Eigenvalues Eigenvectors and Differential Equations

Section 5.4 (Systems of Linear DifferentialEquation); Eigenvalues and Eigenvectors

July 1, 2009

2 2 Systems of Linear Differential Equations

Todays Session

2 2 Systems of Linear Differential Equations

Todays Session

A Summary of This Session:

2 2 Systems of Linear Differential Equations

Todays Session

A Summary of This Session:(1) Finding the eigenvalues and eigenvectors of a 2 2 matrix.

2 2 Systems of Linear Differential Equations

Todays Session

A Summary of This Session:(1) Finding the eigenvalues and eigenvectors of a 2 2 matrix.(2) 2 2 linear, first-order, systems of differential equations

2 2 Systems of Linear Differential Equations

Todays Session

A Summary of This Session:(1) Finding the eigenvalues and eigenvectors of a 2 2 matrix.(2) 2 2 linear, first-order, systems of differential equations(3) Phase-plane method

2 2 Systems of Linear Differential Equations

Todays Session

A Summary of This Session:(1) Finding the eigenvalues and eigenvectors of a 2 2 matrix.(2) 2 2 linear, first-order, systems of differential equations(3) Phase-plane method

2 2 Systems of Linear Differential Equations

Todays Session

A Summary of This Session:(1) Finding the eigenvalues and eigenvectors of a 2 2 matrix.(2) 2 2 linear, first-order, systems of differential equations(3) Phase-plane method

2 2 Systems of Linear Differential Equations

Motivation

We are interested in solving systems of first order differentialequations of the form:

x = f (x , y)

y = g(x , y)

2 2 Systems of Linear Differential Equations

Motivation

We are interested in solving systems of first order differentialequations of the form:

x = f (x , y)

y = g(x , y)

or more generally, systems that look like:

x = f (x , y , t)

y = g(x , y , t)

2 2 Systems of Linear Differential Equations

Motivation

We are interested in solving systems of first order differentialequations of the form:

x = f (x , y)

y = g(x , y)

or more generally, systems that look like:

x = f (x , y , t)

y = g(x , y , t)

In the first case, f (x , y) and g(x , y) do not depend on t. They arecalled autonomous.

2 2 Systems of Linear Differential Equations

Motivation

We are interested in solving systems of first order differentialequations of the form:

x = f (x , y)

y = g(x , y)

or more generally, systems that look like:

x = f (x , y , t)

y = g(x , y , t)

In the first case, f (x , y) and g(x , y) do not depend on t. They arecalled autonomous.In the second case, f (x , y , t) and g(x , y , t)depend on t. They are called non-autonomous.

2 2 Systems of Linear Differential Equations

Examples

Which of the following examples is autonomous?

2 2 Systems of Linear Differential Equations

Examples

Which of the following examples is autonomous?(a):

2 2 Systems of Linear Differential Equations

Examples

Which of the following examples is autonomous?(a):

x = 2x 4x y

y = 2x + 2y2

2 2 Systems of Linear Differential Equations

Examples

Which of the following examples is autonomous?(a):

x = 2x 4x y

y = 2x + 2y2

Answer: first-order, autonomous (not linear), 2 2 system of dfqs

2 2 Systems of Linear Differential Equations

Examples

Which of the following examples is autonomous?(a):

x = 2x 4x y

y = 2x + 2y2

Answer: first-order, autonomous (not linear), 2 2 system of dfqs(b)

x = 2x + 4y t

y = x 2y + sin t

2 2 Systems of Linear Differential Equations

Examples

Which of the following examples is autonomous?(a):

x = 2x 4x y

y = 2x + 2y2

Answer: first-order, autonomous (not linear), 2 2 system of dfqs(b)

x = 2x + 4y t

y = x 2y + sin t

Answer: first-order, non-autonomous (yet linear), 2 2 system ofdfqs

2 2 Systems of Linear Differential Equations

Examples

Which of the following examples is autonomous?(a):

x = 2x 4x y

y = 2x + 2y2

Answer: first-order, autonomous (not linear), 2 2 system of dfqs(b)

x = 2x + 4y t

y = x 2y + sin t

Answer: first-order, non-autonomous (yet linear), 2 2 system ofdfqs

We are interested in qualitative as well as quantitativedescriptions of the solutions.

2 2 Systems of Linear Differential Equations

Finding eigenvalues and eigenvectors of matrices

To find the eigenvalues (and corresponding eigenvectors) of amatrix A means to find the (scalar) values and corresponding(non-zero) vectors v which satisfy the vector equation

Av = v .

In some sense the eigenvectors define the main directions alongwhich the matrix A acts (as a geometric transform).

2 2 Systems of Linear Differential Equations

Finding eigenvalues and eigenvectors of matrices

To find the eigenvalues (and corresponding eigenvectors) of amatrix A means to find the (scalar) values and corresponding(non-zero) vectors v which satisfy the vector equation

Av = v .

In some sense the eigenvectors define the main directions alongwhich the matrix A acts (as a geometric transform).

Example 1: Let A =

(5 21 4

). Find its eigenvalues and

corresponding eigenvectors.

2 2 Systems of Linear Differential Equations

Finding eigenvalues and eigenvectors of matrices

To find the eigenvalues (and corresponding eigenvectors) of amatrix A means to find the (scalar) values and corresponding(non-zero) vectors v which satisfy the vector equation

Av = v .

In some sense the eigenvectors define the main directions alongwhich the matrix A acts (as a geometric transform).

Example 1: Let A =

(5 21 4

). Find its eigenvalues and

corresponding eigenvectors.

We let v =

(x

y

).

2 2 Systems of Linear Differential Equations

Example,contd

The equationAv = v .

means:

5x + 2y = x

x 4y = y

2 2 Systems of Linear Differential Equations

Example,contd

The equationAv = v .

means:

5x + 2y = x

x 4y = y

or

(5 )x + 2y = 0

x + (4 )y = 0

2 2 Systems of Linear Differential Equations

Example,contd

The equationAv = v .

means:

5x + 2y = x

x 4y = y

or

(5 )x + 2y = 0

x + (4 )y = 0

This system of equations describes the intersection of two lineswhich go through the origin. In order to have a non-zero solution,the determinant must be zero (this follows from Cramers rule). So (5 ) 2

1 (4 )

= 02 2 Systems of Linear Differential Equations

Example,contd

The equationAv = v .

means:

5x + 2y = x

x 4y = y

or

(5 )x + 2y = 0

x + (4 )y = 0

This system of equations describes the intersection of two lineswhich go through the origin. In order to have a non-zero solution,the determinant must be zero (this follows from Cramers rule). So (5 ) 2

1 (4 )

= 02 2 Systems of Linear Differential Equations

Example,contd

Therefore(5 ) (4 ) 2 = 0

2 2 Systems of Linear Differential Equations

Example,contd

Therefore(5 ) (4 ) 2 = 0

or2 + 9+ 20 2 = 0

2 2 Systems of Linear Differential Equations

Example,contd

Therefore(5 ) (4 ) 2 = 0

or2 + 9+ 20 2 = 0

That is2 + 9 + 18 = 0

2 2 Systems of Linear Differential Equations

Example,contd

Therefore(5 ) (4 ) 2 = 0

or2 + 9+ 20 2 = 0

That is2 + 9 + 18 = 0

Solving gives: = 3,6.

2 2 Systems of Linear Differential Equations

Example,contd

Therefore(5 ) (4 ) 2 = 0

or2 + 9+ 20 2 = 0

That is2 + 9 + 18 = 0

Solving gives: = 3,6.Now we find the eigenvectors.

2 2 Systems of Linear Differential Equations

Example,contd

For 1 = 3, the system becomes:

2x + 2y = 0

x y = 0

2 2 Systems of Linear Differential Equations

Example,contd

For 1 = 3, the system becomes:

2x + 2y = 0

x y = 0

Both equations lead to: x = y . So we can choose the eigenvector

to be v1 =

(11

).

2 2 Systems of Linear Differential Equations

Example,contd

For 1 = 3, the system becomes:

2x + 2y = 0

x y = 0

Both equations lead to: x = y . So we can choose the eigenvector

to be v1 =

(11

).

For 2 = 6, the system becomes:

x + 2y = 0

x + 2y = 0

2 2 Systems of Linear Differential Equations

Example,contd

For 1 = 3, the system becomes:

2x + 2y = 0

x y = 0

Both equations lead to: x = y . So we can choose the eigenvector

to be v1 =

(11

).

For 2 = 6, the system becomes:

x + 2y = 0

x + 2y = 0

Both equations lead to: x = 2y . So we can choose the

eigenvector to be v2 =

(21

).

2 2 Systems of Linear Differential Equations

Example,contd

For 1 = 3, the system becomes:

2x + 2y = 0

x y = 0

Both equations lead to: x = y . So we can choose the eigenvector

to be v1 =

(11

).

For 2 = 6, the system becomes:

x + 2y = 0

x + 2y = 0

Both equations lead to: x = 2y . So we can choose the

eigenvector to be v2 =

(21

).

2 2 Systems of Linear Differential Equations

Using eigenvalues and eigenfunctions to solve linear firstorder systems

This is an alternative method to the annihilator method whichexplains the nature of the solution obtained.

2 2 Systems of Linear Differential Equations

Using eigenvalues and eigenfunctions to solve linear firstorder systems

This is an alternative method to the annihilator method whichexplains the nature of the solution obtained.

Example 2: Solve:

x = 5x + 2y

y = x 4y

2 2 Systems of Linear Differential Equations

Using eigenvalues and eigenfunctions to solve linear firstorder systems

This is an alternative method to the annihilator method whichexplains the nature of the solution obtained.

Example 2: Solve:

x = 5x + 2y

y = x 4y

Here is how we solve it:

1. Find the matrix A corresponding to this linear sytems and putthe equation in matrix form v = Av .

2 2 Systems of Linear Differential Equations

Using eigenvalues and eigenfunctions to solve linear firstorder systems

This is an alternative method to the annihilator method whichexplains the nature of the solution obtained.

Example 2: Solve:

x = 5x + 2y

y = x 4y

Here is how we solve it:

1. Find the matrix A corresponding to this linear sytems and putthe equation in matrix form v = Av .

2. Find the eigenvalues and corresponding eigenvectors of A

2 2 Systems of Linear Differential Equations

Using eigenvalues and eigenfunctions to solve linear firstorder systems

This is an alternative method to the annihilator method whichexplains the nature of the solution obtained.

Example 2: Solve:

x = 5x + 2y

y = x 4y

Here is how we solve it:

1. Find the matrix A corresponding to this linear sytems and putthe equation in matrix form v = Av .

2. Find the eigenvalues and corresponding eigenvectors of A

3. The solution vector

v = c1e1tv1 + c2e

2tv2

2 2 Systems of Linear Differential Equations

Using eigenvalues and eigenfunctions to solve linear firstorder systems

This is an alternative method to the annihilator method whichexplains the nature of the solution obtained.

Example 2: Solve:

x = 5x + 2y

y = x 4y

Here is how we solve it:

1. Find the matrix A corresponding to this linear sytems and putthe equation in matrix form v = Av .

2. Find the eigenvalues and corresponding eigenvectors of A

3. The solution vector

v = c1e1tv1 + c2e

2tv2

2 2 Systems of Linear Differential Equations

Example 2, contd

We already did half of the work in Example 1. From there, we

know A =

(5 21 4

).

2 2 Systems of Linear Differential Equations

Example 2, contd

We already did half of the work in Example 1. From there, we

know A =

(5 21 4

).

The eigenvalues and corresponding eigenvectors are: 1 = 3,

v1 =

(11

)

2 2 Systems of Linear Differential Equations

Example 2, contd

We already did half of the work in Example 1. From there, we

know A =

(5 21 4

).

The eigenvalues and corresponding eigenvectors are: 1 = 3,

v1 =

(11

)

and 2 = 6,v2 =

(21

).

2 2 Systems of Linear Differential Equations

Example 2, contd

We already did half of the work in Example 1. From there, we

know A =

(5 21 4

).

The eigenvalues and corresponding eigenvectors are: 1 = 3,

v1 =

(11

)

and 2 = 6,v2 =

(21

).

Therefore the solution vector is given by:

v = c1e3t

(11

)+ c2e

6t

(21

)

2 2 Systems of Linear Differential Equations

Example 2, contd

We already did half of the work in Example 1. From there, we

know A =

(5 21 4

).

The eigenvalues and corresponding eigenvectors are: 1 = 3,

v1 =

(11

)

and 2 = 6,v2 =

(21

).

Therefore the solution vector is given by:

v = c1e3t

(11

)+ c2e

6t

(21

)

This means: x(t) = c1e3t 2c2e

6t and y(t) = c1e3t + c2e

6t .

2 2 Systems of Linear Differential Equations

Example 2, contd

Lets graph this using pplane(http://math.rice.edu/dfield/dfpp.html). What do you observe?

2 2 Systems of Linear Differential Equations

Example 3

Find the eigenvalues and corresponding eigenvectors of the matrix

A =

(3 44 3

)and use them to write down the solution to

x = 3x + 4y

y = 4x + 3y

2 2 Systems of Linear Differential Equations

Example 3

Find the eigenvalues and corresponding eigenvectors of the matrix

A =

(3 44 3

)and use them to write down the solution to

x = 3x + 4y

y = 4x + 3y

Make sure to plot the phase plane.

2 2 Systems of Linear Differential Equations

Example 3

Find the eigenvalues and corresponding eigenvectors of the matrix

A =

(3 44 3

)and use them to write down the solution to

x = 3x + 4y

y = 4x + 3y

Make sure to plot the phase plane.

2 2 Systems of Linear Differential Equations

Answer to Example 3

The eigenvalues and corresponding eigenvectors are: 1 = 7,

v1 =

(11

)

2 2 Systems of Linear Differential Equations

Answer to Example 3

The eigenvalues and corresponding eigenvectors are: 1 = 7,

v1 =

(11

)

and 2 = 1,v2 =

(11

).

2 2 Systems of Linear Differential Equations

Answer to Example 3

The eigenvalues and corresponding eigenvectors are: 1 = 7,

v1 =

(11

)

and 2 = 1,v2 =

(11

).

Therefore the solution vector is given by:

v = c1e7t

(11

)+ c2e

t

(11

)

2 2 Systems of Linear Differential Equations

Answer to Example 3

The eigenvalues and corresponding eigenvectors are: 1 = 7,

v1 =

(11

)

and 2 = 1,v2 =

(11

).

Therefore the solution vector is given by:

v = c1e7t

(11

)+ c2e

t

(11

)

This means: x(t) = c1e7t c2e

t and y(t) = c1e7t + c2e

t .

2 2 Systems of Linear Differential Equations

Example 3, Phase Plane

2 2 Systems of Linear Differential Equations