3-higher order de v-vi
TRANSCRIPT
V. Linear ODE of Higher Order.Variation of Parameters
Learning Objective
At the end of the module, you should be able to solve linear differential equations of higher order using Variation of parametersmethod.
Generated by Foxit PDF Creator © Foxit Softwarehttp://www.foxitsoftware.com For evaluation only.
Derivation of the method
)()()()( 012 xgyxayxayxa
)(
)(
)(
)(
)(
)(
22
0
2
1
xa
xgy
xa
xay
xa
xay
)()()( xfyxQyxPy The standardform
V. Linear ODE of Higher Order.Variation of Parameters
Generated by Foxit PDF Creator © Foxit Softwarehttp://www.foxitsoftware.com For evaluation only.
Derivation of the method
)()()()( 2211 xyxuxyxuyp Let
where )(1 xy )(2 xyand
are the fundamental set of solutions on some interval I of the homogeneous DE.
V. Linear ODE of Higher Order.Variation of Parameters
Generated by Foxit PDF Creator © Foxit Softwarehttp://www.foxitsoftware.com For evaluation only.
Derivation of the method
)()()()( 2211 xyxuxyxuyp
22221111 uyyuuyyuyp
22221111 yuyuyuyuyp
22112211 uyuyyuyu
2211 uyuy
Generated by Foxit PDF Creator © Foxit Softwarehttp://www.foxitsoftware.com For evaluation only.
Derivation of the method
22221111 yuyuyuyu
)()()( xfyxQyxPy
22112211 uyuyPyuyuP 2211 yuyuQ
)(xf
)( 1111 QyyPyu )( 2222 QyyPyu
2211 yuyu 2211 uyuy 2211 uyuyP
2211 uyuy
2211 yuyu 2211 uyuy 2211 uyuyP
Generated by Foxit PDF Creator © Foxit Softwarehttp://www.foxitsoftware.com For evaluation only.
Derivation of the method
02211 uyuy
)(2211 xfuyuy
W
xfy
W
Wu
)(211
W
xfy
W
Wu
)(122 and
By the Cramer’s Rule:
where)(
0 ,
)(
0 ,
1
12
2
21
21
21
xfy
yW
yxf
yW
yy
yyW
Generated by Foxit PDF Creator © Foxit Softwarehttp://www.foxitsoftware.com For evaluation only.
Summary of the Variation of Parameters method
)()()()( 012 xgyxayxayxa
• Find the complementary solution
• Compute the Wronskian
2211 ycycyc
))(),(( 21 xyxyW
Generated by Foxit PDF Creator © Foxit Softwarehttp://www.foxitsoftware.com For evaluation only.
)()()( xfyxQyxPy
• Write the DE in standard form
• Compute
W
xfy
W
Wu
)(211
W
xfy
W
Wu
)(122 and
Summary of the Variation of Parameters method
Generated by Foxit PDF Creator © Foxit Softwarehttp://www.foxitsoftware.com For evaluation only.
1u• Integrate and
• Finally
pc
p
yyy
yuyuy
2211
2u
The general solution:
Summary of the Variation of Parameters method
Generated by Foxit PDF Creator © Foxit Softwarehttp://www.foxitsoftware.com For evaluation only.
Example
Solve
.)1(44 2xexyyy
Generated by Foxit PDF Creator © Foxit Softwarehttp://www.foxitsoftware.com For evaluation only.
Find 044 yyy
cy
0442 mm
221 mm
Example
Generated by Foxit PDF Creator © Foxit Softwarehttp://www.foxitsoftware.com For evaluation only.
xxc xececy 2
22
1
?py
Example
Generated by Foxit PDF Creator © Foxit Softwarehttp://www.foxitsoftware.com For evaluation only.
2211 yuyuyp
.)1(44 2xexyyy
xxc xececy 2
22
1
Standard form
xx xeyey 22
21 , From yc
f
Example
Generated by Foxit PDF Creator © Foxit Softwarehttp://www.foxitsoftware.com For evaluation only.
Examplexx xeyey 2
22
1 ,
21
21
yy
yyW
)2(2 222
22
xxx
xx
xeee
xeeW
Wronskian
))(2()2( 22222 xxxxx xeexeeeW
xeW 4
Generated by Foxit PDF Creator © Foxit Softwarehttp://www.foxitsoftware.com For evaluation only.
x
x
e
exx4
4)1(
xeW 4
xxxxu 21 )1(
23
23
1
xxu
Example
W
xfyu
)(21
xx xeyey 22
21 ,
Generated by Foxit PDF Creator © Foxit Softwarehttp://www.foxitsoftware.com For evaluation only.
x
x
e
ex4
4)1(
xeW 4
12 xu
xx
u 2
2
2
Example
W
xfyu
)(12
xx xeyey 22
21 ,
Generated by Foxit PDF Creator © Foxit Softwarehttp://www.foxitsoftware.com For evaluation only.
pc yyy
2211 yuyuyy c
xx
u 2
2
223
23
1
xxu
xxxx xexx
exx
xececy 22
223
22
21 )
2()
23(
Example
xxxx ex
ex
xececy 23
22
22
21
62
Generated by Foxit PDF Creator © Foxit Softwarehttp://www.foxitsoftware.com For evaluation only.
Solve
.3csc364 xyy
Example
Generated by Foxit PDF Creator © Foxit Softwarehttp://www.foxitsoftware.com For evaluation only.
Find 0364 yy
.cy0364 2 m
092 m
imim 3,3 21
3,0 ?cy
Example
Generated by Foxit PDF Creator © Foxit Softwarehttp://www.foxitsoftware.com For evaluation only.
xecxecy xxc 3sin3cos 0
20
1
xcxcyc 3sin3cos 21
?py
Example
Generated by Foxit PDF Creator © Foxit Softwarehttp://www.foxitsoftware.com For evaluation only.
2211 yuyuyp
xcxcyc 3sin3cos 21
xyxy 3sin,3cos 21
Example
Generated by Foxit PDF Creator © Foxit Softwarehttp://www.foxitsoftware.com For evaluation only.
xx
xxW
3cos33sin3
3sin3cos
xcxcyc 3sin3cos 21
)3sin3(3cos3 22 xxW
3]3sin3[cos3 22 xxW
Example
Generated by Foxit PDF Creator © Foxit Softwarehttp://www.foxitsoftware.com For evaluation only.
12
1
34
1
11
W
Wu
4
11 W3W
xu12
11
Example
Generated by Foxit PDF Creator © Foxit Softwarehttp://www.foxitsoftware.com For evaluation only.
12
3cot
33sin4
3cos
22
xx
x
W
Wu
x
xW
3sin4
3cos2 3W
xu 3sinln36
12
Example
Generated by Foxit PDF Creator © Foxit Softwarehttp://www.foxitsoftware.com For evaluation only.
pc yyy
2211 yuyuyy c
xxxxxcxcy 3sin)3sinln36
1(3cos)
12
1(3sin3cos 21
Example
Generated by Foxit PDF Creator © Foxit Softwarehttp://www.foxitsoftware.com For evaluation only.
Extending to Higher Order
Solve)()()()()( 0123 xgyxayxayxayxa
• Find the complementary solution
• Compute the Wronskian
332211 ycycycyc
))(),(),(( 321 xyxyxyW
Generated by Foxit PDF Creator © Foxit Softwarehttp://www.foxitsoftware.com For evaluation only.
)()()()( xfyxRyxQyxPy
• Write the DE in standard form
• Compute
W
Wu 1
1 W
Wu 2
2 and W
Wu 3
3
Extending to Higher Order
Generated by Foxit PDF Creator © Foxit Softwarehttp://www.foxitsoftware.com For evaluation only.
321
321
321
yyy
yyy
yyy
W
32
32
32
1
)(
0
0
yyxf
yy
yy
W
31
31
31
2
)(
0
0
yxfy
yy
yy
W
)(
0
0
21
21
21
3
xfyy
yy
yy
W
Extending to Higher Order
Generated by Foxit PDF Creator © Foxit Softwarehttp://www.foxitsoftware.com For evaluation only.
,1u• Integrate and
• Finally
pc yyy
2u
The general solution:
3u
Extending to Higher Order
332211 yuyuyuyp
Generated by Foxit PDF Creator © Foxit Softwarehttp://www.foxitsoftware.com For evaluation only.
Solve
xyy tan
Example
Generated by Foxit PDF Creator © Foxit Softwarehttp://www.foxitsoftware.com For evaluation only.
0 yy
cy 03 mm
0)1( 2 mm
imimm 321 ,,0
xcxccyc cossin 321
Example
Generated by Foxit PDF Creator © Foxit Softwarehttp://www.foxitsoftware.com For evaluation only.
xyy tan
py
xcxccyc cossin 321
xx
xx
xx
W
cossin0
sincos0
cossin1
Example
1cossin
sincos1
xx
xxW
Generated by Foxit PDF Creator © Foxit Softwarehttp://www.foxitsoftware.com For evaluation only.
xyy tan
xxx
xx
xx
W
cossintan
sincos0
cossin0
1
xxx
xxxW tan
sincos
cossintan1
Example
Generated by Foxit PDF Creator © Foxit Softwarehttp://www.foxitsoftware.com For evaluation only.
xyy tan
xx
x
x
W
costan0
sin00
cos01
2
x
x
xx
xW
cos
sin
costan
sin01
2
2
Example
Generated by Foxit PDF Creator © Foxit Softwarehttp://www.foxitsoftware.com For evaluation only.
xyy tan
xx
x
x
W
tansin0
0cos0
0sin1
3
xxx
xW sin
tansin
0cos13
Example
Generated by Foxit PDF Creator © Foxit Softwarehttp://www.foxitsoftware.com For evaluation only.
xx
W
Wu tan
1
tan11
xdxu tan1
Example
xu cosln1
Generated by Foxit PDF Creator © Foxit Softwarehttp://www.foxitsoftware.com For evaluation only.
x
xx
x
W
Wu
cos
sin
1cos
sin2
2
22
dxx
xdx
x
xu
cos
cos1
cos
sin 22
2
xxxu
dxxxu
sintansecln
)cos(sec
2
2
Example
Generated by Foxit PDF Creator © Foxit Softwarehttp://www.foxitsoftware.com For evaluation only.
xx
W
Wu sin
1
sin33
xdxu sin3
Example
xxu cos)cos(3
Generated by Foxit PDF Creator © Foxit Softwarehttp://www.foxitsoftware.com For evaluation only.
332211 yuyuyuyp
pc yyy
Complete it.
Example
Generated by Foxit PDF Creator © Foxit Softwarehttp://www.foxitsoftware.com For evaluation only.
VI-The Cauchy-Euler Equation-
At the end of the section, students should be able to identify and solve a
Cauchy-Euler equation.
Learning Outcome
Generated by Foxit PDF Creator © Foxit Softwarehttp://www.foxitsoftware.com For evaluation only.
The Cauchy-Euler Equation
A linear DE of the form
)(... 011
11
1 xgyadx
dyxa
dx
ydxa
dx
ydxa
n
nn
nn
nn
n
where the coefficients areconstants.
01,...,, aaa nn
Generated by Foxit PDF Creator © Foxit Softwarehttp://www.foxitsoftware.com For evaluation only.
2nd–order Cauchy-Euler equation
)(2
22 xgcy
dx
dybx
dx
ydax
where the coefficients areconstants.
cba ,,
Generated by Foxit PDF Creator © Foxit Softwarehttp://www.foxitsoftware.com For evaluation only.
Homogeneous
02
22 cy
dx
dybx
dx
ydax
where the coefficients areconstants.
cba ,,
Generated by Foxit PDF Creator © Foxit Softwarehttp://www.foxitsoftware.com For evaluation only.
Method of Solution
02
22 cy
dx
dybx
dx
ydax
Let be the form ofsolution.
mxy
Generated by Foxit PDF Creator © Foxit Softwarehttp://www.foxitsoftware.com For evaluation only.
02
22 cy
dx
dybx
dx
ydax
mxy
1 mmxy2)1( mxmmy
Substituteinto theDE.
Method of Solution
Generated by Foxit PDF Creator © Foxit Softwarehttp://www.foxitsoftware.com For evaluation only.
The Auxiliary Equation
0)1( cbmmam
0)(2 cmabam
Generated by Foxit PDF Creator © Foxit Softwarehttp://www.foxitsoftware.com For evaluation only.
Case I- Distinct Real Roots
If are two distinct real rootsof the auxiliary equation, then thegeneral solution is
21,mm
.21
21mm xcxcy
Generated by Foxit PDF Creator © Foxit Softwarehttp://www.foxitsoftware.com For evaluation only.
Case II- Repeated Real Roots
If are two repeated real rootsof the auxiliary equation, then thegeneral solution is
21,mm
.ln11
21 xxcxcy mm
Generated by Foxit PDF Creator © Foxit Softwarehttp://www.foxitsoftware.com For evaluation only.
Case III- Conjugate Complex Roots
If are two conjugate complex roots of the auxiliary equation, then thegeneral solution is
imim 21 ,
)].lnsin()lncos([ 21 xcxcxy
.21ii xcxcy
or
Generated by Foxit PDF Creator © Foxit Softwarehttp://www.foxitsoftware.com For evaluation only.
Example
Solve
.0422
22 y
dx
dyx
dx
ydx
Generated by Foxit PDF Creator © Foxit Softwarehttp://www.foxitsoftware.com For evaluation only.
.0422
22 y
dx
dyx
dx
ydx
0)(2 cmabam
The Auxiliary Equation:
04)12(1 2 mm
4 ,2 ,1 cba
0432 mm
Example
Generated by Foxit PDF Creator © Foxit Softwarehttp://www.foxitsoftware.com For evaluation only.
0432 mm
0)1)(4( mm
1 ,4 21 mm
Example
.12
41
xcxcy
Generated by Foxit PDF Creator © Foxit Softwarehttp://www.foxitsoftware.com For evaluation only.
Solve
.0842
22 y
dx
dyx
dx
ydx
Example
Generated by Foxit PDF Creator © Foxit Softwarehttp://www.foxitsoftware.com For evaluation only.
.0842
22 y
dx
dyx
dx
ydx
The Auxiliary Equation:
01)48(4 2 mm
0144 2 mm
0)12( 2 m
Example
1 ,8 ,4 cba
Generated by Foxit PDF Creator © Foxit Softwarehttp://www.foxitsoftware.com For evaluation only.
2
1 ,
2
121 mm
Example
.ln11
21 xxcxcy mm
xxcxcy ln2
1
22
1
1
Generated by Foxit PDF Creator © Foxit Softwarehttp://www.foxitsoftware.com For evaluation only.
Solve
.2
1)1(,1)1(,0174 2 yyyyx
Example
Generated by Foxit PDF Creator © Foxit Softwarehttp://www.foxitsoftware.com For evaluation only.
0174 2 yyx
The Auxiliary Equation
017)40(4 2 mm
01744 2 mm
)4(2
)17)(4(416)4( m
Example
17 ,0 ,4 cba
Generated by Foxit PDF Creator © Foxit Softwarehttp://www.foxitsoftware.com For evaluation only.
)4(2
)17)(4(416)4( m
8
16)4( im
imim 22
1 ,2
2
121
Example
Generated by Foxit PDF Creator © Foxit Softwarehttp://www.foxitsoftware.com For evaluation only.
imim 22
1 ,2
2
121
)].lnsin()lncos([ 21 xcxcxy
)]ln2sin()ln2cos([ 212
1
xcxcxy
Example
Generated by Foxit PDF Creator © Foxit Softwarehttp://www.foxitsoftware.com For evaluation only.
)]ln2sin()ln2cos([ 212
1
xcxcxy
.2
1)1( ,1)1( yy
1)1( y
1))]1ln(2sin())1ln(2cos([)1( 212
1
cc
11 c
Example
Generated by Foxit PDF Creator © Foxit Softwarehttp://www.foxitsoftware.com For evaluation only.
)]ln2sin()ln2cos([ 212
1
xcxcxy
2
1)1( y
)]ln2cos(2
)ln2sin(2
[
)]ln2sin()ln2cos([2
1
212
1
212
1
xx
cx
x
cx
xcxcxy
Example
Generated by Foxit PDF Creator © Foxit Softwarehttp://www.foxitsoftware.com For evaluation only.
2
1)1( y
2
1)]1ln2cos(
1
2)1ln2sin(
1
2[)1(
)]1ln2sin()1ln2cos([)1(2
1
212
1
212
1
cc
cc
Example
2
12
2
12 c
02 c
Generated by Foxit PDF Creator © Foxit Softwarehttp://www.foxitsoftware.com For evaluation only.
)]ln2[cos(2
1
xxy
)]ln2sin()ln2cos([ 212
1
xcxcxy
The solution is
Example
Generated by Foxit PDF Creator © Foxit Softwarehttp://www.foxitsoftware.com For evaluation only.
Solve
.0875 23 yyxyxyx
Example
Generated by Foxit PDF Creator © Foxit Softwarehttp://www.foxitsoftware.com For evaluation only.
.0875 23 yyxyxyx
Let the solution bemxy
1 mmxy
2)1( mxmmy
3)2)(1( mxmmmy
Example
Generated by Foxit PDF Creator © Foxit Softwarehttp://www.foxitsoftware.com For evaluation only.
0875 23 yyxyxyx
087
)1)((5)2)(1)((1
2233
mm
mm
xxmx
xmmxxmmmx
Example
0]87
)1)((5)2)(1)([(
m
mmmmmxm
Generated by Foxit PDF Creator © Foxit Softwarehttp://www.foxitsoftware.com For evaluation only.
0]842[ 23 mmmxm
0 0842 23 mmm
Example
0)4)(2( 2 mm
imimm 2 ,2 ,2 321
Generated by Foxit PDF Creator © Foxit Softwarehttp://www.foxitsoftware.com For evaluation only.
The general solution
)]ln2sin()ln2cos([ 3202
1 xcxcxxcy
)ln2sin()ln2cos( 322
1 xcxcxcy
Example imimm 2 ,2 ,2 321
Generated by Foxit PDF Creator © Foxit Softwarehttp://www.foxitsoftware.com For evaluation only.
NON HOMOGENEOUS
)(2
22 xgcy
dx
dybx
dx
ydax
where the coefficients areconstants.
cba ,,
Generated by Foxit PDF Creator © Foxit Softwarehttp://www.foxitsoftware.com For evaluation only.
)(2
22 xgcy
dx
dybx
dx
ydax
The general solution:
pc yyy
NON HOMOGENEOUS
Generated by Foxit PDF Creator © Foxit Softwarehttp://www.foxitsoftware.com For evaluation only.
py
Find by solving the homogeneous DE
Find using Variation of Parametersmethod
cy
pc yyy
Generated by Foxit PDF Creator © Foxit Softwarehttp://www.foxitsoftware.com For evaluation only.
Example
.42 24
2
22 xxy
dx
dyx
dx
ydx
.12
41
xcxcyc
py
Generated by Foxit PDF Creator © Foxit Softwarehttp://www.foxitsoftware.com For evaluation only.
Example .12
41
xcxcyc
41 xy 1
2 xy
23
14
4
xx
xxW 222 54 xxx
Generated by Foxit PDF Creator © Foxit Softwarehttp://www.foxitsoftware.com For evaluation only.
Example
.42 24
2
22 xxy
dx
dyx
dx
ydx
Standard form
142 2
22
2
xyxdx
dy
xdx
yd
1)( 2 xxf
Generated by Foxit PDF Creator © Foxit Softwarehttp://www.foxitsoftware.com For evaluation only.
Example4
1 xy 12
xy
22
1
11
0
xx
xW 1 xx
1)( 2 xxf
)(5
1
5
)( 31
2
11'
1
xx
x
xx
W
Wu
)2
1(ln
5
121x
xu
Generated by Foxit PDF Creator © Foxit Softwarehttp://www.foxitsoftware.com For evaluation only.
Example4
1 xy 12
xy
14
023
4
2
xx
xW 46 xx
1)( 2 xxf
)(5
1
524
2
462'
2 xxx
xx
W
Wu
)35
(5
1 35
2
xxu
Generated by Foxit PDF Creator © Foxit Softwarehttp://www.foxitsoftware.com For evaluation only.
Example 41 xy
12
xy
)35
(5
1 35
2
xxu )
2
1(ln
5
121x
xu
2211 uyuyyp
352ln
5
1 2424 xxxxxyp
56
5ln
5
1 424 xxxxyp
Generated by Foxit PDF Creator © Foxit Softwarehttp://www.foxitsoftware.com For evaluation only.