Ch7: Linear Systems of Differential Equations

Example: System of DE

Sec(7.1+7.2): First-order Systems

)3sin(4022''

26''

tyxy

yxx

Independent variable: tdependent variables: x, y

2ed order

)(

)(

ty

tx

Solutions

Example:

yxy

yxx

22'

26'

First-order

ytxy

ytxx222'''

26''

3ed-order zxz

zyxy

zyxx

'

'

2'

First-order

Order of the system

Ch7: Linear Systems of Differential Equations

Example:

yxy

yxx

22'

26'

3

2

22'

26'

yxy

yxx

zxz

zyxy

zyxx

'

'

2'

Linear system

yzxz

zyxy

zxyxx

'

'

2'

zexz

zyxy

zytxx

t

'

'

2' 2

Sec(7.1+7.2): First-order Systems

Ch7: Linear Systems of Differential Equations

Example:

yxy

yxx

22'

26'

zxz

zyxy

zyxx

'

'

2'Matrix Form

zexz

zyxy

zytxx

t

'

'

2' 2

y

x

y

x

22

26

'

'

z

y

x

z

y

x

101

111

121

'

'

'

z

y

x

e

t

z

y

x

t01

111

121

'

'

' 2

AXX ' AXX ' AXX '

Sec(7.1+7.2): First-order Systems

Ch7: Linear Systems of Differential Equations

Example:

yxy

tyxx

22'

26'

zxz

zyxy

zyxx

'

'

2'

Homog and Non-homg

tt ezexz

zyxy

tzytxx

2

22

'

'

2'

022

26

'

' t

y

x

y

x

z

y

x

z

y

x

101

111

121

'

'

'

tt e

t

z

y

x

e

t

z

y

x

2

22

0

01

111

121

'

'

'

homognon

'

FAXX

homog

' AXX

homognon

'

FAXX

Sec(7.1+7.2): First-order Systems

Ch7: Linear Systems of Differential Equations

t

t

ty

tx

sin

cos2

)(

)(

Solution

Example:

xy

yx

2

1'

2'

0

2

)0(

)0(

y

xIVP

Practical Importance:

High-orderSystem Converted

First-orderSystem

Example: tyyyy sin5'2''3'''

''

'

3

2

1

yx

yx

yx

''''

'''

''

3

2

1

yx

yx

yx

txxxx

xx

xx

sin523'

'

'

1233

32

21

First-order System

Sec(7.1+7.2): First-order Systems

Ch7: Linear Systems of Differential Equations

Example:

'

'

4

3

2

1

yx

yx

xx

xx

Transform into first-order system

yxy

xyx

)1(''

)1(''

314

43

132

21

)1('

'

)1('

'

xxx

xx

xxx

xx

Example:

'

'

4

3

2

1

yx

yx

xx

xx

Transform into first-order system

tyxy

yxx

''

''

txxx

xx

xxx

xx

314

43

312

21

'

'

'

'

tx

x

x

x

x

x

x

x

0

0

0

0101

1000

0101

0010

'

'

'

'

4

3

2

1

4

3

2

1

Sec(7.1+7.2): First-order Systems

Ch7: Linear Systems of Differential Equations

Example: Consider the first-order linear system of DE

y

x

y

x

13

24

'

'

t

t

t

t

e

etX

e

etX

5

5

22

2

1

2)( ,

3)(

Sec(7.1+7.2): First-order Systems

Verify that the vector functions

are both solutions of (*)

(*)

Ch7: Linear Systems of Differential EquationsSec(7.1+7.2): First-order Systems

Def:1 11 1 12 2 1 1

2 21 1 22 2 2 2

1 1 2 2

( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

n n

n n

n n n nn n n

x t a t x a t x a t x f t

x t a t x a t x a t x f t

x t a t x a t x a t x f t

Matrix Form:

1 11 12 1 1 1

1 21 22 2 1 2

1 1 2 1 1

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

n

n

n n nn

x a t a t a t x f t

x a t a t a t x f t

x a t a t a t x f t

System of linear first-order DE

' X A X F If F=0 homogeneous system

If F 0 non-homogeneous system

Therorem ( Existence of a Unique Solution)

0

all entries of ( ) are cont on

all entries of ( ) are cont on

t I

A t I

F t I

0 0

' ( ) ( ) (*)

( )

X A t X F t

X t X

There exists a unique solution of IVP(*)

Ch7: Linear Systems of Differential EquationsSec(7.1+7.2): First-order Systems

Therorem ( Principle of Superposition)

solutionsn are Let 21 n,X, , XX AXX 'Consider the sys of DE: (*)

solution also 2211 nn XcXc Xc

Example:

y

x

y

x

13

24

'

'

t

t

t

t

e

etX

e

etX

5

5

22

2

1

2)( ,

3)(

are both solutions of (*)

(*)

t

t

t

t

e

ec

e

ectX

5

5

22

2

23

2

3)( solution of (*)

DEF ( Wronskian)

solutionsn are Let 21 n,X, , XX AXX 'Consider the sys of DE: (*)

nnn

n

n

xx

xx

XXXW

1

111

21 ),,,(

their wronskian is the nxn determinant

Example:

y

x

y

x

13

24

'

'

t

t

t

t

e

etX

e

etX

5

5

22

2

1

2)( ,

3)(

Find W(X1,X2)

(*)

Ch7: Linear Systems of Differential EquationsSec(7.1+7.2): First-order Systems

THM ( Wronskian)

solutionsn are Let 21 n,X, , XX AXX 'Consider the sys of DE: (*)

0),,,( 21 nXXXW

Example:

y

x

y

x

13

24

'

'

t

t

t

t

e

etX

e

etX

5

5

22

2

1

2)( ,

3)(

Linearly dependent or independent ??

(*)

tindependenlinearly

21 n,X, , XX

Ch7: Linear Systems of Differential EquationsSec(7.1+7.2): First-order Systems

THM ( general solution for Homog)

indeplin are 21 n,X, , XX

AXX 'Consider the sys of DE: (*)

Example:

y

x

y

x

13

24

'

'

t

t

t

t

e

etX

e

etX

5

5

22

2

1

2)( ,

3)(

Find the general solution for (*)

(*)

nn XcXc XctX 2211)(solutions are 21 n,X, , XX The general sol for (*) is

Example:

y

x

y

x

13

24

'

'

Solve IVP

(*)

1

1

)0(

)0(

y

x

Ch7: Linear Systems of Differential EquationsSec(7.1+7.2): First-order Systems

THM ( general solution for non-Homog)

indeplin are 21 n,X, , XX

FAXX 'Consider the sys of DE: (*)

Example:

t

t

y

x

y

x

3

14

13

24

'

'

t

t

t

t

e

etX

e

etX

5

5

22

2

1

2)( ,

3)(

Find the general solution for (*)

(*)

nnc XcXc XctXwhere 2211)(

(**) solutions are 21 n,X, , XX The general sol for (*) is

Example: Solve IVP

1

1

)0(

)0(

y

x

AXX ' (**)

(*)for sol particular pX)()()( tX tXtX pc

Sol for Homog

0)(

ttX P Particular sol for non-Homog

t

t

y

x

y

x

3

14

13

24

'

'

How to solve the system of DE

System of Linear First-Order DE(constant Coeff) ' X AX

Distinct realEigenvalues

(7.3)

repeated realEigenvalues

(7.5)

complexEigenvalues

(7.3)

System of Linear First-Order DE(Non-homog) ' X AX F

Variation of ParametersE

igen

valu

e M

eth

od