I try to estimate the decay rate r of a substance x. I do have time course measurements of x in a system where x is produced with a constant rate c. So I would like to set up an equation describing the concentration as a function of time, and use this equation in a regression analysis to estimate the two unknown variables r and c.
I looked around and only found Concentration of Solution as Time Progresses which is similar but not quite the same.
Here is what I came up with so far. The change of concentration x at time t is in my model
x'(t) = c - r * x(t)
a steady influx of c and a proportional decay of r*x. If I solve this equation (using wolfram alpha as a black box I have to admit) I get
x(t) = c/r + k*exp(-rt)
This result does not behave as expected. I expect a function describing the linear increase coupled with an exponential decrease term. The former is missing in the results I obtain.
Could somebody please point me to the error in reasoning I obviously suffer from?
thank you very much.