Population Growth Model?

📌 CONCEPT: The population growth model is based on the fact that the rate of change of a population's size is proportional to the current population, which can be represented mathematically as dP/dt = kP, where P is the population size, t is time, and k is the growth rate constant.

📐 RULE / FORMULA: The differential equation to model population growth is given by dP/dt = kP, where k is the growth rate constant. In this case, k = 0.02 and the initial population P(0) = 1000.

💡 WORKED EXAMPLE: Let's derive the differential equation for the given population growth model. Given that the initial population P(0) = 1000 and k = 0.02, we have dP/dt = 0.02P. This is a simple first-order linear differential equation, which can be solved using the separation of variables method.

⚠️ COMMON MISTAKE: Students often forget to include the initial condition P(0) = 1000 in the differential equation, which is essential for finding the actual population size at any given time t.