estimate reaction rate from time course concentration measurments via regression

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.

• It should be $x(t) = \frac{c}{r}+ke^{-rt}$ – Paul Oct 11 '18 at 11:38
• Yes of course, you are right. That was a typo. I changed the question accordingly. – fabian Oct 11 '18 at 11:42
• The solution can also be written as $x(t) = \frac{c}{r}(1-e^{-rt})+x_0e^{-rt}$ where $x_0$ is the initial value of $x$. If t is small then $(1-e^{-rt})$ is approximately $rt$ giving you your intuitive linear term for small t. – Paul Oct 11 '18 at 11:57