I am working for a mathematical model in which I intend to fit the model itself to statistical data to determine the values that my parameters must have. I am dealing with equations of the sort:
$$0.001=\frac{e^be^{-c}}{1-\int_0^9{e^{at}}dt}$$ and I wish to find all the combinations of a, b c greater or equal to $0$ that may satisfy this condition (I know that there should only be one). I have thought of a brute force approach, but it would be unrealistic to solve this problem in that way.
There has to exist some sort of function to solve this. I have been reading on data annealing, but it doesnt do the trick, and normal data fitting doesn't really make sense, since i don't have "points" to which I must find an equation, but rather i just need to find those three parameters.
As @podiki states, the equation only depends on the difference of $b$ and $c$, let's call this $d$. Then the equation is equivalent to:
$$0.001 = \frac{e^d}{1 - \int_0^9 e^{at}dt }$$
You can easily evaluate the integral:
$$\int_0^9 e^{at}dt = \frac 1 a (e^{9a} - 1).$$
Therefore the equation is equivalent to:
$$e^d = \frac{0.001}{a} (e^{9a} - 1).$$
Its unique solution is:
$$d = \ln \left( \frac{0.001}{a} (e^{9a} - 1) \right),$$
which is well definied, if and only if $a \ne 0$.