section 3.1 : systems of two linear algebraic...

Section 3.1 : Systems of Two Linear AlgebraicEquations

Chapter 3 : Systems of Two First Order Equations

Math 2552 Differential Equations

Section 3.1 Slide 1

Section 3.1

TopicsWe will cover these topics in this section.

1. Solving first order separable differential equations.

ObjectivesFor the topics covered in this section, students are expected to be able todo the following.

1. Determine the eigenvalues and eigenvectors of a matrix

2. Solve a system of two linear equations

3. Characterize a linear system in terms of the number of solutions,and whether the system is consistent or inconsistent.

4. Characterize a linear system in terms of its eigenvalues.

Section 3.1 Slide 2

Examples

1. Solve the linear system, and determine whether the lines intersect,are parallel, or are coincident.

x1 − 2x2 = −1−x1 + 3x2 = 3

2. Determine the eigenvalues and eigenvectors of the matrices.

a) A =

(5 22 1

)b) B =

(−5 −55 −5

)

Section 3.1 Slide 3

Section 3.2 : Systems of Two First Order LinearDifferential Equations



Section 3.2 Slide 4

Section 3.2


1. Systems of two first order linear DEs


1. Express a set of linear ODEs as a matrix equation

2. Characterize a DE by its dimension and whether it is homogeneous

3. Confirm that a given vector function is a solution to a DE system.

4. Identify critical points of a system of DEs.

5. Covert a second order DE into a system of DEs.

6. Sketch component plots of a solution to a DE.

The textbook also introduces phase diagrams in this section, we willintroduce them in a later part of this course.

Section 3.2 Slide 5

Foxes and Rabbits

Foxes and rabbits live together on an island. The foxes (predators) eatthe rabbits, rabbits (prey) eat vegetables.

Photo credits: www.flickr.com/photos/mattyfioner, www.flickr.com/photos/43158397@N02

Suppose

� x1(t) = number of foxes on the island

� x2(t) = number of rabbits on the island

� k foxes are removed from the island per day, k > 0

For simplicity, assume x1 and x2 are continuous and differentiable.

Section 3.2 Slide 6

A Linear System of DEs

Populations of the two species can be modeled using a system of linearODEs.

dx1dt

= ax1 + bx2 − k

dx2dt

= cx1 + dx2

We can write this as a matrix equation:

expressing two linear DEs as a matrix equation is our 1st learning objective

Section 3.2 Slide 7

Dimension and Homogeneity

d~x

dt= P (t)~x+ ~g(t)

Our linear system

� is first order system of dimension two, because ~x has two elements.

� is non-homogeneous because ~g(t) 6= ~0.

If ~g(t) = ~0 for all t, the system is homogeneous.

dimension and homogeneity covers our 2nd learning objective

Section 3.2 Slide 8

Solutions to a System

A solution to a two-dimensional system

d~x

dt= P (t)~x+ ~g(t) (1)

is a set of two functions, x1(t) and x2(t) that satisfy the system for all ton some interval.

Suppose we are given values for a, b, c, d,~g so that

d~x

dt=

(20 0−10 30

)~x (2)

It can be shown that there are two solutions to this system, which are

~u1(t) = eλ1t ~v1, ~u2(t) = eλ2t ~v2

λ1 and λ2 are the eigenvalues of P , and ~v1 and ~v2 are the correspondingeigenvectors. Identify one solution, and show that it satisfies (2).

Section 3.2 Slide 9

Critical Points

Critical points, or equilibrium solutions of a system of DEs correspondto points where

d~x

dt= ~0

For example, the critical points of

d~x

dt=

(20 0−10 30

)~x (3)

are:

Section 3.2 Slide 10

Second Order Equations

We can always convert a second order linear DEs to a system of firstorder linear DEs.

Example: convert the second order equation to a system of first orderdifferential equations.

d2y

dt2− sin(t)

dy

dt+ 7y = et cos t+ 1


Note: Component Plots

Plots of solutions u1 and u2 vs t are component plots of atwo-dimensional system of ODEs.

Example: a solution to

d~x

dt=

(20 0−10 30

)~x

is:~u(t) = c1~u1 + c2~u2 =


Section 3.3 : Homogeneous Linear Systems withConstant Coefficients




Section 3.3


1. Systems of two first order linear DEs


1. Solve first order homogeneous linear systems.

2. Sketch component plots and phase portraits of linear systems ofdifferential equations.

The textbook also introduces the Wronskian and discusses uniqueness.We’ll briefly explore these topics and go through them in more detail inSection 6.2.


Compartment Model

A tank is divided into two cells. Each cell is filled with a fluid, and smallopening allows the fluid to flow freely between the cells.

Assume: height of fluid in a cell changes at a rate proportional to thedifference between fluid height in that cell and the fluid height in theother cell.Questions:

1. What happens to the system after a long period of time?

2. Construct a linear system for the fluid level heights.

3. Solve the linear system.

4. Determine whether the solution is unique.

5. Sketch component plots and a phase portrait of the system.


Object Motion

Suppose that the position of a moving object is given by y(t), and

d2y

dt2+ 2

dy

dt+ αy = 0, α ∈ R

1. Transform the given differential equation into a system of first orderequations. Express the system using a matrix equation of the form

~x′ = A~x, where ~x =

(x1x2

)2. Express the eigenvalues of A in terms of α.

3. Let α = −3.

A) Determine the eigenvectors of A.B) Write down the solution to the first order system.C) Sketch the phase portrait and classify the critical points of the

system.

4. Repeat question (3) with α = 34 (if time permits).


Checking Your Work

� As you are completing your homework, studying for exams, orcompleting other courses that use this class as a pre-requisite, youmay want to use a computer-generated solution to check your workfor accuracy.

� You can obtain direction fields in the free version of WolframAlphawith streamplot. Try giving this to WolframAlpha:

streamplot[{y,−2y + 3x}, {x,−1, 1}, {y,−1, 1}]


Solution Uniqueness

Consider ~x ′ = A~x, where A is a 2× 2 matrix with eigenvalues λ1 andλ2, eigenvectors ~v1 and ~v2.

A) Write down a solution to the system. Assume λ1 and λ2 are realand distinct.

B) If for some t0 ∈ R, that ~x(t0) =

(b1b2

). Construct a matrix equation

that can be used to solve for the arbitrary constants you used in (A).

C) Is the matrix you constructed in (B) invertible?


The Wronskian of two Vector Functions

Given vector functions ~x1(t) =

(x11x21

)and ~x2(t) =

(x12x22

), the

determinant

W [~x1, ~x2](t) =∣∣~x1 ~x2

∣∣ = ∣∣∣∣x11 x12x21 x22

∣∣∣∣is the Wronskian of ~x1(t) and ~x2(t).

Definition

Example: the Wronskian of vector functions ~x1 = et(11

)and

~x2 = e2t(23

)is:


Linear Independence

Vector functions ~x1(t) and ~x2(t) are said to be linearly independenton an interval of t when

W [~x1, ~x2](t) 6= 0

everywhere on that interval.

Definition

Example: for what values of t are the vector functions ~x1 = et(11

)and

~x2 = e2t(23

)linearly independent?


Fundamental Set of Solutions

If ~x1 and ~x2 are two linearly independent solutions to ~x ′ = A~x,where A is real 2× 2, then ~x1 and ~x2 are a fundamental set.

Definition

Example: the vectors ~x1 = et(11

)and ~x2 = e2t

(23

)are a

fundamental set of solutions to ~x ′ = A~x, where A =

(−1 2−3 4

).


Section 3.4 : Complex Eigenvalues




Section 3.4


1. Systems of two first order linear DEs for the complex eigenvalue case


1. Solve first order homogeneous linear systems that have complexeigenvalues

2. Sketch component plots and phase portraits of linear systems ofdifferential equations for complex eigenvalues


Example

Determine the general solution to

~x ′ = A~x =

(−1 2−1 −3

)~x

Sketch the phase portrait and classify the critical point.


Summary

Suppose we have

~x(t) = c1eλ1t~v1 + c1e

λ2t~v2

where λ1, λ2 are complex. We wish to express ~x as a real-valued vectorfunction.

General procedure:

1. Compute eigenvalues λ = α± iβ2. Compute eigenvector, ~v, for λ = α+ iβ

3. Set ~v = ~a+ i~b

4. General solution is ~x(t) = c1~u+ c2 ~w, where

~u = eαt(~a cosβt−~b sinβt)

~w = eαt(~a sinβt+~b cosβt)


Section 3.5 : Repeated Eigenvalues




Section 3.5


1. Systems of two first order linear DEs for the repeated eigenvalue case


1. Solve first order homogeneous linear systems that have repeatedeigenvalues

2. Sketch component plots and phase portraits of linear systems ofdifferential equations for repeated eigenvalues


Object Motion

The motion of an object moving in the xy-plane is given by ~r(t), where

~r(t) =

(x(t)y(t)

)The velocity of the object is constrained by

dx

dt= −x+ ky (4)

dy

dt= −y (5)

Assume k ∈ R. At time t = 0, our object is located at ~r(0) =

(12

).

Take a few minutes on your own to solve this initial value problem.Create a rough sketch of the phase portrait (i.e. - trajectories in thexy-plane). How your answer depend on the value of k?


Example

Construct the general solution to the system.

~x ′ =

(1 −11 3

)~x


Summary

Suppose we would like to solve

d~x(t)

dt= A~x

where A is a 2× 2 defective matrix.

General procedure to solve this system:

1. Identify eigenvalue λ and corresponding eigenvector ~v

2. Solve (A− λI)~w = ~v for ~w

3. General solution is ~x(t) = c1~x1 + c2~x2, where

~x1 = eλt~v

~x2 = eλt(t~v + ~w)


Section 3.6 : A Brief Introduction to NonlinearProblems




Section 3.6


1. First order nonlinear autonomous systems.


1. Solve first order systems of non-linear autonomous systems byconverting them into a DE that is exact, Bernoulli, separable, orhomogeneous.


Autonomous First Order Systems

A nonautonomous first order system has the form

~x′ = f(t, ~x)

An autonomous first order system has the form

~x′ = f(~x)

In this section we focus on autonomous systems.


Object Motion

The motion of an object moving in the xy-plane is given by ~r(t), where

~r(t) =

(x(t)y(t)

)In previous lectures, the velocity of the object, ~r ′, was constrained bylinear equations.

Suppose the velocity of the object is constrained by

x′(t) = −xy (6)

y′(t) = x2 (7)

1. Identify the critical points of the system.

2. Identify a function, H(x, y) = c that satisfies the above constraints.

3. Plot the phase portrait, level curves of H, and the trajectory of anobject that passes through the point (x, y) = (4, 0).


section 3.1 : systems of two linear algebraic...

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