TITLE: Is it possible to solve the two unknowns of this function given the area, the base length and a ratio between the start and end points?
QUESTION [1 upvotes]: I am trying to calculate the values of $a$ and $b$ in the following function:
$$
f(x) = -e^{ax} + b + 1.
$$
There are a few "rules" in play:

$\int\limits_{0}^{n} f(x)\, \mathrm{d}x= v$
$f(n) = rb$

where

$v$ is the total area of the subgraph of $f$,
$n$ is the length of the base of this subgraph
$r$ is the desired ratio between the start ($v_{0}$) and end ($v_{n}$) points (ex. 0.5)

Is it possible to derive equations to calculate $a$ and $b$, given $v$, $n$ and $r$? If so, would you be willing to show me how?

REPLY [1 votes]: The constraints are written 
$$\begin{cases}\dfrac{1-e^{an}}a+(b+1)n=v,\\-e^{an}+b+1=rb.\end{cases}$$
You can eliminate $b$ using the second equation,
$$b=\frac{1-e^{an}}{r-1},$$
which you plug in the first, giving a nasty nonlinear equation in $an$
$$\dfrac{1-e^{an}}{an}+\frac{1-e^{an}}{r-1}=\frac vn-1.$$
This must be solved by a numerical method. (There is a slight hope that it can be done with Lambert's W function, but I have not investigated.)
{"set_name": "stack_exchange", "score": 1, "question_id": 3328679}