Section 1.7. Autonomous Differential Equations
Objective:
1. Definition of autonomous differential equations and their applications
2. The behavior of the solution of an autonomous differential equation
In this section, we are moving forward to another type of differential equations. The differential equation is a first order nonlinear differential equation. We will not be able to find its general solution but we are able to decide the solution behavior as [latex]t[/latex] approaches infinity. We starts with the direction field of a first order differential equation. Direction field, way of graphically representing the solutions of a first-order differential equation without actually solving the equation. The equation [latex]y'=f(x,y)[/latex] gives a direction, [latex]y'[/latex], associated with each point [latex](x,y)[/latex] in the plane that must be satisfied by any solution curve passing through that point. The direction field is defined as the collection of small line segments passing through various points having a slope that will satisfy the given differential equation at that point. https://www.wolframalpha.com/input/?i=vector+field+plot
Example 1: [latex]y'=-2y[/latex] draw the direction field of the ODE.
Exercise 1: [latex]y'=3y[/latex] draw the direction field of the ODE.
Example 2: [latex]y'=x-2y[/latex] draw the direction field of the ODE.
Exercise 2: [latex]y'=2x+y[/latex] draw the direction field of the ODE.
Definition: An autonomous differential equation is an equation of the form [latex]y'=f(y)[/latex]. Equilibrium solutions of a general first order autonomous equation [latex]y'=f(y)[/latex] can be found by locating roots of [latex]f(y)=0.[/latex] These roots of [latex]f(y)[/latex] are called critical points.
Example 3: Logistic Growth: }We want to choose growth rate [latex]h(y)[/latex] so that (a) [latex]h(y)=r\gt 0[/latex] when [latex]y[/latex] is small, (b) [latex]h(y)[/latex] decreases as [latex]y[/latex] grows larger, and (c) [latex]h(y)\lt 0[/latex] when [latex]y[/latex] is sufficiently large. The simplest such function is [latex]h(y)=r-ay[/latex], where [latex]r,a\gt 0[/latex]. Our differential equation then becomes [latex]y'=(r-ay)y[/latex] with [latex]r,a\gt 0[/latex].
Exercise 3: [latex]y'=(3-4y)y[/latex]. Find equilibrium solutions of the ODE and draw its direction field.
Definition: An equilibrium solution of a general first order autonomous equation [latex]y'=f(y)[/latex] could be a stable solution or unstable solution or semi-stable solution depending on the behavior of the direction field near the solution.
Example 4: [latex]y'=(y-1)^{2}(9-y^{2})[/latex]. Find equilibrium solutions of the ODE and classify their stability.
Exercise 4: [latex]y'=y^{2}(y^{2}-4)[/latex]. Find equilibrium solutions of the ODE and classify their stability.
Group Work
1. An epidemic spreads through a population at a rate proportional to the product of the number of people already infected and the number of people susceptible, but not yet infected. Therefore, if [latex]S[/latex] denotes the total population of susceptible people and [latex]I=I(t)[/latex] denotes the number of infected people at time [latex]t[/latex], then [latex]I'=rI(S-I),[/latex] where [latex]r[/latex] is a positive constant. Assuming that [latex]I(0)=I_{0}[/latex], find [latex]I(t)[/latex] for [latex]t\gt0[/latex], and show that [latex]\text{lim}_{t\rightarrow\infty}I(t)=S.[/latex]
2. Consider the first order autonomous equation
\[y’=y(1+y)^{2}(4-y) \]
Answer the following questions without solving the equation.
(a) (3 points) Find all equilibrium solutions.
(b) (6 points) For each equilibrium solution, classify its stability.
Justify your answer.
(c) (3 points) If [latex]y(1)=3[/latex], what is [latex]y(0)[/latex]?
(d) (3 points) If [latex]y(-2)=5[/latex], what is [latex]\lim_{t\rightarrow\infty}y(t)[/latex]?
(e) (3 points) Suppose [latex]y(0)=a[/latex], and [latex]\lim_{t\rightarrow\infty}y(t)=0[/latex].
Find all possible values of [latex]a[/latex].