Question

(2) Many physical and biological phenomena follow simple exponentiation laws. For example, suppose you have a petri dish with a population of NO 1.000 bacterium at t = 0 s (initial condition N(0)-No), and after some time, the population has grown. We need an equation of motion for the population size. Empirically, the time rate of change of the population size is proportional to the population size N(t). dt a) Using the method of infinitesimals, find a solution for this equation. b) Then show that it is indeed a solution to this first order differential equation. c) Finally, draw a graph of your solution indicating all interesting points. Please explain what your solution means using its graph. The growth rate is denoted by γ

Answer 1

dN (t)/dt = N (t)_(1) Rightarrow dN (t)/N(t) = gamma dt Rightarrow ln N (t) = gamma t + ln c Rightarrow ln N (t) - ln c = gamma t Rightarrow N (t) = c e^gamma t Applying initial condition that at t = 0 N (t) = N_0 So C = N Therefore N (t) = N_0 e^gamma t_(2) (b): Substituting (2) in (1) in L.H.S. d/dt N_0 gamma t = N_0 gamma e^gamma t = gamma N (t) = R.H.S.