Page 91 - Schaum's Outline of Differential Equations

74 LINEAR DIFFERENTIAL EQUATIONS: THEORY OF SOLUTIONS [CHAR 8

LINEARLY INDEPENDENT SOLUTIONS
A set of functions {ji(x), y 2(x), ..., y n(x)} is linearly dependent on a <x < b if there exist constants c x,
c 2, ... , c n, not all zero, such that

Example 8.1. The set {x, 5x, 1, sin x] is linearly dependent on [-1, 1] since there exist constants c l = -5, c 2 = 1, c 3 = 0,
and c 4 = 0, not all zero, such that (8.7) is satisfied. In particular,

Note that c 1 = c 2 = • • • = c n = 0 is a set of constants that always satisfies (8.7). A set of functions is linearly
dependent if there exists another set of constants, not all zero, that also satisfies (8.7). If the only solution to
(8.7) is c 1 = c 2 = • • • = c n = 0, then the set of functions {yi(x), y 2(x), ..., y n(x)} is linearly independent on
a < x < b.
Theorem 8.2. The wth-order linear homogeneous differential equation L(y) = 0 always has n linearly inde-
pendent solutions. If Ji(x), y 2(x), ..., y n(x) represent these solutions, then the general solution
of L()0 = 0 is

where c 1, c 2, ... , c n denote arbitrary constants.

THE WRONSKIAN
The Wronskian of a set of functions {z\(x), Z 2(x), ..., Z n(x)} on the interval a<x<b, having the property
that each function possesses n — 1 derivatives on this interval, is the determinant

Theorem 8.3. If the Wronskian of a set of n functions defined on the interval a < x < b is nonzero for at
least one point in this interval, then the set of functions is linearly independent there. If the
Wronskian is identically zero on this interval and if each of the functions is a solution to the
same linear differential equation, then the set of functions is linearly dependent.
Caution: Theorem 8.3 is silent when the Wronskian is identically zero and the functions are not known to be
solutions of the same linear differential equation. In this case, one must test directiy whether Eq. (8.7) is satisfied.

NONHOMOGENEOUS EQUATIONS

Let}^ denote any particular solution of Eq. (8.5) (see Chapter 3) and lety h (henceforth called the homogeneous
or complementary solution) represent the general solution of the associated homogeneous equation L(y) = 0.
Theorem 8.4. The general solution to L(y) = (j)(x) is