Population Growth Beginning in the next section we will see that differential equations can be used to describe or model many different physical systems. In this problem suppose that a model of the growing population of a small community is given by the initial-value problem where P is the number of individuals in the community and time t is measured in years. How fast—that is, at what rate—is the population increasing at t = 0? How fast is the population increasing when the population is 500?

Solution:-Step1Given thatWe have to find how fast and at what rate the population increasing at t = 0 and how fast is the population increasing when the population is 500Step2We haveThe initial-value problemWhere time t is measured in years and P is the number of individuals in the community.Step3Now=Rate at which population increases at “t “ time.At t=0The rate of growth of population is = = 0.15 = 15+20=35Therefore, the population increasing at t = 0 is 35.Step4When the population is 500The rate of growth of population is = = 75+20=95Therefore, the population increasing when the population is 500 is 95.