Equilibrium solutions A differential equation of the form y′ ( t ) = f ( y ) is said to be autonomous ( the function f depends only on y ) . The constant function y = y 0 is an equilibrium solution of the equation provided f ( y 0 ) = 0 ( because then y′ ( t ) = 0 and the solution remains constant for all t ) . Note that equilibrium solutions correspond to horizontal lines in the direction field. Note also that for autonomous equations, the direction field is independent of t. Carry out the following analysis on the given equations. a. Find the equilibrium solutions. b. Sketch the direction field, for t ≥ 0 . c. Sketch the solution curve that corresponds to the initial condition y (0) = 1 . 41. y ′ ( t ) = y ( y – 3)
Equilibrium solutions A differential equation of the form y′ ( t ) = f ( y ) is said to be autonomous ( the function f depends only on y ) . The constant function y = y 0 is an equilibrium solution of the equation provided f ( y 0 ) = 0 ( because then y′ ( t ) = 0 and the solution remains constant for all t ) . Note that equilibrium solutions correspond to horizontal lines in the direction field. Note also that for autonomous equations, the direction field is independent of t. Carry out the following analysis on the given equations. a. Find the equilibrium solutions. b. Sketch the direction field, for t ≥ 0 . c. Sketch the solution curve that corresponds to the initial condition y (0) = 1 . 41. y ′ ( t ) = y ( y – 3)
Solution Summary: The author explains that the equilibrium solution of the given differential equation is y(t)=3 — the horizontal line segments represent equilibrium solutions in the directional fields.
Equilibrium solutionsA differential equation of the form y′(t) = f(y) is said to be autonomous (the function f depends only on y). The constant function y = y0 is an equilibrium solution of the equation provided f(y0) = 0 (because then y′(t) = 0 and the solution remains constant for all t). Note that equilibrium solutions correspond to horizontal lines in the direction field. Note also that for autonomous equations, the direction field is independent of t. Carry out the following analysis on the given equations.
a.Find the equilibrium solutions.
b.Sketch the direction field, for t ≥ 0.
c.Sketch the solution curve that corresponds to the initial condition y (0) = 1.
Assume that N(t) denotes the density of an insect species at time t and P(t) denotes the density of its
predator at time t. The insect species is an agricultural pest, and its predator is used as a biological control
agent. Their dynamics are given below by the system of differential equations. Complete parts (a) through
(c).
dN
= 7N - 5PN
dt
dP
= 4PN - P
dt
.....
(a) Explain why
dN
= 7N describes the dynamics of the insect in the absence of the predator.
dt
If there are no predators present, then P(t) =
for all t. Substitute P =
in the given differential
dN
equations to get
dt
So in the absence of the predators, the above equation describes the
dynamics of the insect population.
dN
Solve the equation,
dt
N(t) =
(Type an expression using t as the variable.)
Describe what happens to the insect population in the absence of the predator.
In the absence of the predator, the insect population
Consider a system of differential equations describing the progress of a disease in a population, given by
In our particular case, this is:
a) Find the nullclines (simplest form) of this system of differential equations.
The x-nullcline is y =
x = 3-3xy - 1x
y = 3xy - 2y
where x (t) is the number of susceptible individuals at time t and y(t) is the number of infected individuals at time t. The number of individuals is counted in
units of 1,000 individuals.
The y-nullclines are y =
and x =
b) There are two equilibrium points that are biologically meaningful (i.e. whose coordinates are non-negative). They are (x1,9₁) and (x2, y2), where we
order them so that x1 < x2.
The equilibrium with the smaller x-coordinate is (x₁, y₁) :
The equilibrium with the larger x-coordinate is (x2, y₂) =
where A =
0
sin (a)
c) The linearization of the system of differential equations at the equilibrium (ï1, Y₁) gives a system of the form
(+)
¹ (~),
f
əx
∞
a
Ω
= A
x'
G
E
= F(x, y) for a vector-valued function…
Find the equilibrium solutions and determine which are stable and which are unstable.
y' = 9y – 3y?
O y=-3 (stable); y = 0 (unstable)
y = 0 (unstable); y=3 (stable)
y = 0 (stable); y = 3 (unstable)
y=-3 (unstable); y = 0 (stable)
Thomas' Calculus: Early Transcendentals (14th Edition)
Knowledge Booster
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, computer-science and related others by exploring similar questions and additional content below.
01 - What Is A Differential Equation in Calculus? Learn to Solve Ordinary Differential Equations.; Author: Math and Science;https://www.youtube.com/watch?v=K80YEHQpx9g;License: Standard YouTube License, CC-BY
Higher Order Differential Equation with constant coefficient (GATE) (Part 1) l GATE 2018; Author: GATE Lectures by Dishank;https://www.youtube.com/watch?v=ODxP7BbqAjA;License: Standard YouTube License, CC-BY