18.03 Lecture Notes - Lecture 1: Doubling Time
Recitation Handout
Review separation of variables (SoV)
!Rewrite as and then separate y and t to get
!Once the variables have been separates, we can integrate both sides
!!If F'(y) = 1/h(y) and G'(t) = g(t), then
!!Remember to check separates what happens when h(y) = 0 since in this case you cannot divide
Example:
!Solution:
Modeling natural growth
!The oryx population has a positive natural growth rate of k/years and there is assumed a constant harvesting rate of α oryxes
!per year. So let x(t) be the number of oryxes at time t
!The oryxes growth is measured by the constant k (some positive number)
!!x( t + Δt ) = x(t) + kx(t)Δt
!The oryxes harvest rate is measured by the constant a (some positive number)
!!x (t + Δt ) = x(t) - aΔt
!By combining these equations we get x (t + Δt ) = x(t) +kx(t)Δt - aΔt
!Now we rewrite this to think about the rate of change
!!And as Δt approaches 0,
!Special case: there is no harvest, so a = 0
!!In this case we get the equation
!Question: what is the doubling time of this system?
!!Start by finding the population at time t = 0, which is some constant C
!!Then set the equation equal to 2C
!The solution of is
!!You can use x(0) to find the value of C
Document Summary
Rewrite as and then separate y and t to get. Once the variables have been separates, we can integrate both sides. If f"(y) = 1/h(y) and g"(t) = g(t), then. Remember to check separates what happens when h(y) = 0 since in this case you cannot divide. The oryx population has a positive natural growth rate of k/years and there is assumed a constant harvesting rate of oryxes per year. So let x(t) be the number of oryxes at time t. The oryxes growth is measured by the constant k (some positive number) The oryxes harvest rate is measured by the constant a (some positive number) x( t + t ) = x(t) + kx(t) t x (t + t ) = x(t) - a t. By combining these equations we get x (t + t ) = x(t) +kx(t) t - a t. Now we rewrite this to think about the rate of change.
