Study Guide
Block 2 : Ordinary D i f f e r e n t i a l Equations
U n i t 8: The U s e o f Power S e r i e s
Overview
The method of v a r i a t i o n of p a r a m e t e r s q u a r a n t e e s u s t h e g e n e r a l
s o l u t i o n of L ( y ) = f ( x ) once w e know t h e g e n e r a l s o l u t i o n of
L ( y ) = 0. There i s a v e r y g e n e r a l c l a s s o f e q u a t i o n s of t h e
form L ( y ) = 0 f o r which w e can n o t o n l y b e s u r e t h e g e n e r a l
s o l u t i o n e x i s t s b u t f o r w h i c h w e can a l s o c o n s t r u c t t h e general
s o l u t i o n i n t h e form o f a power series. Among o t h e r t h i n g s ,
t h e r e f o r e , t h i s u n i t s u p p l i e s us with another important applic a t i o n of power series.
.
Study Guide
Block 2: O r d i n a r y D i f f e r e n t i a l E q u a t i o n s
U n i t 8: The U s e o f Power S e r i e s
2.
=
Lecture 2.060
S t u d y Guide
Block 2 : O r d i n a r y D i f f e r e n t i a l E q u a t i o n s
U n i t 8: The U s e o f Power S e r i e s
2.
L e c t u r e 2.060 c o n t i n u e d
Study Guide
Block 2: Ordinary D i f f e r e n t i a l Equations
U n i t 8: The Use of Power S e r i e s
3 . Do t h e E x e r c i s e s .
(Every e x e r c i s e i n t h i s U n i t i s d e s i g n a t e d
a s a learning exercise.
The main r e a s o n i s t h a t t h e m a t e r i a l i s
n o t covered i n t h e t e x t ) .
Note #1: A l l e x e r c i s e s a r e g i v e n i n t h e form L ( y )
= 0.
The r e a s o n f o r t h i s i s t h a t w e can use v a r i a t i o n of parameters t o s o l v e L ( y ) = f ( x ) once t h e g e n e r a l s o l u t i o n of L ( y ) = 0 i s known. Note # 2 : To h e l p s u p p l y you w i t h a b r i e f review of convergence a s w e l l a s w i t h a few m a n i p u l a t i v e d e v i c e s which a r e h e l p f u l i n t h e s t u d y of s e r i e s s o l u t i o n s , w e have i n c l u d e d a t t h e end of t h e e x e r c i s e s a s p e c i a l p r e f a c e a s a "preamble" t o t h e s o l u t i o n s of the exercises i n t h i s unit.
Feel f r e e t o read t h i s preface before you b e g i n working on t h e e x e r c i s e s . 4. Exercises :
a. R e w r i t e
m
i n an e q u i v a l e n t form which i n v o l v e s a s i n g l e i n f i n i t e s e r i e s .
b. Determine each of t h e c o e f f i c i e n t s a o , a l , . . . a n , . . . ,
if
c. L e t
and u s e t h e power series t e c h n i q u e t o f i n d t h e g e n e r a l s o l u t i o n
o f (1 - x ) dy/dx
+
y = 0.
Study Guide
Block 2: Ordinary D i f f e r e n t i a l Equations
Unit 8: The Use of Power S e r i e s
Use t h e power s e r i e s technique, l e t t i n g
m
t o f i n d t h e g e n e r a l s o l u t i o n of (1
-
x)dy/dx
-y
-
2xy = 0 i n t h e form
= 0.
2.8.3(L)
Find t h e g e n e r a l s o l u t i o n of dy/dx
where a.
is a r b i t r a r y ,
2.$.4(L)
Find a l l t h e s o l u t i o n s of
which can be w r i t t e n i n t h e form
Read t h e o p t i o n a l n o t e which follows t h e s o l u t i o n of t h i s
e x e r c i s e f o r a deeper look a t what happened h e r e .
Use s e r i e s t o f i n d t h e g e n e r a l s o l u t i o n of
(1
-
2
x )y"
-
xyt
+
y = 0 , where 1x1 < 1
i n t h e form
m
w i t h a.
and a 1 a r b i t r a r y .
2.8.5
Study Guide
Block 2 : Ordinary D i f f e r e n t i a l Equations
Unit 8: The Use of Power S e r i e s
Find t h e g e n e r a l s o l u t i o n of (1 + x 2 ) y n
-
xyf -b y = 0 i n t h e
form
with a,
and al a r b i t r a r y .
a. Find t h e g e n e r a l s o l u t i o n of y" - xy
where a.
= 0 i n t h e form
and al a r e a r b i t r a r y .
b. Find t h e p a r t i c u l a r s o l u t i o n , y = f (x) , of t h e equation
y" - xy = 0 , given t h a t f ( 0 ) = 0 and f ' ( 0 ) = 1.
c. With f (x) a s i n p a r t (b) compute f (1) t o t h e n e a r e s t hundredth.
Find a l l s o l u t i o n s of x 3y" + xy' - y = 0 which a r e a n a l y t i c a t
x = 0 ( i . e . , f i n d a l l s o l u t i o n s which have t h e form
Preface:
P a r t 1 - A Brief Review of Convergence
Suppose { f n ( x ) : n = 1, 2 , 3 , ...I
d e f i n e d on some common i n t e r v a l , I .
1.
i s a sequence of f u n c t i o n s
D e f i n i t i o n #1
The sequence { f n ( x ) ) i s s a i d t o be pointwise convergent on I
if
l i m f n ( x o )
n + ~
Study Guide
Block 2 : Ordinary D i f f e r e n t i a l Equations
U n i t 8: The U s e of Power S e r i e s
e x i s t s f o r e a c h xo
2.
E
I.
I f t h e sequence { f n ( x ) ) i s p o i n t w i s e convergent on I , w e
d e f i n e t h e l i m i t f u n c t i o n , f ( x ) , of t h i s sequence by
f (X ) =
0
l i m f n ( x o ) , f o r each xo € 1 -
f n (xO), o u r " t o l e r a n c e l i m i t " ,
N o t i c e t h a t i n computing
That i s , t o s a y t h a t f ( x o ) = Jig f n ( x o ) E , u s u a l l y depends on xo.
> N +
means t h a t g i v e n E > O t h e r e e x i s t s N such t h a t n ] f (x0) - f n (xo) I < E ; and f o r t h e same E , d i f f e r e n t v a l u e s o f x,
may d e t e r m i n e d i f f e r e n t v a l u e s o f N. 3.
I f I had a f i n i t e number o f p o i n t s (which it d o e s n t t s i n c e
an i n t e r v a l i s a connected segment of t h e x - a x i s , and hence
h a s i n f i n i t e l y many p o i n t s ) , t h e f a c t t h a t N dependent on xo
4.
would be i r r e l e v a n t s i n c e w e c o u l d examine e a c h N and t h e n
choose t h e g r e a t e s t .
But w i t h a n i n f i n i t e s e t t h e r e need n o t
be a g r e a t e s t member. For example, i f 0 < xo < 1 and f o r a
g i v e n x0, N = 1/x0, t h e n a s xo + 0, N +
=J.
T h i s i n i t s e l f i s n o t bad. What is bad i s t h a t c e r t a i n " s e l f e v i d e n t " r e s u l t s need n o t b e t r u e . By way of i l l u s t r a t i o n ,
r e c a l l t h e example i n P a r t 1 of o u r c o u r s e i n which w e d e f i n e d
f n on [0,11 by
Then, s i n c e A&g f n ( x ) = 0 , i f 0 < x < 1 and 1, i f x = 1, t h e
l i m i t f u n c t i o n f ( x ) e x i s t s and i s i n k e d g i v e n by
Thus, e a c h member of t h e sequence f n ( x ) i s continuous on [ O f 1 1
b u t t h e l i m i t f u n c t i o n f i s d i s c o n t i n u o u s a t x = 1.
T h i s v i o l a t e s o u r " i n t u i t i v e " f e e l i n g t h a t t h e l i m i t of a
sequence of c o n t i n u o u s f u n c t i o n s s h o u l d be a continuous f u n c t i o n .
Study Guide
Block 2: Ordinary D i f f e r e n t i a l Equations
U n i t 8: The U s e of Power S e r i e s
S i n c e t h e a f o r e mentioned t r o u b l e a r o s e because N depended
on x 0 , w e t r y t o e l i m i n a t e t h e t r o u b l e by d e f i n i n g a s t r o n g e r *
t y p e of convergence i n which N w i l l n o t depend on xo. T h i s
l e a d s t o D e f i n i t i o n #2.
5.
Definition #2
{ f n ( x ) i s s a i d t o be uniformly c o n v e r g e n t t o f (x) on I i f f o r
e v e r y E P O t h e r e e x i s t s an N s u c h t h a t n
N + If ( x ) - f n ( x ) I <
f o r e v e r y x €1.
E
Thus, i n uniform convergence t h e c h o i c e of N depends o n l y on
t h e c h o i c e of E n o t on t h e c h o i c e o f xo. T h i s " s l i g h t " rnodific a t i o n i s enough t o q u a r a n t e e t h e f o l l o w i n g r e s u l t s .
6.
I£ ( f n ( x ) } converges un3,formly t o f Cxl on a 5 x 5 bC= [a,bl a T,)
and each f n i s c o n t i n u o u s on [ a , b ] , t h e n f i s a l s o continuous
on [ a , b l .
I f { f n ( x ) 1 converges uniformly t o f ( x ) on [ a , b l
x a ta,bl
, then
f o r each
,
I f { f n ( x ) } i s a sequence of c o n t i n u o u s l y d i f f e r e n t i a b l e f u n c t i o n s
which converge point-wise t o f ( x ) on [ a , b l and i f { f n l ( x )
converges uniformly on [ a , b ] , t h e n f 1( x ) e x i s t s on [ a r b ] and may
be computed by f (x) = l i m f n l (x)
n-tw
.
While w e have t h u s f a r been t a l k i n g a b o u t sequences of
f u n c t i o n s , n o t i c e t h a t o u r r e s u l t s a l s o apply t o series s i n c e
e v e r y s e r i e s i s a sequence of p a r t i a l sums.
T h a t i s , when w e w r i t e
7.
00 * R e c a l l t h a t " s t r o n g e r " r n e a n s a d e f i n i t i o n which i n c l u d e s t h e
previous one.
In o t h e r words, e v e r y t h i n g t h a t obeys t h e
s t r o n g e r d e f i n i t i o n o b e y s t h e weaker o n e , b u t n o t e v e r y t h i n g
which o b e y s t h e weaker one o b e y s t h e s t r o n g e r o n e .
Study Guide
Block 2 : Ordinary D i f f e r e n t i a l Equations
U n i t 8: The U s e o f Power S e r i e s
what w e mean i s ;
lim
k-
( x ) where F~ ( x ) = f l ( X I
+. . .+
f k (x) =
C
f n (x)
n= l
One v e r y i m p o r t a n t series i s t h e power series
The u s e f u l n e s s o f power series s t e m s from t h e f o l l o w i n g theorem:
Theorem 1:
I f t h e power series
converges f o r some v a l u e of x = x0'
t h e n i t converges a b s o l u t e l y
f o r e v e r y x such t h a t 1 x b0 , and t h i s convergence i s a l s o
uniform on e v e r y c l o s e d i n t e r v a l d e f i n e d by 1x1 5 IxlI<Ixol.
1
/XI<
converges uniformly and a b s o l u t e l y f o r
R, w e can t a k e t h e
same l i b e r t i e s w i t h t h e power series a s we c o u l d have t a k e n
w i t h polynomials.
T h a t i s , by t h e p r o p e r t i e s o f a b s o l u t e
convergence w e can add series term-by-term,
w e can r e - a r r a n g e
t h e terms, etc.
For example, i f
converges a b s o l u t e l y w e may, i f we w i s h , rewrite t h e series a s
t h e sum o f two series, one of which c o n t a i n s t h e terms i n
which t h e exponents a r e even and t h e o t h e r , i n which t h e
exponents a r e odd.
iO
a,(x - b ) n b u t a s
*More g e n e r a l l y , o n e u s u a l l y t a l k s a b o u t
u s u a l we t a k e t h e s p e c i a l c a s e b = 0 s i m p f y f o r c o n v e n i e n c e .
Study Guide
Block 2: Ordinary D i f f e r e n t i a l Equations
Unit 8: The U s e of Power S e r i e s
That i s ,
.
.
C
anxn =
n=0
C.
n even
anxn+
C
n odd
%xn
etc.
Moreover by uniform convergence, w e nay d i f f e r e n t i a t e and i n t e g r a t e power s e r i e s term-by-term.
9.
I n p a r t i c u l a r , t h e r e s u l t s w e s h a l l use most i n t h i s Unit
are:
C
n-0
n=O
anxn
+
a n xn
5
C bnxn .C
(a,
n=0
n=0
=
C
n=O
bnxn
++
g,
+
n
bn)x
= bn f o r each n. Preface: P a r t 2
Now t h a t we have reviewed t h e theory behind power s e r i e s , we
should l i k e t o p r e s e n t two computational techniques t h a t a r e
employed very o f t e n when we look f o r s e r i e s s o l u t i o n s of
Study Guide
Block 2 : O r d i n a r y D i f f e r e n t i a l Equations
U n i t 8: The U s e of Power S e r i e s
d i f f e r e n t i a l equations.
The f i r s t , and p r o b a b l y e a s i e r , o f t h e s e t e c h n i q u e s i n v o l v e s
adding series when t h e summation s t a r t s a t d i f f e r e n t v a l u e s of
n. For example, suppose w e want t o e x p r e s s
a s a s i n g l e series.
What
we o b s e r v e i s t h a t i f t h e f i r s t t e r m
had been
we c o u l d have simply added t h e two series term-by-tern
C (a,
n= 2 n
+ bn)x
.
Re, t h e r e f o r e , c o n v e r t into
by " s p l i t t i n g o f f " i t s f i r s t two terms.
Thus,
That i s ,
t o obtain
Study Guide
Block 2 : Ordinary D i f f e r e n t i a l Equations
Unit 8: The Use of Power S e r i e s
Notice t h a t t h e procedure used h e r e i s a p p l i c a b l e t o a l l f i n i t e
sums q u i t e i n g e n e r a l . Namely,
W e use a b s o l u t e convergence when w e allow o u r s e l v e s t h e luxury
of a s s u r i n g t h a t we may regroup terms a t w i l l , even when t h e
sum involves i n f i n i t e l y many terms.
The second technique involves summing two s e r i e s which d o n t t
" l i n e up1' term-by-term.
express
For example, suppose we want t o
a s a s i n g l e s e r i e s , Granted t h a t t h e summations do n o t begin
a t t h e same v a l u e of n , t h e r e i s even a worse problem i n t h e
sense t h a t t h e g e n e r a l term i n t h e f i r s t sum involves xn while
n-1
i n t h e second sum t h e g e n e r a l term involves x
.
The key t o t h i s problem l i e s i n t h e f a c t t h a t given,
we may r e p l a c e c, by cnmr provided w e add r t o both our lower
and upper l i m i t s of summation. That i s
Study Guide
Block 2: Ordinary D i f f e r e n t i a l Equations
U n i t 8: The Use of Power S e r i e s
A s a more c o n c r e t e example,
To check t h i s , we need o n l y o b s e r v e t h a t
while
Thus, r e t u r n i n g t o (1) w e n o t i c e t h a t t o change x
n- 1 i n t o xn w e
T h e r e f o r e , w e r e p l a c e e a c h n i n C b xn-I by
n
1 and a d j u s t f o r t h i s by s u b t r a c t i n g 1 from each of o u r
must add 1 t o n.
+
n
l i m i t s of summation.
T h a t is:
Using (21, (1) becomes
which, i n t u r n , i s
A s a f i n a l p o i n t , w e s h o u l d o b s e r v e t h a t " l i n i n g up" exponents
u s u a l l y s h i f t s t h e summation t o s t a r t a t a d i f f e r e n t v a l u e of
n.
The p r o p e r p r o c e d u r e i s f i r s t t o " l i n e up" t h e exponents
*Again, a s a quick check,
m
C
n-1 bnx
n-1
= bl
+
b2x
+
b3x2
+
b4x3
+
...;
while Study Guide
Block 2 : Ordinary D i f f e r e n t i a l Equations
U n i t 8: The U s e of Power S e r i e s
and t h e n t o a l t e r t h e lower l i m i t of t h e summation ( i f n e c e s s a r y ) .
For example, g i v e n
w e e i t h e r r a i s e n by 2 i n t h e f i r s t series o r lower n by 2 i n
t h e second series, w e must add 2 t o t h e l i m i t s of summation.
This y i e l d s
s o t h a t ( 3 ) becomes
W e now " s p l i t o f f " t h e f i r s t t e r m i n t h e f i r s t series i n ( 4 ) t o
obtain
Note
We could have e l e c t e d t o raise n by 2 i n t h e f i r s t series i n
( 3 ) . That i s , w e could have w r i t t e n
whereupon ( 3 ) would become
W e would t h e n rewrite
Study Guide
Block 2: Ordinary D i f f e r e n t i a l Equations
U n i t 8: The U s e o f Power S e r i e s
s o t h a t (6) becomes
a, I t i s l e f t f o r you t o check t h a t (5) and (7) a r e t h e same, each
being
Further d r i l l is l e f t t o the exercises.
MIT OpenCourseWare
http://ocw.mit.edu
Resource: Calculus Revisited: Complex Variables, Differential Equations, and Linear Algebra
Prof. Herbert Gross
The following may not correspond to a particular course on MIT OpenCourseWare, but has been
provided by the author as an individual learning resource.
For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.