Let the population of a species of birds surviving at a time ' t ' be governed by the differential equation dp dt - p = -100 . If p(0) = 50 , then p(- _e 2) is equal to

Step-by-step solution

The given differential equation is dp dt - p = -100 This can be written as dp p - 100 = dt Integrating both sides, we get: dp p - 100 = dt |p - 100| = t + C p - 100 = A e^t p(t) = 100 + A e^t Given p(0) = 50 , substituting t = 0 : 50 = 100 + A e^0 A = -50 Thus, the population function is p(t) = 100 - 50 e^t We need to find p(- _e 2) : p(- _e 2) = 100 - 50 e^ - _e 2 Since e^ - _e 2 = e^ _e (1/2) = 1 2 : p(- _e 2) = 100 - 50 ( 1 2 ) = 100 - 25 = 75 Answer: 75