The population p of the city at time t is given by d p d t = p 2 -100 . If initial population is 100 then p =

Step-by-step solution

The differential equation dp dt = p 2 - 100 with initial condition p(0) = 100 is separable. Rewriting gives dp dt = p - 200 2 , leading to dp p - 200 = 1 2 dt . Integrating both sides yields |p - 200| = 1 2 t + C . Exponentiating produces |p - 200| = e^ C e^ t/2 . The general solution is p(t) = 200 + Ae^ t/2 , where A is a constant. Applying the initial condition p(0) = 100 gives 100 = 200 + A , so A = -100 . Thus, the particular solution is p(t) = 200 - 100e^ t/2 . This corresponds to option B . B