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Chapter 1 Introduction 1.1 Preliminaries Definition (Differential equation) A differential equation (DE) is an equation involving a function and its deriva- tives. Differential equations are called partial differential equations (PDE) or or- dinary differential equations (opr) according to whether or not they contain partial derivatives. The order of a differential equation is the highest order derivative occurring. A solution (or particular solution) of a differential equa- tion of order n consists of a function defined and n times differentiable on a domain D having the property that the functional equation obtained by substi- tuting the function and its n derivatives into the differential equation holds for every point in D. Example 1.1. An example of a differential equation of order 4, 2, and 1 is given respectively by 3 A. (#) + iy-2s de det Pe Oy aa + Bp 7% yy! =1. * 2 CHAPTER 1. INTRODUCTION Example 1.2. The function y= sin(z) is a solution of dy\> dy = — = i: 3 ( z) + 5 + y = 2sin(2) +c0s" (x) on domain R; the function z = e* cos(y) is a solution of 022 Ox 2p SG Ba? * dy? on domain R?; the function y = 2/2 is a solution of yy! =2 on domain (0,00). * Although it is possible for a DE to have a unique solution, e.g., y =0 is the solution to (y’)” + y? = 0, or no solution at all, eg., (y’)? + y? = —1 has no solution, most DE’s have infinitely many solutions. Example 1.3. The function y = 4a + C on domain (—C/4, 00) is a solution of yy’ = 2 for any constant C. x Note that different solutions can have different domains. The set of all solutions to a DE is call its general solution. 1.2 Sample Application of Differential Equations A typical application of differential equations proceeds along these lines: Real World Situation L Mathematical Model Solution of Mathematical Model t Interpretation of Solution 4 CHAPTER 1. INTRODUCTION To solve for y, we proceed as y-2= vi, (y—2)° =y, (irreversible step) y—4y+4=y, ye —5yt+4=0, (y¥-Dy-4) =9. Thus, the set y € {1,4} contains all the solutions. We quickly see that y = 4 satisfies Equation (1.1) because 4= V44+2—4=242—54=4, while y = 1 does not because L=V1+2—=1=3. So we accept y= 4 and reject y = 1. r 
