# How to write this as a differential equation?

I was solving differential equations problems from Mary Boas book, and there was a problem that I couldn't write it as a differential equation

Solve the equation for the rate of growth of bacteria if the rate of increase is proportional to the number present but the population is being reduced at a constant rate by the removal of bacteria for experimental purposes.

How can I add the removal rate to my differential equation (I write the increase rate as $$\dfrac{dN}{dt} = KN(t)$$where $$K$$ is a constant, and $$N(t)$$ is the number of bacteria at any time).

We should add the term for the removal at a constant rate $$K_2>0$$ that is
$$\dfrac{dN(t)}{dt} = KN(t)-K_2$$
• At the end of the book, the final answer is $N = N_0e^{Kt}-(R/K)(e^{Kt}-1)$, however, if I just add it like that I get $N = \frac{N_0e^{Kt}+R}{K}$ – David Scott Nov 26 '18 at 18:23
• You also need to adjust for the initial conditions $N(0)=N_0$, check that! – user Nov 26 '18 at 18:26