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How can I solve the equation $$c_1+c_2 e^{c_3 x}+e^{c_4 x}-e^{c_5 x}=0,$$
where $c_1,\ldots,c_5$ are real numbers? I encountered this equation when I was solving a maximization problem.
i can say only that all $c_i$s are real x is positive and i want an analytic solution

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  • $\begingroup$ Do you know the values of each $c_i$? $\endgroup$ – 1110101001 Sep 25 '14 at 5:18
  • $\begingroup$ Is the equation true for all $x$ and we are attempting to find the $c_i$? Or are we assuming we know the c's and are attempting to find x in terms of these constants? $\endgroup$ – Paul Sundheim Sep 25 '14 at 5:26
  • $\begingroup$ $x=0, c_2=-c_1$ is an obvious solution. $\endgroup$ – UserX Sep 25 '14 at 5:43
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    $\begingroup$ Such an equation have no general closed-form solution. However, if all the $c_i$'s are integer multiplies of some number then the equation can be written as a polynomial equation which may have an analytical solution. $\endgroup$ – Winther Sep 25 '14 at 5:43
  • $\begingroup$ Otherwise, Newton method. If you want, provide a set of $c_i$'s. $\endgroup$ – Claude Leibovici Sep 25 '14 at 5:51

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