diff equation 7 sys 2012 fall
TRANSCRIPT
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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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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
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
Linear Systems of Differential Equations
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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 ='
Linear Systems of Differential Equations
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Linear Systems of Differential Equations
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
= + + + +
= + + + +
= + + + +
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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Linear Systems of Differential Equations
'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
Linear Systems of Differential Equations
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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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Linear Systems of Differential Equations
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
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
= + + + +
= + + + +
= + + + +
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