Ordinary Differential Equations of Order > 1

Abstract

Many differential equations arising in engineering and the physical sciences, particularly in the study of vibratory or oscillatory phenomena, have the form

$$$$

(1)

in which α_n, α_n-1,…, α_l, α_o and f are given functions of x. Such an equation is known as a linear ordinary differential equation of order n. In this chapter it will sometimes be convenient to write a linear equation in the abbreviated form

$$$$

(2)

where L denotes a linear differential operator:

$$$$

The linearity property of L is expressed by the easily confirmed result

$$$$

where c₁ and c₂ are any constants and y and z any functions of x possessing derivatives up to the nth order.