1550078481-Ordinary_Differential_Equations__Roberts_

8 Ordinary Differential Equations

we introduce another notation, y(^2 ), for the second derivative. The tenth
derivative in this notation is y(^10 ), which is both easy to read and write. The
kth derivative of y is written y(k). Throughout the text, we will use all three
notations for the derivative of a function. (Caution: When first using this
new notation, one sometimes mistakenly writes y^2 for the second derivative
instead of y(^2 ). Of course, y^2 is "y squared" and not the second derivative of
y.)

In this text, we will deal mainly with ordinary differential equations. How-
ever, ordinary and partial differential equations are both subdivided into two
large classes, linear equations and nonlinear equations, depending on whether
the differential equation is linear or nonlinear in the unknown function and
its derivatives.

DEFINITIONS Linear and Nonlinear n-th Order Ordinary

Differential Equations

The general n-th order ordinary differential equation can be writ-

ten symbolically as

(6) F( x, y, y (1) ' ... ' y (n)) _ -^0.

Ann-th order ordinary differential equation is linear, if it can be written

in the form

(7) ao(x )y(n) + a1 (x )y(n-l) + · · · + an(x)y = g(x ).

The functions ak(x) are called the coefficient functions.

A nonlinear ordinary differential equation is an ordinary differential

equation which is not linear.

It follows from the definition that for an ordinary differential equation to be
linear it is necessary that:

1. Each coefficient function ak ( x) depends only on the dependent variable
   x and not on the independent variable y.


2. The independent variable y and all of its derivatives y(k) occur alge-
   braically to the first degree only. That is , the power of each term in-

volving y and its derivatives is l.

3. There are no terms which involve the product of either the independent
   variable y and any of its derivatives or two or more of its derivatives.