Page 171 - DMTH504_DIFFERENTIAL_AND_INTEGRAL_EQUATION

Differential and Integral Equation

Notes Having seen that this method works, we can now state a theorem that gives the method a
rigorous foundation.
Theorem: If L is a non-singular, linear differential operator defined on a closed interval [a, b] and
subject to unmixed boundary conditions at both endpoints, then
(i) L has an infinite sequence of real eigenvalues , ,..., which can be ordered so that
0 1
| } < | |<...<| |<...
0 1 n
and
lim | n |
n

(ii) The eigenfunctions that correspond to these eigenvalues form a basis for C[a, b], and the
series expansion relative to this basis of a piecewise continuous function y with piecewise
continuous derivative on [a, b] converges uniformly to y on any subinterval of [a, b] in
which y is continuous.
We will not prove this result here. Instead, we return to the equation, Ly = y, which defines the
eigenfunctions and eigenvalues. For a self-adjoint, second order. Linear differential operator,
this is

d dy
x
x
p ( ) q ( )y , y ...(16)
dx dx
which, in its simplest form, is subject to the unmixed boundary conditions
,
y(a) + y (a) = 0, y(b) + y (b) = 0, ...(17)
1 2 1 2
with 2 1 2 2 0 and 2 1 2 2 0 to avoid a trivial condition. This is an example of a Sturm
Liouville system, and we will devote the unit II for study of the properties of the solutions of
such systems.

Self Assessment

1. Consider the linear second order differential equation

2
d y dy
x (1 x ) y 0
dx 2 dx
Show that the Sturm Liouville form of the above equation is
x
x
(xe y ) + e y = 0, for x > 0
2. Show that the equation
2
d y A ( ) dy [ B ( ) C ( )]y 0
x
x
x
dx 2 dx
can be written in self-adjoint form by defining
x
p(x) = exp A ( )dx

what are q(x), r(x) in terms of A, B, C?

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