13.1. DEFINITIONS

A linear differential equation is the differential equation in which the dependent variable and its derivatives occur only in the first degree and are not multiplied together. Thus, the general linear differential equation of the n ^th order is of the form where P ₁, P ₂, , P _{n 1}, P _n and X are functions of x only.

A linear differential equation with constant coefficients is of the form

where a ₁, a ₂, , a _{n 1}, a _n are constants and X is either a constant or a function of x only.

13.2. THE OPERATOR D

The part of the symbol may be regarded as an operator such that when it operates on y, the result is the derivative of y.

Similarly, may be regarded as operators.

For brevity, we write

Thus, the symbol D is a differential operator or simply an operator.

Written in symbolic form, equation (1) becomes

or

where

i.e., f (D) is a polynomial in D.

The operator D can be treated as an algebraic quantity.

Thus

The polynomial f(D) can be factored by ordinary rules of algebra and the factors may be written in any order.

13.3. THEOREMS

Theorem 1. If y = y ₁, y = y ₂, , y = y _n are n...