405
Internat. J. Math. & Math. Sci.
Vol. 9 No. 2 (1986) 405-408
RESEARCH NOTES
AN INTEGRAL EQUATION ASSOCIATED WITH
LINEAR HOMOGENEOUS DIFFERENTIAL EQfJATIONS
A.K. BOSE
Department of Mathematical Sciences
Clemson University
Clemson, South Carolina 29631 U.S.A.
(Received March 7, 1985)
ABSTRACT.
y
Associated with each linear homogeneous differential equation
n-I
(n)
Z
ai(x)y
i=O
equation
(i)
of order
x
f(x O) + I
x
f(x)
0
n
on the real line, there is an equivalent integral
x
u
h(u)du + f
x
[I
x
0
G
0
f(x)
which is satisfied by each solution
n-1
(u v)a 0(v)f(v)dv]du
of the differential equation.
KEY WORDS AND PHRASES. Linear homogeneous differential equations, Integral equations,
Initial value problems, Variation of parameters formula, Uniform convergence.
1980 MATHEMATICS SUBJECT CLASSIFICATION CODE. 34A
i.
INTRODUCTION.
Let
be a positive integer, I
n
I.
be the class of all functions continuous on
y
(n)
a
n-I (x)y
(n-l)
R
be an interval on the real line
+
+
and
C(1)
Let
aO(x)y
x
(I I)
I
be any n-th order (normalized) ordinary linear homogeneous differential equation, where
a
i(x)
(n-l).
e C(I), i=0,I,2
The purpose of this article is to derive an equivalent integral equation satisfied
by the solutions of the linear homogeneous differential equation (I.I).
2.
MAIN RESULTS.
be a solution of (l.l) defined on
f(x)
Let
THEOREM.
I
and x
0
e I.
Then
f(x)
is also a solution of the integral equation
x
x
f(x)
where
h(x)
f(x
O)
u
+ f h(u)du + I If
x
x
0
0
x
Gn_l(U,V)ao(v)f(v)dv]du,
(2.1)
0
is the unique solution of the
(n-l)-th order linear homogeneous
differential equation
y
(n-l)
an_ (x)y(n-2)
+
+
a
l(x)y
x
I
(2.2)
A.K. BOSE
406
satisfying the initial conditions
Y(X0)
and G
f
n-I (x,u)
(x0),
Y (x 0)
f (x
0)
Y
Green’s
is the well-known
(n-2) (x
0)
f
(n-l)
(x0),
Function associated with the homogeneous
equation (2.2)
PROOF OF THE THEOREM.
In order to deduce the integral equation (2. I), we will
use the well-known Variation of Parameters formula
X
h(x) + f
y(x)
x
Gn_
(x,u) (u)du
(2.3)
0
solving uniquely the non-homogeneous initial value problem
y
(n-l)
y(x
0)
a
(x)y (n-2) +
n-1
(Xo),
f
Y (x
0)
+
f (x
a
+ (x)
l(x)y
0)
Y
(n-2)
(Xo)
f(n-l) (x 0
for each (x) c C(1).
Consider the sequence of functions:
fl (x), f2(x)
fk(x)
I, where
defined on
X
fl(x)
+ I h(u)du
f(x0)
f2(x)
x
fl(x)
fk(x)
Clearly, for each
k,
X
U
+ I
Ef
x
fl (x)
0
0
x
X
U
+ I
[I
x
fk(x0)
0
x
G
n-1
(u,v)a0(v)f
(v)dv3du
0
(2.4)
Gn_
(u,V)ao(V)fk_l (v)dv]du
0
f(Xo) fk(x)
is differentiable on
I
and for k
2
X
fk(x)
h(x) + I
x
Gn_l(X,u)a0(u)fk_l(U)du.
(2.5)
0
Using (2.3), we conclude that, for each k a 2,
fk(x)
is the unique solution of the
non-homogeneous initial value problem
y
(n-l)
a
n-l
(x)y (n-2) +
+
a
l(x)y
+
aO(x)fk_ l(x)
(2.6)
Hence each
(x0),
Y(X0)
f
fk(x)
n
e C (I).
Y (x 0)
f (x
0)
Y
(n-2)
(x 0)
(n-l)
f
(Xo).
Both the sequences {fk(x)} and {fk(x)} converge uniformly on
every compact subset on the interval I. To see this, let B be a compact subset of
I. Then there exists a closed and bounded interval [a,b] such that B
I and
[a,b]
INTEGRAL EQUATION ASSOCIATED WITH DIFFERENTIAL EQUATIONS
x
0
407
[a,b].
e
M
Let
IGn-l(U’v)a0(v)l
max
s
Ifk+ l(x)
Ifk+ l(x)
fk(x)
-<
Mksk(b-a)2k/2k!
fk(x)
_<
Mksk(b-a)k/k!
fi+ l(x)
using recursively the bounds for
the series
Ifl(v)
max
for each u,v
[a,b].
One can
[a,b],
now see very easily that for each x e
fi(x)
Mksk(b-a)2k/2k!, Mksk(b-a)k/k!
1,2,3,
i
Since each of
converges, we conclude by Weierstrass’
M-test that each of the series of functions
+
fl(x)
+
fk(x)), fl(x)
(fk+l (x)
k=1
converges uniformly on [a,b] and hence on
g(x) e C
such that
B.
k=l
(fk+l (x)
fk(x))
Therefore there is a function
I(I)
for all
fl (x)
+
fl(x)
+
.
I
k=l
k-I
(fk+l (x)
fk (x))
(fk+l(X)
fk(x))
In particular
x e I.
g(x O)
lira
k+=
lira
k=
fk(x)
g(x)
fk(x)
g (x)
f(Xo).
Also from (2.5) we get by taking limit as
k-
x
h(x) + I
g (x)
x
G
n-i
(x
U)ao(u)g(u)du
x e I
(2.7)
0
Hence
g(x)
f(Xo)
x
x
u
+ I
h(u)du + I
[
x
x
0
x
0
Gn_l(U,V)ao(v)g(v)dv]du
(2.8)
0
Again, relation (2.7) implies by (2.3) that g (x) is the unique solution of the
initial value problem
Therefore
Y(n-l)
an_l(x)y(n-2) +
Y (x
f (x
n
g(x) e C (I)
g
(n)
In other words,
y
f(x)
O)
Y
(x
O)
+
l(x)y
f
aO(x)g(x
(n-l)
(Xo).
and
(x)gn-I (x)
+
+
a
g(x)
is the unique solution of the homogeneous initial value problem
a
n-i
n-I
(x)y
O)
f(Xo),
g(x)
for all
(n-l)
Y (x
f(x
0)
(x) +
O)
x e I.
x
f(x)
(n-l)
a
(x)
(n)
y(x
Hence
O)
+
+
x
0
+
f (x
aO(x)g(x)
a0(x)y
O)
Y
x e I
x e I
(n-l)
(x
O)
f
(n-l)
(Xo).
Therefore, by (2.8)
x
u
h(u)du + I
[
x
x
0
G
0
n-I (u,v)ao(v)f(v)dv]du
A.K. BOSE
408
This completes the proof.
REMARK.
The above proof clearly shows how a solution of a linear homogeneous
equation with prescribed initial values can be constructed out Qf a solution
and the
Green’s
Function
G
n-I
(x u)
h(x)
of a lower order homogeneous linear equation.
This is specially significant in case of second order homogeneous equations, as
ceA-X’t
solutions
and the
Green’s
function
Gl(X,U)
e
A(x)-A(u)
x
[A(x)
a
f
x
l(u)du],
of first order homogeneous equation
y
a
l(x)y
are readily
0
available.
REFERENCES
I.
CODDINGTON, E.A.
2.
WIDDER, D.V.
An Introduction to Ordinary Differential Equat.!ons
Cliffs, New Jersey: Prentice-Hall, Inc.
Inc.
Advanced Calculus,
Englewood Cliffs, New Jersey:
Englewood
Prentice-Hall,