I am stuck in trying to solve the following:
Given two points $(x_{1}; y_{1})$ and $(x_{2}; y_{2})$, to determine the parameters $a$ and $b$ in the equation:
$$y=\frac{e^{a+bx} - e^{a}}{1+e^{a+bx}}.$$
In other words, I have two unknowns and two set of points, so there's for sure a solution, but how can the following system of equation be rewritten explicitly on $a$ and $b$?
$$\left\{\begin{matrix} y_{1}&=\frac{e^{a+bx_{1}} - e^{a}}{1+e^{a+bx_{1}}}\\ y_{2}&=\frac{e^{a+bx_{2}} - e^{a}}{1+e^{a+bx_{2}}} \end{matrix}\right.$$
Should I implement an equation solver instead? In which case, do you have any suggestion on the method?
Thanks very much, I hope it's not a too stupid question :)