Diff Equation 7 Sys 2012 FALL

8/12/2019 Diff Equation 7 Sys 2012 FALL

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Outline Eigenvalues and Eigenvectors

Homogeneous Systems of Equationswith Constant Coefficients

Nonhomogeneous Systems ofEquations: Method of Variation of

Parameters

Reduction of order


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Nth Order ODEs reduces to a Linear 1st

Order Systems

An arbitrary nth order equation

Is transformed into a system of n first order

equations, by defining

)1()( ,,,,, = nn yyyytFy K

)1(

321 ,,,, ==== nn yxyxyxyx K

1 2

2 3

1 2( , , , )n n

x x

x x

x F t x x x

=

=

=M

K


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Practical Importance:

High-order

System

Converted

First-order

System

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

Linear Systems of Differential Equations


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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

xxxx

=

=

==

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


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Example:

yxy

yxx

22'26'

=+=

3

2

22'

26'

yxy

yxx

=+=

zxzzyxy

zyxx

= +=

+=

''

2'

Linear system

yzxzzyxy

zxyxx

=+=

+=

''

2'

zexz

zyxy

zytxx

t=+=

+=

'

'

2' 2



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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

XX =' AXX =' XX ='



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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



= + + + +

= + + + +

= + + + +

KK

KK

M

KK

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

= +

K

K

M M M M M M

L

System of linear first-order DE

'X A X F= +


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'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(*)


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Example:

yxy

tyxx

22'26'

=++=

zxz

zyxy

zyxx

=

+=

+=

'

'

2'

Homog and Non-homg

ttezexz

zyxy

tzytxx

2

22

'

'

2'

+=

+=

+=

+

=

022

26

'

' t

y

x

y

x

=

z

y

x

z

y

x

101

111

121

'

'

'

+

=

tte

t

z

y

x

e

t

z

y

x

2

22

0

01

111

121

'

'

'

homognon

'

+= FAXX

homog

' AXX =

homognon

'

+= FAXX



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System of Equations: First-Order Linear

Differential Equations - Substitution

Consider the 2x2 system of linear homogeneous differential

equations (with constant coefficients)

x'(t) = ax(t) + by(t)y'(t) = cx(t) + dy(t)

We can solve this system using what we know:

1. Isolatey(t) in the first equation =>y(t) =x'(t)/b

ax(t)/b.

2. Differentiate thisy(t) equation =>y'(t) =x"(t)/b

ax'(t)/b.

3. Solve forx(t) andy'(t)

=>x"(t)

(a + d)x'(t) + (ad

bc)x(t) = 0.

4. Go back to step 1. Solve fory(t) in terms ofx'(t) andx(t).


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Eigenvalue and Eigenvector:

Eigenvalues and Eigenvectors

The number is said to be an eigenvalue of the nxn matrix Aprovided there exists a nonzero vector v such that

v is called an eigenvector of the matrix A. v is associated with theeigenvalue

vAv =

Characteristic Equation: ( )P A I = It is a polynomial of order n. ( A is nxn)


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Sec(7.1+7.2): First-order Systems

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




= + + + +

= + + + +

= + + + +

KK

KK

M

KK

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

= +

K

K

M M M M M M

L

System of linear first-order DE

'X A X F= +


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First-order homogeneous Systems

THM ( Wronskian)

solutionsnareLet 21 n,X,, XX L

AXX ='Consider the sys of DE: (*)

1 2( , , , ) 0nW X X X Ltindependenlinearly

21 n,X,, XX L


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THM ( general solution for Homog)

indeplinare21 n,X,, XX L

XX ='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 (*)

(*)

nnXcXcXctX +++= L2211)(solutionsare21 n,X,, XX L

The general sol for (*) is

Example:

=

y

x

y

x

13

24

'

'

Solve IVP

(*)

=

1

1

)0(

)0(

y

x

First-order homogeneous Systems


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How to solve the system of DE

System of Linear First-Order DE

(constant Coeff) 'X AX=

Distinct realEigenvalues

repeated realEigenvalues

complexEigenvalues

System of Linear First-Order DE

(Non-homog)

'X AX F= +

Variation of

Parameters

E

igenvalueM

ethod


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Eigenvalue Method

Our Goal: 'X AX=Solve the Homog linear system

Method:Find all eigenvalues of the matrix A1 n ,,, 21 L

Distinct real

Eigenvalues

repeated real

Eigenvalues

complex

Eigenvalues

Solution: 1) Find n linearly independent solutions

2) The general solution is: their linear combination

nXXX ,,, 21 L


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'X AX=

Method: Find all eigenvalues of the matrix A1n ,,, 21 L Distinct real Eigenvalues

Find all eigenvectors of the matrix A2nvvv ,,, 21 L

N-lin. Independent solutions are:

3 111 veX t= 22

2 veX t= ntn veX n=LLL

The general solution is:4nnXcXcXctX +++= L2211)(

Eigenvalue Method


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'X AX=Method:

Find all eigenvalues of the matrix A1i =2,1 Complex conjugate eigenvalues

Find an eigenvector for21v

solution is:

31

1 veX t=

Two-lin independent solutions are:4

i +=1Complex vector

1)]sin()[cos( vtite t +=1

)(ve

ti+=

)real(1 XX = )Imag(2 XX =

Two-lin independent solutions are:52211)( XcXctX +=

Eigenvalue Method


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'X AX=Method:Find all eigenvalues of the matrix A1

i =2,1 Complex conjugate eigenvalues

Find an eigenvector for

2iBBv 211 +=

Two-lin independent solutions are:

3

i +=1

Two-lin independent solutions are:4

2211

)( XcXctX +=

1 1 2

2 2 1

[ cos sin ]

[ cos sin ]

t

t

X B t B t e

X B t B t e

=

= +

Eigenvalue Method


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Fundamental Matrices Suppose that x(1)(t),, x(n)(t) form a fundamental set of

solutions for x'

= P(t

)x on 0),

Ifx < 0,

2 3 4 0x y xy y + =

/ lndx x x=/ ln( )dx x x=

( )

3ln( ) 32 2

2 4 4

2 2

( )

( )

1 ln

xe x

y x x dx x dxx x

x dx x xx

= =

= =

Example 2

when

x

(, 0)

( ) 2 21 2 lny x c x c x x= +


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Case

of the Higher Order Linear DE

( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( 1)

1 1 0( )n n

n na x y x a x y x a x y x a x y g x

+ + + + =L

Solution of the associated homogeneous equation:

1 1 2 2 3 3( ) ( ) ( ) ( )c n ny c y x c y x c y x c y x= + + + +LL

Theparticular solution

is assumed as:

1 1 2 2 3 3( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )p n ny u x y x u x y x u x y x u x y x= + + + +LL

( ) kk

Wu x

W = ( ) ( )k ku x u x dx=


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( ) k

k

W

u x W =

1 2 3

1 2 3

1 2 3

( 1) ( 1) ( 1) ( 1)

1 2 3

n

n

n

n n n n

n

y y y y

y y y y

W y y y y

y y y y

=

L

L

L

M M M O M

L

1 2 1 1

1 2 1 1

( 2) ( 2) ( 2) ( 2) ( 2)

1 2 1 1

( 1) ( 1) ( 1) ( 1) ( 1)

1 2 1 1

0

0

0

( )

k k n

k k n

k

n n n n n

k k n

n n n n n

k k n

y y y y y

y y y y y

W

y y y y y

y y y f x y y

+

+

+

+

=

L L

L L

M M O M M M O M

L L

L L

( ) ( ) ( )/ nf x g x a x=


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0

0

0

( )f x

MWk: replace the kth

column of Wby

For example, when n = 3,

2 3

1 2 3

2 3

0

0

( )

y y

W y y

f x y y

=

1 3

2 1 3

1 3

0

0

( )

y y

W y y

y f x y

=

1 2

3 1 2

1 2

0

0

( )

y y

W y y

y y f x

=

( ) ( )( )ng xf xa x=


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Process

of the Higher Order Case

Step 2-1 standard form

Step 2-2 Calculate W, W1

, W2

, ., Wn

11

Wu

W

= 22W

uW

=Step 2-3

Step 2-4 .( )1 1u u x dx= ( )2 2u u x dx=

( ) ( ) ( ) ( ) ( ) ( ) ( )1 1 2 2p n ny x u x y x u x y x u x y x= + + +LLStep 2-5

( ) ( )

( ) ( )

( )( )

( )( )

( )( )

( ) ( 1)1 1 0( )n nn

n n n n

a x a x a x g xy x y x y x y

a x a x a x a x + + + + =L

nn

Wu

W

=

( )n nu u x dx=


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Non-homogeneous Cauchy-Euler

Equation

( ) ( ) ( )( ) 1 ( 1)1 1 0( )n n n n

n na x y x a x y x a xy x a y g x

+ + + + =L

not constant coefficients

but the coefficients ofy(k)

(x) have the form ofak is some constantk

ka x

associated homogeneous

equation

particular solution

( ) ( )( ) 1 ( 1)1

1 0( ) 0

n n n n

n na x y x a x y x

a xy x a y

+ +

+ + =L

k


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Guess the solution asy(x) =xm

, then

Associated homogeneous equation of the Cauchy-Euler equation( ) ( )( ) 1 ( 1)1 1 0( ) 0

n n n n

n na x y x a x y x a xy x a y

+ + + + =L

( )

( )

( )

1 11

2 2

2

1

1

0

( 1)( 2) 1

( 1)( 2) 2

( 1)( 2) 3

0

n m n

n

n m nn

n m n

n

m

m

a x m m m m n x

a x m m m m n x

a x m m m m n x

a xmx

a x

+

+

+ +

+ + + +

+

+ =

LL

LL

LL

M


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( )

( )( )

1

2

1

0

( 1)( 2) 1

( 1)( 2) 2( 1)( 2) 3

0

n

n

n

a m m m m n

a m m m m na m m m m n

a m

a

+

+ ++ +

++ =

LL

LL

LL

M

auxiliary function

with constant

coefficient

0 0 0 02 1ln (ln ) (ln, , , ),m m m m k

x x x x x x x

LL

( ) ( ) ( )( )

( ) ( ) ( )

( )

2

1

2

1

, , , ,cos ln cos ln ln cos ln (ln )

cos ln (ln )

sin ln sin ln ln sin ln (ln )

sin ln (ln

, , , ,

)

k

k

x x x x x x x x

x x x

x x x x x x x x

x x x

LL

LL

31 2, , , , km mm m

x x x xLL

A.

B.

C.


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Example 1

(text page 164)

( )2

2 ( ) 4 0x y x xy x y =

Example 2 (text page 164)( )24 8 ( ) 0x y x xy x y + + =

Example 3

(text page 165)

( )24 17 0x y x y + = ( )1 1y = ( )112

y =


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Example 4

( ) ( )

3 25 7 ( ) 8 0x y x x y x xy x y + + + =

( )( )22 4 0m m+ + =

( )( ) ( )1 2 5 1 7 8 0m m m m m m + + + =

auxiliary function

3 2 23 2 5 5 7 8 0m m m m m m + + + + =3 22 4 8 0m m m+ + + =


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Example 5

( )2 43 ( ) 3 2 xx y x xy x y x e + =

auxiliary function

( )1 3 3 0m m m + =3

1 2cy c x c x= +

2 4 3 0m m + =

2 3m =1

1m =

Step 1

solution of the associated homogeneous equation

Step 2-2

Particular solution

3

3

22

1 3

x xW x

x

= =

3

5

1 22

02

2 3xxx

Wx

exe

x= =

211

xWu x e

W= =

2

3

2

02

1 2

x

xx e

xW x e= =

22

xWu e

W= =Step 2-3


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2 2xu u dx e= =

2

1 1 2 2x x x

u u dx x e xe e= = + Step 2-4

2

1 1 2 2 2 2x x

py u y u y x e xe= + =

Step 3 3 21 2 2 2

x xy c x c x x e xe= + +

Step 2-5


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N-th order linear DE

Constant Coeff variable Coeff

Homog(find yp)NON-HOMOG

(find yp)

Variational of

Parameters

Cauchy-Euler In General

Power Series

Series Solutions of Linear Equations


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Sec 6.1: Solution about Ordinary Points

Sec 6.2: Solution about Singular Points

Sec 6.1.1: Review of Power Series

Sec 6.1.2: Power Series Solutions


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Power SeriesRepresenting Function by series

( ) ( ) ( ) ( )

21 2

0

2

1 2

0

power series

An expression of the form

is a .

An expression of the form

i

centered at

s

0

a

n nn o n

n

n n

n o n

n

c x c c x c x c x

c x a c c x a c x a c x

x

a

=

=

= + + + + +

= + + + + +

=

Power SDefinition eries

L L

L L

( ). The term is

the

power series cent

; the number of is the

ered at

.

n

nc xa a

a

x =

nth terms center

Power Series


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( )

=

>

0

There are three possibilities for with respect to

convergence:

1. There is a positive number such that the series diverges fo

r

n

n

na x a

R

x a R

The Convergence Theorem for Power Seriesheore 5

0

Identity Property


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Identity Property

0

( ) 0nnn

c x a

=

=If0nc =

:Example

1( ) xf x e=

2 ( ) sinx x=

3 ( ) cosx x=

2 3

1( ) 11! 2! 3!

x x xx = + + + +KK

3 5

2 ( )3! 5!

x xx x= + KK

2 4

3( ) 1 2! 4!

x x

f x = + KK

R =

Any polynomial such as 2 33 2 4 6x x x+ +

Definition: ( )f x Is analytic at 0x

IF: Can be represented by power series centerd at 0x( )x

(i.e) 00

( ) ( )nnn

f x c x x

=

= with R>0 0x x R

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Diff Equation 7 Sys 2012 FALL