2.5 Autonomous Equations

What is an autonomous ODE

• A first order ODE $y^{\prime }=f(t,y)$ is called autonomous if the right hand side f doesn’t depend on t. that is $y^{\prime }=f(y)$

• autonomous DEs are separable and can be solved

Stable fixed points

• when the derivatives go to the sero it

Population growth models

• model 1: exponential growth and decay

  • simple model: assume population $y(t)$ changes at a rate $r$ proportional to the number of people (y) with no explicit time dependence ( autonomous

  • rate constant r can be positive or negative
  • the corresponding IVP is
    • $y^{\prime }=ry,y(0)=y_0$
  • the ode is separable and can be solved analytically
    • $y(t)=y_0{e}^{rt}$
  • this model sucks because infinite growth makes no sense
    • it works for smaller t

• model 2: logistic growth

  • this replaces $y^{\prime }=ry$ with

    • $y^{\prime }=ry\left(1-\frac{y}{K}\right),y(0)=y_0$
  • r is called the intrinsic growth rate if r>0
  • K>0 is called the carrying capacity (the maximum population)

Phase line plot:

• the phase line for autonomous DE $y^{\prime }=f(y)$is a simple way to visualize equ