Post on 31-Dec-2015
MAT 3237Differential Equations
Section 18.4
Series Solutions Part I
http://myhome.spu.edu/lauw
HW
No WebAssign Do18.4 #1 Do not turn in
Why Power Series Solutions?
Many DE cannot be solved explicitly in terms of finite combinations of elementary functions.
Example
Quantum Mechanics
2 0y xy y
Example
Quantum Mechanics
2 20
2
2 11
1
1 3 7 (4 5)1
2! (2 )!
1 5 9 (4 3)
(2 1)!
n
n
n
n
ny c x x
n
nc x x
n
2 0y xy y
Extract Information (e.g. from Approximation)
2 0y xy y
32 2
02
22 1
11
1 3 7 (4 5)1
2! (2 )!
1 5 9 (4 3)
(2 1)!
n
n
n
n
ny c x x
n
nc x x
n
Quantum Mechanics
Review
Power Series Differentiate Power Series
Recall: Index Shifting Rules
6
2
24
0
25
1
2 11iii
iii
Recall: Index Shifting Rules
6
2
24
0
25
1
2 11iii
iii
decrease the index by 1
increase the i in the summation by 1
Recall: Index Shifting Rules
6
2
24
0
25
1
2 11iii
iii
increase the index by 1
decrease the i in the summation by 1
Example
7 6 8
3 2 4
1 1
1 2k k k
k k k
k k k
Definition
A Power Series is of the form
2 30 1 2 3
0
nn
n
c x c c x c x c x
Two representations of
1 21 2 3
1 21 2
0
13
( )
0 2 3
2 3
nn
n
nn
n
f x
nc x c c x c x
nc x c c x c x
Theorem (Identity Property)
If is convergent on some interval , and
then
0
nn
n
c x
0
for all in I0nn
n
c x x
0 for all nc n
Theorem
2 30 1 2 3
0
0
( )
(0)
0
0
nn
n
f x c x c c x c x c x
f c
ncn allfor 0
?Why
0
for all in I0nn
n
c x x
Theorem
1 21 2 3
1
1
0( ) 2 3
(0) 0
nn
n
f x nc x c c x c x
f c
ncn allfor 0
?Why
0
0 for all in Inn
n
c x x
Theorem
2
2
22 3 4
2
( ) ( 1)
2(1) 3(2) 4(3)
(0)
0
0
nn
n
f x n n c x
c c x c x
f c
ncn allfor 0
?Why
0
0 for all in Inn
n
c x x
Example 1
Solve the following DE by power series
( 4) 3 0x y y
Example 1
Solve the following DE by power series
( 4) 3 0x y y
ExpectationWe see the pattern that , for 0
( 2)!
2 4 !
n n
n
nc
n
Summary
To combine the summations, we need• same power of x and same index ranges
Use • different representations of the derivative• index shifting
Find the relationship between the coefficients and look for patterns• do not “collapse” numbers
The lowest coefficient(s) remains (fix by I.C.)