Including the mean in differential equation

I have a differential equation in the form: $ $ \frac{1}{C} \frac{dC}{dt} = a – C – \mu_C $ $

where C is a random variable. Is it possible to derive an exact solution to this equation to get the time-dependence of the variable C and its distribution? I know that we can do so without the average term at the end but am not sure how to deal with the mean due to its time dependence.