chapter 5 higher-order linear differntial equations
DESCRIPTION
CHAPTER 5 Higher-Order Linear Differntial Equations. Second-order DE:. (all). Second-order linear DE:. (a,c,d,e). Note: A,B,C,F function of x only. Second-order homogeneous linear DE:. Note: F(x)=0. (d,e). Second-order homogeneous linear DE (with constant coefficients):. (d). - PowerPoint PPT PresentationTRANSCRIPT
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)
CHAPTER 5 Higher-Order Linear Differntial Equations
Example
0'3''4 ) yyyd
Second-order homogeneous linear DE:
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
CHAPTER 5 Higher-Order Linear Differntial Equations
Second-order homogeneous linear DE:
Example 06'5'' yyyxexy 2
1 )( xexy 32 )(
Consider the homogeneous 2ed-order linear DE:
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
CHAPTER 5 Higher-Order Linear Differntial Equations
Consider the homogeneous 2ed-order linear DE:
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
CHAPTER 5 Higher-Order Linear Differntial Equations
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
CHAPTER 5 Higher-Order Linear Differntial Equations
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
04'' yy
xexy 21 )(
xexy 22 )(
02 CBrAr (**)
221 , rr
042 r
2 ,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
CHAPTER 5 Higher-Order Linear Differntial Equations
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
04'4'' yyy
xexy 21 )(
xxexy 22 )(
02 CBrAr (**)
221 , rr
0442 rr
2 ,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
CHAPTER 5 Higher-Order Linear Differntial Equations
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:
1 Find the characteristic equation
Find the roots of (**)
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
Consider the homogeneous nth-order linear DE (with constant coeff):
00)1(
1)(
yayaya nn
nn
How to y1, y2, .. yn:
1 Find the characteristic equation
Find the roots of (**)
Example
033 )1()2()3()4( yyyy
xexy )(2xxexy )(3
(**)
2nrrr ,, , 21
033 234 rrrr
1 ,0 4321 rrrr
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
0)1( 3 rr
xexxy 24 )(
xxx excxececcy(x) 24321
issolution general the
CHAPTER 5 Higher-Order Linear Differntial Equations
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
04'' yy
)2cos()( 01 xexy x
02 CBrAr (**)
221 , rr
042 r
irir 2 ,2 21
Distinct real repeated real 2 non-real
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
Consider the homogeneous nth-order linear DE (with constant coeff):
00)1(
1)(
yayaya nn
nn
How to y1, y2, .. yn:
1 Find the characteristic equation
Find the roots of (**)
(**)
2nrrr ,, , 21
Distinct real repeated real non-real
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
CHAPTER 5 Higher-Order Linear Differntial Equations
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
CHAPTER 5 Higher-Order Linear Differntial Equations
Consider the homogeneous nth-order linear DE:0)()()( 0
)1(1
)( yxayxayxa n
nn
n (*)
How to solve homog. nth-order linear DE:
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
CHAPTER 5 Higher-Order Linear Differntial Equations
Consider the non-homogeneous nth-order linear DE:
)()()()( 0)1(
1)( xfyxayxayxa n
nn
n (**)
How to solve 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 3Example xyy 124'' :solve
solution. particular a is 3 given that x (x) y p