 0<\phi<<1\\ 0<r<<1\\ r>(\gamma+\phi)\frac{\alpha}{\alpha-1}\\ and\quad x>0, \end{align*} where "<<" means much greater (alternatively, one may assume that $\delta < \alpha, |\gamma|<\alpha,\phi<\alpha$ and $r<\alpha$). Here, "closed-form solution" can be understood in very broad sense, meaning that all kind of special functions are allowed (to be precise, analytic expression for the solution would be enough). Thank you for your time.

Marius
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  • This can't be solved exactly unless you have a general method for solving $e^x + ae^{bx} + c = 0$, because you pick your parameters to give any of those values (up to, maybe, sign). And that sounds very unlikely to have a nice expression indeed. Would you be satisfied with a good approximation, e.g. a Taylor series in one of your parameters? – Alex Meiburg Dec 02 '17 at 12:33
  • Thank you Alex! In that case, I would be satisfied if there is (not too complex) good approximation. – Marius Dec 02 '17 at 12:52

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