1550078481-Ordinary_Differential_Equations__Roberts_

N-th Order Linear Differential Equations 181

From linear algebra, we know that this system of equations has a unique so-

lution if and only if the determinant of the squ are matrix in (23) is nonzero.

Of course, the determinant of the square matrix in (23) is the Wronskian

of Y1, Y2, ... , Yn evaluated at xo. Since the functions Y1 (x ), Y2(x), ... , Yn(x)

are linearly independent on I , their Wronskian evaluated at x 0 is nonzero.

Therefore, there exists a unique solution (c 1 , c2, ... , en) to (23) and conse-

quently y(x) = z(x) = C1Y1(x) + c2y2(x) + · · · + CnYn(x) on the interval I.

DEFINITION General Solution

If Y1 (x), Y2(x), ... , Yn(x) are n linearly independent solutions of the n-th

order homogeneous linear differential equation (21) on an interval I , then

the general solution of the DE (21) on I is

where the Ci are arbitrary constants.

EXAMPLE 8 General Solution of a Third-Order,

Homogeneous Linear Differential Equation

Given that the functions ex, xex, and x^2 ex are linearly independent solutions
on ( -oo, oo) of the third-order homogeneous linear differential equation yC^3 ) -
3yC^2 l + 3yCll - y = 0, write the general solution.

SOLUTION

The general solution of the given differential equation on the interval
(-00,00) is y(x) = c 1 ex + c 2 xex + c3x^2 ex where c1,c2, and c3 are arbitrary
constants.