# solve the equation $c_1+c_2 e^{c_3 x}+e^{c_4 x}-e^{c_5 x}=0$

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

• Do you know the values of each $c_i$? – 1110101001 Sep 25 '14 at 5:18
• 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? – Paul Sundheim Sep 25 '14 at 5:26
• $x=0, c_2=-c_1$ is an obvious solution. – UserX Sep 25 '14 at 5:43
• 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. – Winther Sep 25 '14 at 5:43
• Otherwise, Newton method. If you want, provide a set of $c_i$'s. – Claude Leibovici Sep 25 '14 at 5:51