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Section 5.4 (Systems of Linear Differential Equation); Eigenvalues and Eigenvectors July 1, 2009 2 × 2 Systems of Linear Differential Equations
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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
