Suppose a country's population at any time t grows in accordance with the rule dP dt = kP + I where P denotes the population at any time t, k is a positive constant reflecting the natural growth rate of the population, and I is a constant giving the (constant) rate of immigration into the country. If the total population of the country at time t = 0 is P0, find an expression for the population at any time t.

Step-by-step explanation:

Given that:

A country population at any given time (t) is:

where;

P = population at any time t

k = positive constant

I = constant rate of immigration into the country.

Using the method of separation of the variable;

Taking integration on both sides:

By applying the exponential on both sides;

Assume

Then:

When time t = 0, The Total population of the country is also

C - I = kP₀

C =  kP₀ + I

Substituting the value of C back into equation(1), we have: