1550078481-Ordinary_Differential_Equations__Roberts_

Chapter 4

N-th Order Linear Differential

Equations

As we noted earlier, ordinary differential equations are divided into two
distinct classes- linear equations and nonlinear equations. In chapters 2
and 3, we studied a few differential equations which can be solved expli citly
in terms of elementary functions or which can be written as formulas involv-
ing quadratures. In particular, we found that the solution of the first-order
linear differential equation y' = a( x )y + b( x) can be written symbolically as

y(x ) = y 1 (x)(K + v(x)) where K is an arbitrary constant,

y J"' ( )d ix b(t)

1 (x)=e at t and v(x)= Yl(t)dt.

We define higher order linear differential equat ion as follows.

DEFINITIONS Homogeneous and Nonhomogeneous

Linear Differential Equations

Ann-th order linear differential equation is any differential equation

of the form

(1) an(x )y(n) (x) +an-1 (x )y(n-l) (x) + · · · + a1 (x )y(l) (x) + ao(x )y(x) = b(x)

where an(x) is not identically zero. The functions an(x), an_ 1 (x),... , a 1 (x),

and a 0 (x), which are all functions of the independent variable x alone, are

call ed the coefficient functions of (1).

If b(x) = 0, then the linear differential equation (1) is said to be homo-

geneous.

If b(x ) is not the zero function, then (1) is said to be nonhomogeneous.

Observe that equation (1) is linear in y and its derivatives. The differ-
ential equation x^2 y(^3 ) - 2exy(^2 ) + (cos x )y(l) + 7y = tan 4x is a third-order

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