The following two applications involve a type of differential equation which can be solved by separating the variables and using a partial fraction decomposition to help calculate the antiderivatives. The same type of differential equation is also used to model the spread of rumors and diseases as well as some populations and chemical reactions. Logistic Growth: The growth rate of many different populations depends not only on the number of individuals (leading to exponential growth) but also on a "carrying capacity" of the environment. If x is the population at time t and the growth X rate of x is proportional to the product of the M = carrying capacity 3 population and the carrying capacity M minus the population, then the growth rate is described by the differential equation population dx a = k'x'(M x) ind endent variable time t where k and M are constants for a given time species in agiven environment. Fig 1, Logistic growth curve 8.4 Partial Fraction Decomposition Contemporary Calculus 8 36. Let k = 1 and M = 100, and assume the initial population is x(0) = 5 . dx (a) Solve the differential equation a = x(100 x) for x . (b) Graph the population x(t) for 0 s t s 20. (c) When will the population be 20? 50'? 90? 100? (d) What is the population after a "long" time? (Find the limit, as t becomes arbitrarily large, of x .) (e) Explain the shape of the graph in (a) in terms of a population of bacteria. (f) When is the growth rate largest? (Maximize dxfdt .) (g) What is the population when the growth rate is largest? x = x(t) = population at time t