Chapter 1
BASIC NOTIONS
1.1. A problem from differential equations
Suppose we are given the problem of finding a solution of
(1.1) f (x) + f(x) = g(x)
in an interval a x b with the solution satisfying
(1.2) f(a) = 1,f (a) = 0.
(We shall not enter into a discussion as to why anyone would want to solve
this problem, but content ourselves with the statement that such equations
do arise in applications.) From your course in differential equations you will
recall that when g = 0, equation (1.1) has a general solution of the form
(1.3) f(x) = A sin x + B cos x,
where A and B are arbitrary constants. However, if we are interested in
solving (1.1) for g(x) an arbitrary function continuous in the closed inter-
val, not many of the methods developed in the typical course in differential
equations will be of any help. A method which does work is the least pop-
ular and would rather be forgotten by most students. It is the method of
variation of parameters which states, roughly, that one can obtain a solution
of (1.1) if one allows A and B to be functions of x instead of just constants.
Since we are only interested in a solution of (1.1), we shall not go into any
justification of the method, but merely apply it and then check to see if
what we get is actually a solution. So we differentiate (1.3) twice, substitute
into (1.1) and see what happens. Before proceeding, we note that we shall
get one equation with two unknown functions. Since we were brought up
from childhood to believe that one should have two equations to determine
1
http://dx.doi.org/10.1090/gsm/036/01
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