Theorem 2.1.3 From any infinite set of points lying in a
closed and
bounded subset K of Rk one can select an infinite sequence
![]()
order ="0" src="./articles_imgs/8736/the_ge18.gif" > of
distinct points converging to a point 
.
The set [a, b] in R is closed and bounded, so this theorem applies.
Let u be a solution of equation (2.7) other than the identically-zero
solution, and suppose that u has infinitely many zeros in [a, b]; these
are necessarily simple by the preceding paragraph . By the Theorem
2.1.3 one can find a sequence 
of distinct zeros converging to
∈ [a, b]. Since 
, it follows from the continuity of u that
u () = 0. If u' ( ) vanished as well, we would have to conclude that
u is identically zero on [a, b]. This is contrary to our assumption, so
u' ( ) ≠ 0. It follows that there is a neighborhood N of
in [a, b]
such that u ≠ 0 in N, except at itself. But any neighborhood of
contains infinitely many zeros of u. This contradiction proves the
assertion, which we summarize as follows :
Theorem 2.1.4 Let the hypotheses of Theorem 2.1.1 hold, with r ≡
0. Then a nontrivial solution u has at most finitely many zeros on
[a, b].
2.1.1 The homogenous problem and bases
Lemma 2.1.1 has the remarkable consequence that one can define a pair of
solutions of the homogeneous equation u '' + pu' + qu = 0 (eq. 2.7 above),
say 
(x) and 
(x), with the property that any other solution u of that
equation may be written as a linear combination of and on [a, b]:
where 
and 
are constants.
To see this, define a pair of solutions , of the initial-value problem
as follows: is a solution of the homogeneous equation (2.7) which also
satisfies the initial data
whereas is a solution of the homogeneous equation (2.7) which also satisfies
the initial data
Each of these functions is uniquely determined, in view of
the existence and
uniqueness theorems 2.1.1 and 2.1.2. Now let u (x) be any solution whatever
of the homogeneous equation (2.7). Define 
and
, and
define a function v by the formula
Then v is a solution of the homogeneous equation which, because of equations
(2.10) and (2.11), has the same initial data as does u. Therefore v ≡ u. This
proves that equation (2.9) holds.
Example 2.1.1 Let the homogeneous equation be u'' +u = 0, and take the
left-hand endpoint to be a = 0. It is easy to verify that the functions cos x
and
sin x are solutions of the homogeneous equation satisying the initial data of
equations (2.10) and (2.11), respectively. Therefore, any solution u(x) of the
homogeneous equation can be expressed in the form
for some choice of the constants and .
The functions and , defined above as solutions of the homogenous
equation (2.7) together with the initial data (2.10) and (2.11), possess two
important properties. One is their linear independence :
Definition 2.1.1 A set of k functions 
defined on an interval
I
is linearly dependent there if there exist constants
, not all
zero,
such that
at each point of I. A set of functions that is not linearly dependent on I is
said to be linearly independent there.
This definition makes no reference to differential equations . Linear dependence
or independence is a property that any collection of functions may or
may not have.
Example 2.1.2 Consider the functions cos x and sin x of the preceding example.
For the sake of simplicity, assume that the interval is the entire x
axis. If these functions are linearly dependent, then
for
all real values of x. Setting x = 0 we infer that = 0, and then must also
vanish since sin x is not identically zero.
Example 2.1.3 The functions
are
linearly independent on
any interval. To see this we write the expression for linear dependence:
Because this must hold on an interval, we can differentiate any number of
times . If we differentiate k times we immediately infer that
= 0. Returning
to the expression above, for which the highest degree term is now k - 1, we
repeat the procedure and find that 
= 0. Continuing, we find that all the
constants must vanish, i.e., there can be no expression of linear dependence.
An alternative way of drawing this conclusion is proposed in Problem 5
in the problems at the end of this section.
The preceding examples are typical of proofs of linear independence: assume
the opposite and deduce a contradiction in the form that all the coefficients
must vanish. Let's apply this approach to the functions and
satisfying the conditions (2.10) and (2.11), respectively. Suppose that for
some choice of constants and
for all x ∈ [a, b]. By virtue of condition (2.10) we find, by evaluating the
left-hand side at x = a, that = 0. Since the relation of linear dependence
above must hold for all x ∈ [a, b], we are free to differentiate it. Doing so
and
applying the condition (2.11), we then see that = 0. Thus these functions
are linearly independent. Moreover, they are linearly independent by virtue
of the conditions (2.10) and (2.11) without regard to whether they satisfy
the differential equation or, indeed, any other requirement.
However, the second important property of the functions and is precisely
that they are solutions of the linear, homogeneous differential equation
(2.7). Such a linearly independent set of solutions is called a basis:
Definition 2.1.2 A linearly independent pair of solutions of the second-order,
linear, homogeneous equation (2.7) is called a basis of solutions for
that equation.
We made a particular choice of basis (, ) above, but bases are not unique.
The choice made for (, ), by requiring them to satisfy the conditions (2.10,
2.11), is useful for initial data that are supplied at the point x = a, and is
sometimes referred to as the standard basis for problems in which data are
prescribed at x = a. We could, however, have chosen some
point other than
x = a to supply the same initial values of 
and 
(i = 1, 2), and the result
would have been a different but equally valid basis. Alternatively, we could
have chosen different initial values for and
and the result would again
be a basis, unless these values are chosen "peculiarly," in a sense described
below, following Theorem 2.1.7.
These two properties of the pair (, ) (that they are linearly independent
and are solutions of the homogeneous equation 2.7) imply that any
solution of equation (2.7) is a linear combination of them with constant
coefficients,
as in equation (2.9). We now show that this property is common
to all bases of solutions of for the linear, homogeneous equation (2.7):
Theorem 2.1.5 Let 
and 
be a basis of solutions for equation (2.7) and
let u be any solution of that equation. Then there exist constants and
such that
Proof: Pick any point 
in [a, b] and consider the pair of equations
This is a system of two linear equations in the two unknowns and . For
such a system there are two possibilities: either (1) the system (2.13) has a
unique solution (, ) or (2) the corresponding homogeneous system
has a nontrivial solution, i.e., a solution for which and are not both
zero. Suppose that it is this second alternative that prevails. Then the
function 
is a solution of equation (2.7) for which
v () = 0 and v' () = 0. By Lemma 2.1.1 the function v vanishes identically
on the interval [a, b]. But this implies that and
are linearly dependent,
contrary to the hypothesis that they are a basis. Consequently the second
alternative is not possible, and the first must hold. But then the functions u
and 
are both solutions of the same
initial-value problem, and are
therefore the same by Theorem 2.1.2. 2
Some simple cases, for which solutions are elementary functions, follow.
Example 2.1.4 Return to the equation u''+u = 0 of Example 2.1.1 above; it
has the solutions = cos x and = sin x, which are linearly independent,
as shown in Example 2.1.2. These constitute a basis, the standard basis
relative to the point x = 0. However, consider the pair = cos x
- sin x
and = cos x + sin x. Since = ( + ) /2 and = (
- ) /2, the
relation (2.9) shows that the arbitrary solution u can be written as a linear
combination of and . This pair therefore constitutes a second basis.
Example 2.1.5 It is easy to verify that the equation
u'' - u = 0 
(2.15)
has the solutions = exp x and = exp (-x); these are linearly independent
(cf. Problem 10 in the problem set for this section). On the other hand,
recall that cosh x = ( + ) /2 and sinh x = (
- ) /2. Thus = cosh x
and = sinh x constitute a second basis (it is this latter basis which
represents
a standard basis relative to the origin for this equation.)
A linear vector space V over the real numbers is defined by a set of
axioms. If the space has a finite basis, i.e., a set of (say) n linearly
independent elements (vectors) 
such that any vector v
in V may be written
for some choice of the real numbers 
, then the space is
said to be finite-dimensional (n-dimensional in this case). Otherwise
it is infinite-dimensional.
The set C2[a, b] of real-valued functions possessing continuous second
derivatives on the interval [a, b] satisfies the axioms for a vector
space. It is easily seen to be infinite-dimensional. However the subset
of C2[a, b] consisting of those functions satisfying a linear, homogeneous
differential equation like equation (2.7) is also a vector space,
a subspace of C2[a, b]. This vector space has a basis consisting of
two elements. It follows that the solution space of equation (2.7) is a
two-dimensional, real vector space.
