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Sec 6.2: Solutions About Singular Points. N-th order linear DE. Constant Coeff. variable Coeff. Cauchy-Euler 4.7. Ch 6 Series Point. Homog( find y p ) 4.3. NON-HOMOG (find y p ). Annihilator Approach 4.5. Variational of Parameters 4.6. Ordinary 6.1. Singular 6.2. - PowerPoint PPT Presentation
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N-th order linear DE
Constant Coeff variable Coeff
Homog(find yp)4.3
NON-HOMOG(find yp)
Annihilator Approach
4.5
Variational of Parameters
4.6
Cauchy-Euler4.7
Ch 6 SeriesPoint
Sec 6.2: Solutions About Singular Points
0x
Ordinary6.1
0x
0
nn
n
c x
Singular6.2
0x
0
n rn
n
c x
Singular PointsDefinition:
( )f x Is analytic at 0xIF: Can be represented by power series centerd at 0x( )f x
(i.e) 00
( ) ( )nnn
f x c x x
with R>0 0x x R
Definition:
0x Is an ordinary point of the DE (*)
'' ( ) ' ( ) 0 (*)y P x y Q x y IF: are analytic at
0x( ) and ( )P x Q x
A point that is not an ordinary point of the DE(*) is said to be singular point
Special Case: Polynomial Coefficients
2 1 0( ) '' ( ) ' ( ) 0 a x y a x y a x y
2 0
2 0
( ) 0 singular point( ) 0 ordinary point
a x xa x x
:Example21) ( 25) '' 2 ' 0x y xy y
2 22) (x -4) '' 3( 2) ' 5 0y x y y
3) '' 2 ' 8 0xy xy y
Regular Singular Points
Definition:
0x Is a regular singular point of the DE (*)
'' ( ) ' ( ) 0 (*)y P x y Q x y IF: are analytic at
0x( ) ( )p x and q x
A singular point that is not a regular singular point of the DE(*) is said to be irregular singular point
:Example2 22) ( - 4) '' 3( 2) ' 5 0x y x y y
3) '' 2 ' 8 0xy xy y
20 0where ( ) ( ) ( ) ( ) ( ) ( )p x x x P x and q x x x Q x
Frobenius’ Theorem
:Example
Theorem 6.2: 2 1 0( ) '' ( ) ' ( ) 0 a x y a x y a x y
0
1) There exists at least one solution in the form
( )n rnc x x
IF is a regular
singular point
0x
0
2) The series will convereg at least on some interval 0 -x x R
X=2 is a regular singular point . We can find at least one sol in the form
0
( 2)n rn
n
c x
with 0 2x R
2 2 ( - 4) '' 3( 2) ' 5 0x y x y y
Frobenius’ Theorem
:Example
Theorem 6.2: 2 1 0( ) '' ( ) ' ( ) 0 a x y a x y a x y
0
1) There exists at least one solution in the form
( )n rnc x x
IF is a regular
singular point
0x
0
2) The series will convereg at least on some interval 0 -x x R
X=0 is a regular singular point . We can find at least one sol in the form
0
n rn
n
c x
with 0 x R
'' 2 ' 8 0xy xy y
We need to find all Cn
and r
Frobenius’ Theorem
Theorem 6.2: 2 1 0( ) '' ( ) ' ( ) 0 a x y a x y a x y
0
1) There exists at least one solution in the form
( )n rnc x x
IF is a regular
singular point
0x
0
2) The series will convereg at least on some interval 0 -x x R
What is the difference between •Frobenius Theroem•Theorem for ordinary point
3
Frobenius’ TheoremTheorem 6.2:
2 1 0( ) '' ( ) ' ( ) 0 a x y a x y a x y
0
1) There exists at least one solution in the form
( )n rnc x x
IF is a regular
singular point
0x
0
2) The series will convereg at least on some interval 0 -x x R
Theorem 6.1: Existence of Power Series Solutions
2 1 0( ) '' ( ) ' ( ) 0 a x y a x y a x y
1 2
0 0
1)We can find two-linearly independent ,
in the form ( ) with nn
y y
c x x x x R IF is an
ordinary point
0x
02) R = distance from x to the closest singular point (incl: complex points)
Frobenius’ Theorem
:Example
X=0 is a regular singular point . We can find at least one sol in the form
0
n rn
n
c x
Solve: '' 0xy y We need to find all Cn
and r
1 2 3 40 1 2 3 4
r r r r ry c x c x c x c x c x
1 1 2 30 1 2 3 4' (1 ) (2 ) (3 ) ( 4)r r r r ry rc x r c x r c x r c x r c x
2 10 1 2
1 23 4
'' ( 1) (1 ) (2 )(1 )
+(3 )(2 ) ( 4)(3 )
r r r
r r
y r r c x r rc x r r c x
r r c x r r c x
1 1
0 1 2
2 33 4
'' ( 1) (1 ) (2 )(1 )
+(3 )(2 ) ( 4)(3 )
r r r
r r
xy r r c x r rc x r r c x
r r c x r r c x
Indicial Equations ( indicial roots)
indicial equation is a quadratic equation in r that results from equating the total coefficient of the lowest power of x to zero
:Example '' 0xy y ( 1) 0r r indicial equation
indicial roots 1 21 0r r
Indicial Equations ( indicial roots)
indicial equation is a quadratic equation in r that results from equating the total coefficient of the lowest power of x to zero
:Example '' 0xy y ( 1) 0r r indicial equation
indicial roots 1 21 0r r
'' ( ) ' ( ) 0 (*)y P x y Q x y 2where ( ) ( ) ( ) ( )p x xP x and q x x Q x
0 0a = (0) b = (0)p and q
indicial equation 0 0( 1) 0r r a r b
Indicial Equations ( indicial roots)
:Example
3 '' ' 0xy y y (3 2) 0r r indicial equation
'' ( ) ' ( ) 0 (*)y P x y Q x y 2where ( ) ( ) ( ) ( )p x xP x and q x x Q x
0 0a = (0) b = (0)p and q
indicial equation
0 0( 1) 0r r a r b
Find the indicial roots:
:Example
2 '' (1 ) ' 0xy x y y (2 1) 0r r indicial equation Find the indicial roots:
Method of Solutions
Find the indicial equations and roots: r1 > r2
Case I: 1 2
1 2
positive integer, distinct realr rr r
11
0
n rn
n
y c x
2
20
n rn
n
y c x
Case III: 1 2r r 11
0
n rn
n
y c x
2 sec 4.2y
Case II: 1 2
1 2
=positive integer, distinct realr rr r
11
0
n rn
n
y c x
22
0
n rn
n
y c x
2 sec 4.2y
Method of Solutions
Case III: 1 2r r 11
0
n rn
n
y c x
2 sec 4.2y
Case II: 1 2
1 2
=positive integer, distinct realr rr r
11
0
n rn
n
y c x
22
0
n rn
n
y b x
12 1
0
( ) ln n rn
n
y y x x b x
22 1
0
( ) ln n rn
n
y y x x b x
