~TIVE
SOLUTIONS FOR A CERTAIN SECOND
ORDER DIFFERENTIAL EQUATION'§)
-
An Honors Thesis (ID 499)
by
William J. Wade
Thesis Director
Dr. T. K. Puttaswamy
- -I "_,'~
)+1 -
'\ """, tA L~.:
(
\ Ll c."~\.''''~
,
Ball State University
Muncie, Indiana
May, 1981
-
@The point at 00 is an irregular singular point of our differential equation.
I.
INTRODUCTION
For the past five decades, considerable progress has been made
in studying the behaviour of solutions of equations of the form
dz
=
xr-l
(1.1)
dx
where r, the rank of this system, is a non-negative integer,
~.
!iil and
The unknown z(x) and the constant coefficients Ak are nth order
square matrices and the indicated series is presumed to converge for
all sufficiently large values of \xl.
-
G. D. Birkoff in two papers in 1909 and 1913,
f[l and
iY,
at-
tempted to terminate the infinite series in (1) and produce a new reduced equation of the form
.2:z =
dx
xr-l
(1.2)
by using one of two types of transformations:
Type
I.
z
= B(x)y
where the n 2 elements in matrix B are all analytic in some neighborhood
of x
=00
and the determinant B(<:»)
Type
-
II.
z
I o.
= v(x)y
where v(x) can be represented by a convergent series of the form
2
v.
v(x)
J
where the lead matrix Vo
I
.~
X
0; g is an integer and the determinant
of v(x) is nonvanishing in some neighborhood of infinity,
~ ~
\x\<.CP.
The determinant of the lead matrix Vo may, however, be zero.
G. D. Birkhoff asserted that, if an appropriate transformation
of Type I were used, the s in (2) need not exceed r.
However, in 1953,
Gantmacher,i17, produced a counter-example showing that Birkhoff had
made an error.
In a paper in
1963, H. L. Turrittin
L27 attempted
to make clear
to what extent Birkhoff was right and to what extent he was wrong.
For
instance, there is always some cutoff, i.e., form (2) can always be
reached by a suitable transformation of Type I for some sufficiently
large finite value of s.
If r
= 0,
and this is the case covered by the
counte~example
of Gantmacher, Birkhoff would have been right if he had made his s one
unit larger and if the more general transformation of Type II were used;
Birkhoff is still correct provided the characteristics of Ao are all
distinct.
When Ao has multiple roots, the situation is complicated and
obscure.
In this Senior Honors thesis, an attempt is successfully made
to effectively solve differential equations of the form
~'
when
+ a(z)y'
+ b(z)y
=
0
3
a(z)
b(z)
and both power series converge for
-
-
\z\ > R.
II.
NATURE OF THE PRESENT PROBLEM
Consider the differential equation
y"
where
a(z)
+ a(z)y'
=
t
+ b(z)y
ai z-
=
(2.1)
0
i
l':oO
b(z)
::
L..
()O
,::.0
-
(2.2)
biZ-J..
Here the variable z is complex as the coefficients ai, bi
•••• ) with a 2
-
14 b o -
(i = 0,
1, 2,
(2.3)
The two power series in (2) converge for
\ z\
>R.
In the language of
Fuch's theory, the differential equation (2.1) will have an irregular
singular point at z
= ~.
We effectively solve the differential equation
(2.1) as follows:
We first make the transformation
y
and yi
Then (2.1) can be written as
= Yl
lC
Y2
5
This can be put in the matrix form
=
Y'
°
f
=
and A(z)
where
(2.4)
A(z)y
\ -b(z)
(i
Let ) 1 and
and A2
R,
= 0, 1, 2, •••• ) are 2 x 2 constant matrices, we find that
1:
---
•
\z\ >
If A(z)
where Ai
1)
-a(z)
A2
= 1,
2,
3,
be the eigen values of the matrix Ao·
Then
are roots of the characteristic equation
det (A -
AI) =
°:
= 0,
or
Thus, if
).. 2
+ ao ~
~1 =
+ bo
Ja~
=
-
0.
4
bo
2
and
•
2
which can be written as
A1
6
In view of (2.3), we observe that
).1
and
Ne xt we diagonalize the rna trix Ao.
A2
are distinct.
To do this, we make the
transformation
(2.4) ,
in
=
Then,
This can be written as
=
WI
where B(Z)
-
(2.5)
B(z) w,
_(1 1)_1
~2
- Al
Recalling that A(z) has the representation
Ao
+ Al
z
+ A2
z2
+. • .,
\z\ )R
we obtain that
B(z)
where
-
=
+ • • • ,
(2.6)
7
i
= 1,
2, • • •
Next, we make the transformation
w
= T(z) H in
w'
=
Then,
-
T(z)
+ T'(z)
H'
H
=
B(z)
T(z)
H
which can put in the form
H'
where c(z)
(2.6)
= c(z) H,
= T""l (z)
I
B(z) T(z) - T' (z)
J.
We will choose T(z) such that c(z) is given by this truncated
representation:
c(z)
= Co
+
cl
z' where Co
Let T(z) have the form given by
T(z)
=
z
where TO' Tl' ••• are constant 2 x 2 matrices.
8
Now [T(Z) -I]
=
+•. -) 2
_(!L
Z
T' (z)
~.'E3.3+ ...
+-Z+
)3
+ • • • •
Z
Z
- . . .. -
=
T:rr-2 - • • ••
+
1
zn
So, B(z) T(z) - T' (z)
+ • • •
+ • • •
-
.
Thus,
=
+ 1
zn
Bo
+
~
(B1 + BoT1)
(Bn + Brr-1TI +
+ 12 (B2 + B1T1 + BoT2 - To)
z
9
c(z)
=
+
+ ~
zJ
-l
~ (B" + Bn-l Tl + -
B3 + B2Tl + BIT2 + B0 T3
- - + BoTn - (n-l)
- 2Tl
-
rp..,
~~
T"..2)
(B2 + BITI
+ - -
+ BoT2 - To) +
=
In order to get the desired trunoated form we have to ohoose
and so on.
-1
10
We now make the transformation
in (2.6) to obtain
I
e
which
reduces to
+ Dl
=
(: ).-z-k )
0
Here,
=
Do
-
z
=
t: )
l: :)
'
w~·th
P=A 2-~1
2
and
= C1
Dl
+
4
Next, we make the transformation
I
=
C
:
II
=
C
: ) J'
in (2.7) •
) J
We get
-
-
((:
0) 1
P
+-
z
(2.8)
(
PI
P2
P3
P4
)1(:
: ) J
•
11
Where
Eo
~ (~
=
-
=
:
C-: )(: :)( ~ :)
(1 _b) ( 0 0)
o
El
=
(1
o
=
r(: :) (~ :)
-b
1
1
)f
(Pl- bP3
P3
PI
\ P3
0
P2)
P4
=
(0 -bf )
POP
(1
•
b)
1
0
bPI
+ P2 - bP(bP3 + P4)
bP
3
+ P4
•
12
o
where rl
= PI
- b P 3 and r2
=
b P 3 + P4
•
Finally, make the transformation
J'
.-..
=
(2.8),
in
k
We get
,
J
=
z
rl kl
+ rl
z
r l -1
k = (Eo
+
:)
z rl
k
which reduces to
k'
=
(Eo
=
Fo
+
-El
z
+ Fl
z
Here,
Fo
=
Eo
=
~ 12
z
)
(2.9) •
k
(: -:p)
(
0
0
Fl
= El - rlI 2
k
=
p
3
r2 - r l
).
13
We now show that (2.9) is equivalent to the confluent hypergeometric function:
m w
dw
dt
+
t
Let t = A*z, where A* is a complex
dw
dt
=
dw
dz
=
dz
dt
1
A*
w"
=
=
(2.10) •
0
constant.
dw
dz·
.
(2.10) becomes
-
1
i.e.
w"
w"
+
+
I-E-.
\A*Z
(
S
)
-1
m
-A*Z
WI
w') -
- A*
z
1
A*
mA*
z
w =0'
w =
I
o•
Now, let
and
Then
WI
=
k2
wt
= c*kl' = k2
W"
= kl
2
(A* - ~)
=
k2 + niA*
c*
k1.
Z
This can be put in the foI'm
(:~)'
=
(
0
mA*c*
Z
1
c*
A*- .9.
Z
(:: )
-
If
k
=
l~;).
k'
=
then we have
r(: ~*)
which is the form
-
A*
(2.9).
+
BIBLIOGRAPHY OF SOURCES
1.
Birkhoff, G. D. "Singular Points of Ordinary Differential Equations."
Trans. Amer. Math. Soc.,10 (1909), 463-470.
2.
Birkhoff, G. D. "Equivalent Singular Points of Ordinary Linear
Differential Equations." Math. Ann.,74 (1913), 134-139.
3.
Gantmacher, F. R.
1959.
4.
Hukuhara, M. "Sur 1es Points Singuliers des Equations Diffe"rentie11es
Lineaires, IlL" Mem. Fac. Sci, 2 (1942), 127-37.
5.
Tritzinsky, w. J. "Analytic Theory of Linear Differential Equations."
Act. Math., 62 (1934),167-227.
6.
Turri ttin, H. L. "Reduca tion of Ordinary Differential Equations
to the Birkhoff Canonical Form." Trans. Amer. Math. Soc., 107
(19 6 3), 485-507.
The Theory of Matrices, Vol. 2.
New York:
Chelsea,
;-