I have been trying to solve the following relation for t:
$$exp(r_1*t)-exp(r_2*t) = K*(exp(r_1*t_a)-exp(r_2*t_a))$$
In this case, the following assumptions hold:
- t is positive, real, and non-zero. This is the quantity to find.
- t_a is a constant, real, positive, and non-zero.
- r_1 and r_2 are non-zero and constant.
- K is real, positive and non-zero, additionally K cannot be greater than 1. (When K is 1, we have a special case where t = t_a.)
The closest answers I could find to this problem are here:
Is there any possibility of an analytical solution in this case? Any hints on how it could be proven that there is no analytical solution? My big doubt is that I can plot the parent function (this is the simplified expression for one of the parameters in the function) and find the solution by analyzing the parent function. In practicality, t actually has 2 solutions, but I take the larger one. My intuition tells me there could be an analytical solution, but I can't find a way to fully prove whether or not it exists.