Question

Consider a population of bacteria that grows according to the initial value problem dP/dt=P/10, P(0)=300. Find the population size after 40 hours

Answer 1

 dP / dt = P / 10
 We apply separable variables:
 dP / P = dt / 10
 We integrate both sides:
 Ln (P) = t / 10 + C
 We rewrite the equation:
 Exp (Ln (P)) = Exp (t / 10 + C)
 P = Exp (C) * Exp (t / 10)
 P = C * Exp (t / 10)
 We look for the constant using:
 P (0) = 300
 300 = C * Exp (0/10)
 300 = C * 1
 C = 300
 We rewrite the equation:
 P = 300 * Exp (t / 10)
 After 40 hours we have:
 P = 300 * Exp (40/10)
 P = 16379.44501
 Answer:
 the population size after 40 hours is: 
 P = 16379