# Find a differential equation satisfying:

The population in a certain country grows at a rate proportional to the population at time $t$, with proportionality constant of $0.03$. Due to political turmoil, people are leaving the country at a constant rate of $6000$ per year. Assume there's no immigration to the country. Let $P=P (t)$ is population time at $t$, the being in years.

Write the differential equation reflecting the situation.

Solve it for $P (t)$ given that at $t=0$ the population is 3 million. Find $P (t)$.

I'm stuck on the DE. I know it should have a $-6000$ but the $1.03$ is throwing me off.

• Generally, populations grow at a rate proportional to the population itself. That is because, assuming a constant birth rate, the average number of births per child, the increase will be some constant times the current population. That is, dP/dt= kP. With a birth rate of 0.03, the new population will be P+ 0.03P= (1.03)P so k= 1.03. With an additional 6000 leaving you have dP/dt= 1.03P- 6000. – user247327 May 3 '17 at 15:45

## 1 Answer

Consider the discrete system, $P_{t+1} = P_t + 0.03 P_t - 6000$ $(\star)$, then converting this very informally gives

$$P_{t+1} - P_t = 0.03 P_t - 6000$$ so $$\frac{P_{t+1} - P_t}{\delta t} = \frac{0.03 P_t - 6000}{\delta t}$$ Taking the limit as $\delta t \rightarrow 0$ on the LHS gives $$P'(t) = \frac{0.03}{\delta t} P(t) - \frac{6000}{\delta t}$$ Note that the reason I keep the $\delta t$ on the RHS is because they are actually "constants" and will cancel out in the solving process.

The more standard way of doing this is to write $P_{t+1} = P(t+\delta t)$ and note that your $0.03$ and $6000$ are actually $0.03 \delta t$ and $6000 \delta t$. Thus $\frac{P_{t+\delta t} - P_t}{\delta t} = \frac{0.03 \delta t P_t - 6000 \delta t}{\delta t}$. Therefore $$P'(t) = 0.03 P(t) - 6000$$

Solving this system with $P(0) = 3000000$, you get $$P(t) = 200000 + Ce^{0.03t}$$ or with the given initial condition, $$P(t) = 200000 + 2800000e^{0.03t}$$

Because of some potential problems with the conversion, I'd sick to the difference equation $(\star)$ for this problem.