Homogeneous Systems of Linear Differential Equations with Constant - - PowerPoint PPT Presentation

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Homogeneous Systems of Linear Differential Equations with Constant - - PowerPoint PPT Presentation

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Overview Complex Eigenvalues An Example Homogeneous Systems of Linear Differential Equations with Constant Coefficients and Complex Eigenvalues Bernd Schr oder logo1 Bernd Schr oder Louisiana Tech University, College of Engineering

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SLIDE 1

logo1 Overview Complex Eigenvalues An Example

Homogeneous Systems of Linear Differential Equations with Constant Coefficients and Complex Eigenvalues

Bernd Schr¨

  • der

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Homogeneous Systems of Linear Differential Equations with Constant Coefficients and Complex Eigenvalues

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SLIDE 2

logo1 Overview Complex Eigenvalues An Example

Systems of Linear Differential Equations with Constant Coefficients and Complex Eigenvalues

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Homogeneous Systems of Linear Differential Equations with Constant Coefficients and Complex Eigenvalues

slide-3
SLIDE 3

logo1 Overview Complex Eigenvalues An Example

Systems of Linear Differential Equations with Constant Coefficients and Complex Eigenvalues

  • 1. These systems are typically written in matrix form as
  • y′ = A
  • y, where A is an n×n matrix and

y is a column vector with n rows.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Homogeneous Systems of Linear Differential Equations with Constant Coefficients and Complex Eigenvalues

slide-4
SLIDE 4

logo1 Overview Complex Eigenvalues An Example

Systems of Linear Differential Equations with Constant Coefficients and Complex Eigenvalues

  • 1. These systems are typically written in matrix form as
  • y′ = A
  • y, where A is an n×n matrix and

y is a column vector with n rows.

  • 2. The theory guarantees that there will always be a set of n

linearly independent solutions {

  • y1,...,

yn}.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Homogeneous Systems of Linear Differential Equations with Constant Coefficients and Complex Eigenvalues

slide-5
SLIDE 5

logo1 Overview Complex Eigenvalues An Example

Systems of Linear Differential Equations with Constant Coefficients and Complex Eigenvalues

  • 1. These systems are typically written in matrix form as
  • y′ = A
  • y, where A is an n×n matrix and

y is a column vector with n rows.

  • 2. The theory guarantees that there will always be a set of n

linearly independent solutions {

  • y1,...,

yn}.

  • 3. Every solution is of the form

y = c1

  • y1 +···+cn
  • yn.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Homogeneous Systems of Linear Differential Equations with Constant Coefficients and Complex Eigenvalues

slide-6
SLIDE 6

logo1 Overview Complex Eigenvalues An Example

Systems of Linear Differential Equations with Constant Coefficients and Complex Eigenvalues

  • 1. These systems are typically written in matrix form as
  • y′ = A
  • y, where A is an n×n matrix and

y is a column vector with n rows.

  • 2. The theory guarantees that there will always be a set of n

linearly independent solutions {

  • y1,...,

yn}.

  • 3. Every solution is of the form

y = c1

  • y1 +···+cn
  • yn.
  • 4. If A = ΦDΦ−1 and

x solves x′ = D

  • x, then

A(Φ

  • x)

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Homogeneous Systems of Linear Differential Equations with Constant Coefficients and Complex Eigenvalues

slide-7
SLIDE 7

logo1 Overview Complex Eigenvalues An Example

Systems of Linear Differential Equations with Constant Coefficients and Complex Eigenvalues

  • 1. These systems are typically written in matrix form as
  • y′ = A
  • y, where A is an n×n matrix and

y is a column vector with n rows.

  • 2. The theory guarantees that there will always be a set of n

linearly independent solutions {

  • y1,...,

yn}.

  • 3. Every solution is of the form

y = c1

  • y1 +···+cn
  • yn.
  • 4. If A = ΦDΦ−1 and

x solves x′ = D

  • x, then

A(Φ

  • x) = ΦDΦ−1(Φ
  • x)

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Homogeneous Systems of Linear Differential Equations with Constant Coefficients and Complex Eigenvalues

slide-8
SLIDE 8

logo1 Overview Complex Eigenvalues An Example

Systems of Linear Differential Equations with Constant Coefficients and Complex Eigenvalues

  • 1. These systems are typically written in matrix form as
  • y′ = A
  • y, where A is an n×n matrix and

y is a column vector with n rows.

  • 2. The theory guarantees that there will always be a set of n

linearly independent solutions {

  • y1,...,

yn}.

  • 3. Every solution is of the form

y = c1

  • y1 +···+cn
  • yn.
  • 4. If A = ΦDΦ−1 and

x solves x′ = D

  • x, then

A(Φ

  • x) = ΦDΦ−1(Φ
  • x) = ΦD
  • x

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Homogeneous Systems of Linear Differential Equations with Constant Coefficients and Complex Eigenvalues

slide-9
SLIDE 9

logo1 Overview Complex Eigenvalues An Example

Systems of Linear Differential Equations with Constant Coefficients and Complex Eigenvalues

  • 1. These systems are typically written in matrix form as
  • y′ = A
  • y, where A is an n×n matrix and

y is a column vector with n rows.

  • 2. The theory guarantees that there will always be a set of n

linearly independent solutions {

  • y1,...,

yn}.

  • 3. Every solution is of the form

y = c1

  • y1 +···+cn
  • yn.
  • 4. If A = ΦDΦ−1 and

x solves x′ = D

  • x, then

A(Φ

  • x) = ΦDΦ−1(Φ
  • x) = ΦD
  • x = Φ
  • x′

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Homogeneous Systems of Linear Differential Equations with Constant Coefficients and Complex Eigenvalues

slide-10
SLIDE 10

logo1 Overview Complex Eigenvalues An Example

Systems of Linear Differential Equations with Constant Coefficients and Complex Eigenvalues

  • 1. These systems are typically written in matrix form as
  • y′ = A
  • y, where A is an n×n matrix and

y is a column vector with n rows.

  • 2. The theory guarantees that there will always be a set of n

linearly independent solutions {

  • y1,...,

yn}.

  • 3. Every solution is of the form

y = c1

  • y1 +···+cn
  • yn.
  • 4. If A = ΦDΦ−1 and

x solves x′ = D

  • x, then

A(Φ

  • x) = ΦDΦ−1(Φ
  • x) = ΦD
  • x = Φ
  • x′ = (Φ
  • x)′,

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Homogeneous Systems of Linear Differential Equations with Constant Coefficients and Complex Eigenvalues

slide-11
SLIDE 11

logo1 Overview Complex Eigenvalues An Example

Systems of Linear Differential Equations with Constant Coefficients and Complex Eigenvalues

  • 1. These systems are typically written in matrix form as
  • y′ = A
  • y, where A is an n×n matrix and

y is a column vector with n rows.

  • 2. The theory guarantees that there will always be a set of n

linearly independent solutions {

  • y1,...,

yn}.

  • 3. Every solution is of the form

y = c1

  • y1 +···+cn
  • yn.
  • 4. If A = ΦDΦ−1 and

x solves x′ = D

  • x, then

A(Φ

  • x) = ΦDΦ−1(Φ
  • x) = ΦD
  • x = Φ
  • x′ = (Φ
  • x)′,

that is, y = Φ

  • x solves

y′ = A

  • y.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Homogeneous Systems of Linear Differential Equations with Constant Coefficients and Complex Eigenvalues

slide-12
SLIDE 12

logo1 Overview Complex Eigenvalues An Example

Systems of Linear Differential Equations with Constant Coefficients and Complex Eigenvalues

  • 1. These systems are typically written in matrix form as
  • y′ = A
  • y, where A is an n×n matrix and

y is a column vector with n rows.

  • 2. The theory guarantees that there will always be a set of n

linearly independent solutions {

  • y1,...,

yn}.

  • 3. Every solution is of the form

y = c1

  • y1 +···+cn
  • yn.
  • 4. If A = ΦDΦ−1 and

x solves x′ = D

  • x, then

A(Φ

  • x) = ΦDΦ−1(Φ
  • x) = ΦD
  • x = Φ
  • x′ = (Φ
  • x)′,

that is, y = Φ

  • x solves

y′ = A

  • y.
  • 5. Conversely, every solution of

y′ = A

  • y can be obtained as

above.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Homogeneous Systems of Linear Differential Equations with Constant Coefficients and Complex Eigenvalues

slide-13
SLIDE 13

logo1 Overview Complex Eigenvalues An Example

Systems of Linear Differential Equations with Constant Coefficients and Complex Eigenvalues

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Homogeneous Systems of Linear Differential Equations with Constant Coefficients and Complex Eigenvalues

slide-14
SLIDE 14

logo1 Overview Complex Eigenvalues An Example

Systems of Linear Differential Equations with Constant Coefficients and Complex Eigenvalues

  • 6. So if we can find a representation A = ΦDΦ−1 so that
  • x′ = D
  • x is easy to solve, then

y′ = A

  • y is also easy to solve.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Homogeneous Systems of Linear Differential Equations with Constant Coefficients and Complex Eigenvalues

slide-15
SLIDE 15

logo1 Overview Complex Eigenvalues An Example

Systems of Linear Differential Equations with Constant Coefficients and Complex Eigenvalues

  • 6. So if we can find a representation A = ΦDΦ−1 so that
  • x′ = D
  • x is easy to solve, then

y′ = A

  • y is also easy to solve.
  • 7. An n×n matrix A is called diagonalizable if and only if

there are a diagonal matrix D =      λ1 ··· λ2 . . . ... . . . ··· λn     , and an invertible matrix Φ so that A = ΦDΦ−1.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Homogeneous Systems of Linear Differential Equations with Constant Coefficients and Complex Eigenvalues

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SLIDE 16

logo1 Overview Complex Eigenvalues An Example

Systems of Linear Differential Equations with Constant Coefficients and Complex Eigenvalues

  • 6. So if we can find a representation A = ΦDΦ−1 so that
  • x′ = D
  • x is easy to solve, then

y′ = A

  • y is also easy to solve.
  • 7. An n×n matrix A is called diagonalizable if and only if

there are a diagonal matrix D =      λ1 ··· λ2 . . . ... . . . ··· λn     , and an invertible matrix Φ so that A = ΦDΦ−1.

  • 8. If D is a diagonal matrix, then the solutions of

x′ = D

  • x are

eλ1t e1,...,eλnt en, because the individual equations are of the form x′

j = λjxj.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Homogeneous Systems of Linear Differential Equations with Constant Coefficients and Complex Eigenvalues

slide-17
SLIDE 17

logo1 Overview Complex Eigenvalues An Example

Systems of Linear Differential Equations with Constant Coefficients and Complex Eigenvalues

  • 6. So if we can find a representation A = ΦDΦ−1 so that
  • x′ = D
  • x is easy to solve, then

y′ = A

  • y is also easy to solve.
  • 7. An n×n matrix A is called diagonalizable if and only if

there are a diagonal matrix D =      λ1 ··· λ2 . . . ... . . . ··· λn     , and an invertible matrix Φ so that A = ΦDΦ−1.

  • 8. If D is a diagonal matrix, then the solutions of

x′ = D

  • x are

eλ1t e1,...,eλnt en, because the individual equations are of the form x′

j = λjxj. (λj can be real or complex.)

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Homogeneous Systems of Linear Differential Equations with Constant Coefficients and Complex Eigenvalues

slide-18
SLIDE 18

logo1 Overview Complex Eigenvalues An Example

Systems of Linear Differential Equations with Constant Coefficients and Complex Eigenvalues

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Homogeneous Systems of Linear Differential Equations with Constant Coefficients and Complex Eigenvalues

slide-19
SLIDE 19

logo1 Overview Complex Eigenvalues An Example

Systems of Linear Differential Equations with Constant Coefficients and Complex Eigenvalues

  • 9. That means, if A = ΦDΦ−1 and D is a diagonal matrix,

then the solution of y′ = A

  • y is

y = eλ1tΦ1 +···+eλntΦn, where Φj denotes the jth column of the matrix Φ.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Homogeneous Systems of Linear Differential Equations with Constant Coefficients and Complex Eigenvalues

slide-20
SLIDE 20

logo1 Overview Complex Eigenvalues An Example

Systems of Linear Differential Equations with Constant Coefficients and Complex Eigenvalues

  • 9. That means, if A = ΦDΦ−1 and D is a diagonal matrix,

then the solution of y′ = A

  • y is

y = eλ1tΦ1 +···+eλntΦn, where Φj denotes the jth column of the matrix Φ.

  • 10. The columns of Φ satisfy

AΦj

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Homogeneous Systems of Linear Differential Equations with Constant Coefficients and Complex Eigenvalues

slide-21
SLIDE 21

logo1 Overview Complex Eigenvalues An Example

Systems of Linear Differential Equations with Constant Coefficients and Complex Eigenvalues

  • 9. That means, if A = ΦDΦ−1 and D is a diagonal matrix,

then the solution of y′ = A

  • y is

y = eλ1tΦ1 +···+eλntΦn, where Φj denotes the jth column of the matrix Φ.

  • 10. The columns of Φ satisfy

AΦj = ΦDΦ−1Φj

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Homogeneous Systems of Linear Differential Equations with Constant Coefficients and Complex Eigenvalues

slide-22
SLIDE 22

logo1 Overview Complex Eigenvalues An Example

Systems of Linear Differential Equations with Constant Coefficients and Complex Eigenvalues

  • 9. That means, if A = ΦDΦ−1 and D is a diagonal matrix,

then the solution of y′ = A

  • y is

y = eλ1tΦ1 +···+eλntΦn, where Φj denotes the jth column of the matrix Φ.

  • 10. The columns of Φ satisfy

AΦj = ΦDΦ−1Φj = ΦD

  • ej

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Homogeneous Systems of Linear Differential Equations with Constant Coefficients and Complex Eigenvalues

slide-23
SLIDE 23

logo1 Overview Complex Eigenvalues An Example

Systems of Linear Differential Equations with Constant Coefficients and Complex Eigenvalues

  • 9. That means, if A = ΦDΦ−1 and D is a diagonal matrix,

then the solution of y′ = A

  • y is

y = eλ1tΦ1 +···+eλntΦn, where Φj denotes the jth column of the matrix Φ.

  • 10. The columns of Φ satisfy

AΦj = ΦDΦ−1Φj = ΦD

  • ej = Φλj
  • ej

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Homogeneous Systems of Linear Differential Equations with Constant Coefficients and Complex Eigenvalues

slide-24
SLIDE 24

logo1 Overview Complex Eigenvalues An Example

Systems of Linear Differential Equations with Constant Coefficients and Complex Eigenvalues

  • 9. That means, if A = ΦDΦ−1 and D is a diagonal matrix,

then the solution of y′ = A

  • y is

y = eλ1tΦ1 +···+eλntΦn, where Φj denotes the jth column of the matrix Φ.

  • 10. The columns of Φ satisfy

AΦj = ΦDΦ−1Φj = ΦD

  • ej = Φλj
  • ej = λjΦj.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Homogeneous Systems of Linear Differential Equations with Constant Coefficients and Complex Eigenvalues

slide-25
SLIDE 25

logo1 Overview Complex Eigenvalues An Example

Systems of Linear Differential Equations with Constant Coefficients and Complex Eigenvalues

  • 9. That means, if A = ΦDΦ−1 and D is a diagonal matrix,

then the solution of y′ = A

  • y is

y = eλ1tΦ1 +···+eλntΦn, where Φj denotes the jth column of the matrix Φ.

  • 10. The columns of Φ satisfy

AΦj = ΦDΦ−1Φj = ΦD

  • ej = Φλj
  • ej = λjΦj.

Nonzero vectors with this property are called

  • eigenvectors. λj is called an eigenvalue.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Homogeneous Systems of Linear Differential Equations with Constant Coefficients and Complex Eigenvalues

slide-26
SLIDE 26

logo1 Overview Complex Eigenvalues An Example

Systems of Linear Differential Equations with Constant Coefficients and Complex Eigenvalues

  • 9. That means, if A = ΦDΦ−1 and D is a diagonal matrix,

then the solution of y′ = A

  • y is

y = eλ1tΦ1 +···+eλntΦn, where Φj denotes the jth column of the matrix Φ.

  • 10. The columns of Φ satisfy

AΦj = ΦDΦ−1Φj = ΦD

  • ej = Φλj
  • ej = λjΦj.

Nonzero vectors with this property are called

  • eigenvectors. λj is called an eigenvalue.
  • 11. Eigenvalues can be computed by solving the equation

det(A−λI) = 0, where I is the identity matrix.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Homogeneous Systems of Linear Differential Equations with Constant Coefficients and Complex Eigenvalues

slide-27
SLIDE 27

logo1 Overview Complex Eigenvalues An Example

Systems of Linear Differential Equations with Constant Coefficients and Complex Eigenvalues

  • 9. That means, if A = ΦDΦ−1 and D is a diagonal matrix,

then the solution of y′ = A

  • y is

y = eλ1tΦ1 +···+eλntΦn, where Φj denotes the jth column of the matrix Φ.

  • 10. The columns of Φ satisfy

AΦj = ΦDΦ−1Φj = ΦD

  • ej = Φλj
  • ej = λjΦj.

Nonzero vectors with this property are called

  • eigenvectors. λj is called an eigenvalue.
  • 11. Eigenvalues can be computed by solving the equation

det(A−λI) = 0, where I is the identity matrix.

  • 12. Corresponding eigenvectors are computed with systems of

equations A

  • v = λj
  • v

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Homogeneous Systems of Linear Differential Equations with Constant Coefficients and Complex Eigenvalues

slide-28
SLIDE 28

logo1 Overview Complex Eigenvalues An Example

Systems of Linear Differential Equations with Constant Coefficients and Complex Eigenvalues

  • 9. That means, if A = ΦDΦ−1 and D is a diagonal matrix,

then the solution of y′ = A

  • y is

y = eλ1tΦ1 +···+eλntΦn, where Φj denotes the jth column of the matrix Φ.

  • 10. The columns of Φ satisfy

AΦj = ΦDΦ−1Φj = ΦD

  • ej = Φλj
  • ej = λjΦj.

Nonzero vectors with this property are called

  • eigenvectors. λj is called an eigenvalue.
  • 11. Eigenvalues can be computed by solving the equation

det(A−λI) = 0, where I is the identity matrix.

  • 12. Corresponding eigenvectors are computed with systems of

equations A

  • v = λj
  • v or, more commonly (A−λjI)
  • v =

0.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Homogeneous Systems of Linear Differential Equations with Constant Coefficients and Complex Eigenvalues

slide-29
SLIDE 29

logo1 Overview Complex Eigenvalues An Example

Systems of Linear Differential Equations with Constant Coefficients and Complex Eigenvalues

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Homogeneous Systems of Linear Differential Equations with Constant Coefficients and Complex Eigenvalues

slide-30
SLIDE 30

logo1 Overview Complex Eigenvalues An Example

Systems of Linear Differential Equations with Constant Coefficients and Complex Eigenvalues

  • 13. But if an eigenvalue is complex, we might still want to

have a real-valued solution.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Homogeneous Systems of Linear Differential Equations with Constant Coefficients and Complex Eigenvalues

slide-31
SLIDE 31

logo1 Overview Complex Eigenvalues An Example

Systems of Linear Differential Equations with Constant Coefficients and Complex Eigenvalues

  • 13. But if an eigenvalue is complex, we might still want to

have a real-valued solution.

  • 14. Let

v be an eigenvector of A for the eigenvalue λ = α +iβ with β = 0.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Homogeneous Systems of Linear Differential Equations with Constant Coefficients and Complex Eigenvalues

slide-32
SLIDE 32

logo1 Overview Complex Eigenvalues An Example

Systems of Linear Differential Equations with Constant Coefficients and Complex Eigenvalues

  • 13. But if an eigenvalue is complex, we might still want to

have a real-valued solution.

  • 14. Let

v be an eigenvector of A for the eigenvalue λ = α +iβ with β = 0. Then v is an eigenvector of A for the eigenvalue λ, where v is obtained from v by replacing every component with its complex conjugate.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Homogeneous Systems of Linear Differential Equations with Constant Coefficients and Complex Eigenvalues

slide-33
SLIDE 33

logo1 Overview Complex Eigenvalues An Example

Systems of Linear Differential Equations with Constant Coefficients and Complex Eigenvalues

  • 13. But if an eigenvalue is complex, we might still want to

have a real-valued solution.

  • 14. Let

v be an eigenvector of A for the eigenvalue λ = α +iβ with β = 0. Then v is an eigenvector of A for the eigenvalue λ, where v is obtained from v by replacing every component with its complex conjugate.

  • 15. The functions
  • ℜ(
  • v)eαt cos(βt)−ℑ(
  • v)eαt sin(βt)
  • and
  • ℑ(
  • v)eαt cos(βt)+ℜ(
  • v)eαt sin(βt)
  • are two linearly

independent real solutions of the system of linear differential equations y′ = A

  • y. ℜ(·) and ℑ(·) denote the

real and imaginary parts of the vector, taken componentwise.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Homogeneous Systems of Linear Differential Equations with Constant Coefficients and Complex Eigenvalues

slide-34
SLIDE 34

logo1 Overview Complex Eigenvalues An Example

Solve the System y′ = 3 1 −4 3

  • y

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Homogeneous Systems of Linear Differential Equations with Constant Coefficients and Complex Eigenvalues

slide-35
SLIDE 35

logo1 Overview Complex Eigenvalues An Example

Solve the System y′ = 3 1 −4 3

  • y

det 3−λ 1 −4 3−λ

  • Bernd Schr¨
  • der

Louisiana Tech University, College of Engineering and Science Homogeneous Systems of Linear Differential Equations with Constant Coefficients and Complex Eigenvalues

slide-36
SLIDE 36

logo1 Overview Complex Eigenvalues An Example

Solve the System y′ = 3 1 −4 3

  • y

det 3−λ 1 −4 3−λ

  • =

(3−λ)(3−λ)

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Homogeneous Systems of Linear Differential Equations with Constant Coefficients and Complex Eigenvalues

slide-37
SLIDE 37

logo1 Overview Complex Eigenvalues An Example

Solve the System y′ = 3 1 −4 3

  • y

det 3−λ 1 −4 3−λ

  • =

(3−λ)(3−λ)−1·(−4)

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Homogeneous Systems of Linear Differential Equations with Constant Coefficients and Complex Eigenvalues

slide-38
SLIDE 38

logo1 Overview Complex Eigenvalues An Example

Solve the System y′ = 3 1 −4 3

  • y

det 3−λ 1 −4 3−λ

  • =

(3−λ)(3−λ)−1·(−4) = 9−6λ +λ 2 +4

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Homogeneous Systems of Linear Differential Equations with Constant Coefficients and Complex Eigenvalues

slide-39
SLIDE 39

logo1 Overview Complex Eigenvalues An Example

Solve the System y′ = 3 1 −4 3

  • y

det 3−λ 1 −4 3−λ

  • =

(3−λ)(3−λ)−1·(−4) = 9−6λ +λ 2 +4 = λ 2 −6λ +13

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Homogeneous Systems of Linear Differential Equations with Constant Coefficients and Complex Eigenvalues

slide-40
SLIDE 40

logo1 Overview Complex Eigenvalues An Example

Solve the System y′ = 3 1 −4 3

  • y

det 3−λ 1 −4 3−λ

  • =

(3−λ)(3−λ)−1·(−4) = 9−6λ +λ 2 +4 = λ 2 −6λ +13 λ1,2 = −(−6)± √ 36−4·13 2

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Homogeneous Systems of Linear Differential Equations with Constant Coefficients and Complex Eigenvalues

slide-41
SLIDE 41

logo1 Overview Complex Eigenvalues An Example

Solve the System y′ = 3 1 −4 3

  • y

det 3−λ 1 −4 3−λ

  • =

(3−λ)(3−λ)−1·(−4) = 9−6λ +λ 2 +4 = λ 2 −6λ +13 λ1,2 = −(−6)± √ 36−4·13 2 = 6±4i 2

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Homogeneous Systems of Linear Differential Equations with Constant Coefficients and Complex Eigenvalues

slide-42
SLIDE 42

logo1 Overview Complex Eigenvalues An Example

Solve the System y′ = 3 1 −4 3

  • y

det 3−λ 1 −4 3−λ

  • =

(3−λ)(3−λ)−1·(−4) = 9−6λ +λ 2 +4 = λ 2 −6λ +13 λ1,2 = −(−6)± √ 36−4·13 2 = 6±4i 2 = 3±2i

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Homogeneous Systems of Linear Differential Equations with Constant Coefficients and Complex Eigenvalues

slide-43
SLIDE 43

logo1 Overview Complex Eigenvalues An Example

Eigenvector for λ = 3+2i

3−(3+2i) 1 −4 3−(3+2i) v1 v2

  • =
  • Bernd Schr¨
  • der

Louisiana Tech University, College of Engineering and Science Homogeneous Systems of Linear Differential Equations with Constant Coefficients and Complex Eigenvalues

slide-44
SLIDE 44

logo1 Overview Complex Eigenvalues An Example

Eigenvector for λ = 3+2i

3−(3+2i) 1 −4 3−(3+2i) v1 v2

  • =
  • −2iv1

+ 1v2 = −4v1 − 2iv2 =

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Homogeneous Systems of Linear Differential Equations with Constant Coefficients and Complex Eigenvalues

slide-45
SLIDE 45

logo1 Overview Complex Eigenvalues An Example

Eigenvector for λ = 3+2i

3−(3+2i) 1 −4 3−(3+2i) v1 v2

  • =
  • −2iv1

+ 1v2 = −4v1 − 2iv2 = v2 = 2iv1

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Homogeneous Systems of Linear Differential Equations with Constant Coefficients and Complex Eigenvalues

slide-46
SLIDE 46

logo1 Overview Complex Eigenvalues An Example

Eigenvector for λ = 3+2i

3−(3+2i) 1 −4 3−(3+2i) v1 v2

  • =
  • −2iv1

+ 1v2 = −4v1 − 2iv2 = v2 = 2iv1, v1 := 1

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Homogeneous Systems of Linear Differential Equations with Constant Coefficients and Complex Eigenvalues

slide-47
SLIDE 47

logo1 Overview Complex Eigenvalues An Example

Eigenvector for λ = 3+2i

3−(3+2i) 1 −4 3−(3+2i) v1 v2

  • =
  • −2iv1

+ 1v2 = −4v1 − 2iv2 = v2 = 2iv1, v1 := 1, v2 = 2i

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Homogeneous Systems of Linear Differential Equations with Constant Coefficients and Complex Eigenvalues

slide-48
SLIDE 48

logo1 Overview Complex Eigenvalues An Example

Eigenvector for λ = 3+2i

3−(3+2i) 1 −4 3−(3+2i) v1 v2

  • =
  • −2iv1

+ 1v2 = −4v1 − 2iv2 = v2 = 2iv1, v1 := 1, v2 = 2i, v = 1 2i

  • .

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Homogeneous Systems of Linear Differential Equations with Constant Coefficients and Complex Eigenvalues

slide-49
SLIDE 49

logo1 Overview Complex Eigenvalues An Example

Eigenvector for λ = 3+2i

3−(3+2i) 1 −4 3−(3+2i) v1 v2

  • =
  • −2iv1

+ 1v2 = −4v1 − 2iv2 = v2 = 2iv1, v1 := 1, v2 = 2i, v = 1 2i

  • .

Check:

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Homogeneous Systems of Linear Differential Equations with Constant Coefficients and Complex Eigenvalues

slide-50
SLIDE 50

logo1 Overview Complex Eigenvalues An Example

Eigenvector for λ = 3+2i

3−(3+2i) 1 −4 3−(3+2i) v1 v2

  • =
  • −2iv1

+ 1v2 = −4v1 − 2iv2 = v2 = 2iv1, v1 := 1, v2 = 2i, v = 1 2i

  • .

Check: 3 1 −4 3 1 2i

  • Bernd Schr¨
  • der

Louisiana Tech University, College of Engineering and Science Homogeneous Systems of Linear Differential Equations with Constant Coefficients and Complex Eigenvalues

slide-51
SLIDE 51

logo1 Overview Complex Eigenvalues An Example

Eigenvector for λ = 3+2i

3−(3+2i) 1 −4 3−(3+2i) v1 v2

  • =
  • −2iv1

+ 1v2 = −4v1 − 2iv2 = v2 = 2iv1, v1 := 1, v2 = 2i, v = 1 2i

  • .

Check: 3 1 −4 3 1 2i

  • =

3+2i −4+6i

  • Bernd Schr¨
  • der

Louisiana Tech University, College of Engineering and Science Homogeneous Systems of Linear Differential Equations with Constant Coefficients and Complex Eigenvalues

slide-52
SLIDE 52

logo1 Overview Complex Eigenvalues An Example

Eigenvector for λ = 3+2i

3−(3+2i) 1 −4 3−(3+2i) v1 v2

  • =
  • −2iv1

+ 1v2 = −4v1 − 2iv2 = v2 = 2iv1, v1 := 1, v2 = 2i, v = 1 2i

  • .

Check: 3 1 −4 3 1 2i

  • =

3+2i −4+6i

  • = (3+2i)

1 2i

  • Bernd Schr¨
  • der

Louisiana Tech University, College of Engineering and Science Homogeneous Systems of Linear Differential Equations with Constant Coefficients and Complex Eigenvalues

slide-53
SLIDE 53

logo1 Overview Complex Eigenvalues An Example

Eigenvector for λ = 3+2i

3−(3+2i) 1 −4 3−(3+2i) v1 v2

  • =
  • −2iv1

+ 1v2 = −4v1 − 2iv2 = v2 = 2iv1, v1 := 1, v2 = 2i, v = 1 2i

  • .

Check: 3 1 −4 3 1 2i

  • =

3+2i −4+6i

  • = (3+2i)

1 2i

  • √

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Homogeneous Systems of Linear Differential Equations with Constant Coefficients and Complex Eigenvalues

slide-54
SLIDE 54

logo1 Overview Complex Eigenvalues An Example

General Solution of the System y′ = 3 1 −4 3

  • y

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Homogeneous Systems of Linear Differential Equations with Constant Coefficients and Complex Eigenvalues

slide-55
SLIDE 55

logo1 Overview Complex Eigenvalues An Example

General Solution of the System y′ = 3 1 −4 3

  • y
  • y

=

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Homogeneous Systems of Linear Differential Equations with Constant Coefficients and Complex Eigenvalues

slide-56
SLIDE 56

logo1 Overview Complex Eigenvalues An Example

General Solution of the System y′ = 3 1 −4 3

  • y
  • y

= c1 1

  • e3t cos(2t)

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Homogeneous Systems of Linear Differential Equations with Constant Coefficients and Complex Eigenvalues

slide-57
SLIDE 57

logo1 Overview Complex Eigenvalues An Example

General Solution of the System y′ = 3 1 −4 3

  • y
  • y

= c1 1

  • e3t cos(2t)−

2

  • e3t sin(2t)
  • Bernd Schr¨
  • der

Louisiana Tech University, College of Engineering and Science Homogeneous Systems of Linear Differential Equations with Constant Coefficients and Complex Eigenvalues

slide-58
SLIDE 58

logo1 Overview Complex Eigenvalues An Example

General Solution of the System y′ = 3 1 −4 3

  • y
  • y

= c1 1

  • e3t cos(2t)−

2

  • e3t sin(2t)
  • +c2

2

  • e3t cos(2t)

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Homogeneous Systems of Linear Differential Equations with Constant Coefficients and Complex Eigenvalues

slide-59
SLIDE 59

logo1 Overview Complex Eigenvalues An Example

General Solution of the System y′ = 3 1 −4 3

  • y
  • y

= c1 1

  • e3t cos(2t)−

2

  • e3t sin(2t)
  • +c2

2

  • e3t cos(2t)+

1

  • e3t sin(2t)
  • Bernd Schr¨
  • der

Louisiana Tech University, College of Engineering and Science Homogeneous Systems of Linear Differential Equations with Constant Coefficients and Complex Eigenvalues