1550078481-Ordinary_Differential_Equations__Roberts_

N-th Order Linear Differential Equations 203

and cos Ct where C is an arbitrary constant are linearly independent func-

tions on the interval (-00,00).) Since 81 = sin-/iTLt and 82 = cos-/iTLt

are linearly independent on (-00,00), the general solution of the DE (4) is

B(t) = c1 sin -/iTLt + c2 cos -/iTLt where c 1 and c2 are arbitrary constants.

Now let us consider then-th order homogeneous linear differential equation

with constant coefficients an, an- 1 , ... , a 1 , ao where an-=/:-0. By the represen-
tation theorem for n-th order homogeneous linear differential equations, since
the constants an, an-l, ... , a 1 , ao are all continuous functions on the interval
( -oo, oo) and an -=/:-0, there exists a set containing exactly n linearly inde-
pendent solutions of (5) on (-00,00). Our immediate problem, then, is to
determine a set of n linearly independent solutions to (5) on (-oo, oo).

Both Daniel Bernoulli and Leonhard Euler knew how to solve second or-
der (n=2) homogeneous linear differential equations with constant coefficients
prior to 17 40. Euler was the first to publish his results in 17 43. Following
his approach, we suppose y = erx, where r is an unknown constant (real or
complex), is a solution of (5). Successively differentiating, we find

Substituting into (5), we get

Factoring erx from each term, leads to

Since erx -=f. 0 for any x and any constant r, the function y = erx is a solution
to the differential equation (5) if and only if r satisfies the polynomial equation

(6)

That is, erx is a solution of (5) if and only if r is a root of p(r). Equation (6) is
called the auxiliary equation associated with the differential equation (5).

Distinct Real Roots If the roots r 1 , r 2 , ... , rn of the auxiliary equa-
tion (6) are all real and no two roots are equal, then the functions y 1 (x) = erix,
Y2(x) = er^2 x' ... ' Yn(X) = ernX form a linearly independent set of real-valued
solutions to (5) on the interval (-oo, oo) and the general solution of (5) on
(-oo, oo) is y(x) = C1Y1 (x) + C2Y2 (x) + · · · + CnYn(x) where the Ci 's are ar-
bitrary constants. It is clear from the discussion above that the functions
Yi(x) = erix are all solutions of (5) on (- 00,00). All we need to do is verify
that they are linearly independent on (-00,00). We may do so by showing
that their Wronskian is nonzero at some x o E ( -oo, oo). By definition