Exponential model for population

Question

I am having trouble with a population modeling problem. The first part states:

Assume that Switzerland's population grows at a rate of $0.18$ percent a year and that the $1988$ population was $6.7$ million.

I got the model to be:

$$ y(t)=(6.7×10^6)e ^ {0.0018*t} $$

I am having trouble with the second part of the problem:

If there is a net immigration of $10,000$ people a year into Switzerland, write an expression for the population in year $t$.

I have tried multiple variations of adding $10000$ and $10000t$ to different parts of the equation with no luck.

How do I account for the $10000$ immigrants?

Answer 1

Suppose population grow of a country is with a rate $k$ and a constant immigration $I$ in a year, so the population $P(t)$ satisfies in differential equation $$\frac{dP}{dt}=kP(t)+I$$ where $t$ (uasally) in year and $P(0)$ is initial population in first year. We solve this equation, then $$P(t)=P(0)e^{kt}+\frac{I}{k}e^{kt}-\frac{I}{k}$$