8.1. Introduction

Any equation involving the derivatives, y¢, y², … , of a function y is called a differential equation (DE).  Any function y=f( x )  that satisfies the DE is called a solution.  The order of a DE is the order of the highest derivative in it.  Examples:

(a)     y =y  is 1st order with solutions y=c e x , where c is any constant.

(b)     y +16y=0  is 2nd order with solutions y= c 1 cos4x+ c 2 sin4x , where c1 and c2 are arbitrary constants.

 

Experiences have shown that it is hopeless to seek methods for solving all DE’s.  Instead, it is more fruitful to ask whether a given DE has any solution, and what properties of the solutions, if exist, can be deduced from the DE.  Thus, DEs provide a new source of functions.

 

Results of great generality about the solutions of DEs are usually difficult to obtain.  However, there are a few exceptions, e.g., the linear DEs that are our only concern.