chapter 5 higher-order linear differntial equations second-order de: second-order linear de: note:...

CHAPTER 5 Higher-Order Linear Differntial Equations

Example

xyexya x sin'3'' ) xyyyyyb tan')(cos3'' ) 2

xyyyc sin'3''4 ) 0'3''4 ) yyyd

Second-order DE:

0)'',',,( yyyxG

Second-order linear DE:

)()(')('')( xFyxCyxByxA Note: A,B,C,F function of x only

Second-order homogeneous linear DE:

0)(')('')( yxCyxByxANote: F(x)=0

0'3'' ) yexye x

Second-order homogeneous linear DE (with constant coefficients):

0''' CyByAyNote: A, B, C, are constants

(all)

(a,c,d,e)

(d,e)

(d)


Example

0'3''4 ) yyyd


0)(')('')( yxCyxByxANote: F(x)=00'3'' ) 2 yyexye x

(d,e)

Example 06'5'' yyyxexy 2

1 )( xexy 32 )( are solutions ?? (verify)

Consider the homogeneous 2ed-order linear DE:

0)(')('')( yxCyxByxALet W = the set of all solutions of (*)

(*)

Let F = the set of all real-valued functions

W is a subspace of F dim(W)=2



Example 06'5'' yyyxexy 2

1 )( xexy 32 )(


0)(')('')( yxCyxByxA (*)

W is a subspace of F dim(W)=2

Give me other solutions???

Example 06'5'' yyy tindependenlinearly } ,{ :given that 32 xx ee

for W basis } ,{ 32 xx eexx ececy(x) 3

22

1 issolution general the



0)(')('')( yxCyxByxA (*)

How to solve homog. 2ed-order linear DE:

1 Find two linearly independent solutions for (*)

} )( ),( { 21 xyxy

2 The general solution for (*)

)( )( )( 2211 xycxycxy

Example 06'5'' yyy

xexy 21 )( xexy 3

2 )( xx ececy(x) 3

22

1 issolution general the


Consider the homogeneous 2ed-order linear DE (with constant coeff):

0''' CyByAy (*)

How to y1 & y2:

1 Find the characteristic equation

Find the roots of (**)

Example

06'5'' yyy

xexy 21 )(

xexy 32 )(

02 CBrAr (**)

221 , rr

0652 rr

3 ,2 21 rr

Distinct real repeated real 2 non-real

xr

xr

ey

ey2

1

2

1

xr

xr

xey

ey1

1

2

1

3.5



0''' CyByAy (*)

How to y1 & y2:



Example

04'' yy

xexy 21 )(

xexy 22 )(

02 CBrAr (**)

221 , rr

042 r

2 ,2 21 rr


xr

xr

ey

ey2

1

2

1

xr

xr

xey

ey1

1

2

1

3.5



0''' CyByAy (*)

How to y1 & y2:



Example

04'4'' yyy

xexy 21 )(

xxexy 22 )(

02 CBrAr (**)

221 , rr

0442 rr

2 ,2 21 rr


xr

xr

ey

ey2

1

2

1

xr

xr

xey

ey1

1

2

1

3.5


Consider the homogeneous nth-order linear DE:

0)()()( 0)1(

1)(

yxayxayxa nn

nn (*)

How to solve homog. nth-order linear DE:

1 Find n linearly independent solutions for (*)

} )( , ),( ),( { 21 xyxyxy n

2 The general solution for (*))( )( )( )( 2211 xycxycxycxy nn

Example 0'6''5''' yyy

1)(1 xy xexy 22 )(

xx ececcy(x) 33

221

issolution general the

xexy 33 )(

Consider the homogeneous nth-order linear DE (with constant coeff):

00)1(

1)(

yayaya nn

nn

How to y1, y2, .. yn:



Example

0'4''4''' yyy

xexy 22 )(

xxexy 23 )(

(**)

2nrrr ,, , 21

044 23 rrr

2 ,2 ,0 321 rrr

Distinct real repeated real non-real

xrn

xr

ney

ey

11

rx

rx

rx

rx

exy

exy

xey

ey

34

23

2

1

later

001

1 arara n

nn

n

nrrr ,, , 21 rrr ,, , 1)( 01 xexy


00)1(

1)(

yayaya nn

nn




Example

033 )1()2()3()4( yyyy

xexy )(2xxexy )(3

(**)

2nrrr ,, , 21

033 234 rrrr

1 ,0 4321 rrrr


xrn

xr

ney

ey

11

rx

rx

rx

rx

exy

exy

xey

ey

34

23

2

1

later

001

1 arara n

nn

n

nrrr ,, , 21 rrr ,, , 1)( 01 xexy

0)1( 3 rr

xexxy 24 )(

xxx excxececcy(x) 24321

issolution general the



0''' CyByAy (*)

How to y1 & y2:



Example

04'' yy

)2cos()( 01 xexy x

02 CBrAr (**)

221 , rr

042 r

irir 2 ,2 21


biar 1biar 2

)cos(1 bxey ax

)sin(2 bxey ax)2sin()( 0

1 xexy x)2cos()(1 xxy

)2sin()(1 xxy

2 ,0 ba


00)1(

1)(

yayaya nn

nn




(**)

2nrrr ,, , 21


001

1 arara n

nn

n

, , , biabiabia

)cos(1 bxey ax )sin(2 bxey ax)cos(3 bxxey ax )sin(4 bxxey ax)cos(2

5 bxexy ax )sin(26 bxexy ax

Example

0168 )2()4( yyy

0)4( 22 r

ii 2 ,2

)2cos()(1 xxy )2sin()(2 xxy

2 ,0 ba

)2cos()(3 xxxy )2sin()(4 xxxy


Ploynomial Operator

dx

dyyDy '

2

22 ''

dx

ydyyD

n

nnn

dx

ydyyD )(

03'6''5'''3 yyyyExample Write in operator form

Euler’s Formula

)sin(cos bibeeeeez abiabia

)sin(cos bibebi


Consider the homogeneous nth-order linear DE:0)()()( 0

)1(1

)( yxayxayxa n

nn

n (*)


1 Find n linearly independent solutions for (*)} )( , ),( ),( { 21 xyxyxy n

The general solution for (*))( )( )( )( 2211 xycxycxycxy nn 2

Consider the non-homogeneous nth-order linear DE:

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

1)( xfyxayxayxa n

nn

n (**)

How to solve non-homog. nth-order linear DE:

1Solve the associated homog. DE (*)

Find a particular solution for (**) )(xy p2)( )( )( )( 2211 xycxycxycxy nnc (complementary function)

The general solution for (**) )( )( )( xyxyxy pc 3


Consider the non-homogeneous nth-order linear DE:

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

1)( xfyxayxayxa n

nn

n (**)


1Solve the associated homog. DE (*)

Find a particular solution for (**) )(xy p2)( )( )( )( 2211 xycxycxycxy nnc (complementary function)

The general solution for (**) )( )( )( xyxyxy pc 3Example xyy 124'' :solve

solution. particular a is 3 given that x (x) y p

chapter 5 higher-order linear differntial equations second-order de: second-order linear de: note:...

Documents

order homogeneous linear

edorder linear

e d slide

characteristic equation

constant coefficients

independent solutions

realvalued functions

general solution