Outline

1 First order systems and applications

2 Matrices and linear systems

3 The eigenvalue method for linear systemsDistinct eigenvaluesComplex eigenvalues

4 Multiple eigenvalue solutions

5 A gallery of solution curves of linear systemsReal eigenvaluesComplex eigenvalues

Samy T. Systems Di�erential equations 45 / 92

Example of matrix with repeated rootMatrix:

A =A

1 ≠11 3

B

Characteristic polynomial:

PA(r) = det(A ≠ r Id) = (r ≠ 2)2

Eigenvalues and eigenvectors:

r = 2, v1 =A

1≠1

B

Remark: r = 2 is a double eigenvalue, with 1 eigenvector only.

Samy T. Systems Di�erential equations 46 / 92

Generalized eigenvectors with multiplicity 2

System: xÕ = Ax, with A œ R2,2 and det(A) ”= 0

Situation:A has a double eigenvalue rUnique eigenvector v (up to constant factor)

Recipe to find generalized eigenvectors:1 Find v2 such that (A ≠ r Id)2v2 = 0, but not parallel to v2 Compute v1 = (A ≠ r Id)v23 Then v1, v2 are generalized eigenvectors

Samy T. Systems Di�erential equations 47 / 92

Solving systems with multiplicity 2

Situation:We consider the system xÕ = A xA has a double eigenvalue rGeneralized eigenvectors v1, v2

Corresponding fundamental solutions: We get

x1(t) = v1ert

x2(t) = (v1t + v2) ert

Samy T. Systems Di�erential equations 48 / 92

Example with multiplicity 2 (1)

Equation:

xÕ =A

1 ≠11 3

B

x

Eigenvalues and eigenvector:

r = 2 (multiplicity 2), v =A

1≠1

B

Square of a matrix: We have

A ≠ 2 Id =C≠1 ≠11 1

D

, (A ≠ 2Id)2 = 0

Samy T. Systems Di�erential equations 49 / 92

Example with multiplicity 2 (2)

Applying the recipe to find the generalized eigenvectors: We choose

v2 =C10

D

, v1 = (A ≠ 2Id) v2 =C≠11

D

General solution:

x = c1

A≠1

1

B

e2t + c2

A≠1

1

B

te2t + c2

A10

B

e2t

Samy T. Systems Di�erential equations 50 / 92

Example with multiplicity 2 (3)Asymptotic behavior: As t æ Œ

x(t) æ ŒlimtæŒ

x2(t)x1(t) = ≠1, thus slope ƒ ≠1

x(t) does not approach the asymptote

Graph in the x1x2 plane:

Samy T. Systems Di�erential equations 51 / 92

Generalized eigenvectors with multiplicity 3

Situation:A has a triple eigenvalue rUnique eigenvector v (up to constant factor)

Recipe to find generalized eigenvectors:1 Find v3 such that (A ≠ r Id)3v3 = 0, not parallel to v2 Compute v2 = (A ≠ r Id)v33 Compute v1 = (A ≠ r Id)v24 Then v1, v2, v3 are generalized eigenvectors

Samy T. Systems Di�erential equations 52 / 92

Solving systems with multiplicity 3

Situation:We consider the system xÕ = A xA has a triple eigenvalue rGeneralized eigenvectors v1, v2, v3

Corresponding fundamental solutions: We get

x1(t) = v1ert

x2(t) = (v1t + v2) ert

x3(t) =31

2v1t2 + v2t + v3

4ert

Samy T. Systems Di�erential equations 53 / 92

Example with multiplicity 3 (1)

Equation:

xÕ =

S

WU0 1 2

≠5 ≠3 ≠71 0 0

T

XV x

Aim:

Expression of the general solution to this system

Samy T. Systems Di�erential equations 54 / 92

Example with multiplicity 3 (2)

Eigenvalues and eigenvector:

r = ≠1 (multiplicity 3), v =

S

WU11

≠1

T

XV

Samy T. Systems Di�erential equations 55 / 92

Example with multiplicity 3 (3)

Third power computation: We find

(A + Id)3 = 0

Value for v3: We take

v3 =

S

WU100

T

XV

Samy T. Systems Di�erential equations 56 / 92

Example with multiplicity 3 (4)

Value for v2: We compute

v2 = (A + Id)v3 =

S

WU1 1 2

≠5 ≠2 ≠71 0 1

T

XV

S

WU100

T

XV

We get

v2 =

S

WU1

≠51

T

XV

Samy T. Systems Di�erential equations 57 / 92

Example with multiplicity 3 (5)

Checking value for v1: We compute

u1 = (A + Id)v2 =

S

WU1 1 2

≠5 ≠2 ≠71 0 1

T

XV

S

WU1

≠51

T

XV

We get

v1 =

S

WU≠2≠22

T

XV = ≠2 v

Samy T. Systems Di�erential equations 58 / 92

Example with multiplicity 3 (6)

Fundamental solutions: Recall that

x1(t) = v1e≠t

x2(t) = (v1t + v2) e≠t

x3(t) =31

2v1t2 + v2t + v3

4e≠t

Summarizing values of v1, v2, v3: We have found

v1 =

S

WU≠2≠22

T

XV , v2 =

S

WU1

≠51

T

XV , v3 =

S

WU100

T

XV

Samy T. Systems Di�erential equations 59 / 92

Example with multiplicity 3 (7)

Fundamental solutions in our case: We find

x1(t) =

S

WU≠2≠22

T

XV e≠t

x2(t) =

S

WU≠2t + 1≠2t ≠ 52t + 1

T

XV e≠t

x3(t) =

S

WU≠t2 + t + 1

≠t2 ≠ 5tt2 + t

T

XV e≠t

Samy T. Systems Di�erential equations 60 / 92