Let the population of rabbits surviving at a time t be governed by the differential equation d p t d t = 1 2 p t - 400 . If p ( 0 ) = 100 , then p ( t ) equals

Step-by-step solution

d p t d t = 1 2 p t - 400 p 0 = 100 2 d p t p t - 400 = d t Integrating, ⇒ ∫ 0 p ( t ) dp ( t ) p t - 400 = ∫ 0 t 1 2 dt ⇒ ln p t - 400 0 p t = t 2 ⇒ ln p t - 400 300 = t 2 ⇒ p t - 400 = 300 e t 2 ⇒ 400 - p t = 300 e t 2    ( ∴ p t < 400 ) ⇒ p t = 400 - 300 e t 2