Mathematical Methods for Physicists: A concise introduction - Site Map

DIFFERENTIAL AND INTEGRAL EQUATIONS
Clearly, k ˆ 0, and k is no longer determined by k . But we have, according to
Eq. (11.33),
Z b
a
f …t†u k …t†dt ˆ k ˆ 0;
…11:35†
that is, f …x† is orthogonal to the eigenfunction u k …x†. Thus if ˆ k , the inhomogeneous
equation has a solution only if f …x† is orthogonal to the corresponding
eigenfunction u k …x†. The general solution of the equation is then
X 1 R b
a
u…x† ˆf …x†‡ k u k …x†‡ f …t†u n…t†dt
k u
n n …x†; …11:36†
k
nˆ1 0 0
where the prime on the summation sign means that the term n ˆ k is to be omitted
from the sum. In Eq. (11.36) the k remains as an undetermined constant.
Relation between di€erential and integral equations
We have shown how an integral equation can be transformed into a di€erential
equation that may be easier to solve than the original integral equation. We now
show how to transform a di€erential equation into an integral equation. After we
become familiar with the relation between di€erential and integral equations, we
may state the physical problem in either form at will. Let us consider a linear
second-order di€erential equation
with the initial condition
Integrating Eq. (11.37), we obtain
x 0 ˆ
x 00 ‡ A…t†x 0 ‡ B…t†x ˆ g…t†;
Z t
a
x…a† ˆx 0 ; x 0 …a† ˆx 0 0:
Ax 0 dt
Z t
a
Bxdt ‡
Z t
a
gdt ‡ C 1 :
…11:37†
The initial conditions require that C 1 ˆ x0. 0 We next integrate the ®rst integral on
the right hand side by parts and obtain
x 0 ˆAx
Integrating again, we get
x ˆ
‡
Z t
Z t
a
Z t Z t
a
a
Axdt
a
…B A 0 †xdt ‡
Z t Z t
a
a
Z t
a
gdt ‡ A…a†x 0 ‡ x 0 0:
B…y†A 0
…y† x…y†dydt
g…y†dydt ‡ A…a†x 0 ‡ x0
0
…t a†‡x0 :
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