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Chapter 1 Lecture notes

# Chapter 1 Lecture notes - McGill University Math 263...

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McGill University Math 263: Differential Equations for Engineers CHAPTER 1: INTRODUCTION 1 Definitions and Basic Concepts 1.1 Ordinary Differential Equation (ODE) An equation involving the derivatives of an unknown function y of a single variable x over an interval x ( I ) . 1.2 Solution Any function y = f ( x ) which satisfies this equation over the interval ( I ) is called a solution of the ODE. For example, y = e 2 x is a solution of the ODE y 0 = 2 y and y = sin( x 2 ) is a solution of the ODE xy 00 - y 0 + 4 x 3 y = 0 . In general, ODE has infinitely many solutions, depending on a number of constants. 0-0

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1.3 Order n of the DE An ODE is said to be order n , if y ( n ) is the highest order deriva- tive occurring in the equation. The simplest first order ODE is y 0 = g ( x ) . The most general form of an n -th order ODE is F ( x, y, y 0 , . . . , y ( n ) ) = 0 with F a function of n + 2 variables x, u 0 , u 1 , . . . , u n . The equations xy 00 + y = x 3 , y 0 + y 2 = 0 , y 000 + 2 y 0 + y = 0 are examples of ODE’s of second order, first order and third order respectively with respectively F ( x, u 0 , u 1 , u 2 ) = xu 2 + u 0 - x 3 , F ( x, u 0 , u 1 ) = u 1 + u 2 0 , F ( x, u 0 , u 1 , u 2 , u 3 ) = u 3 + 2 u 1 + u 0 . In general, the solutions of n -th order equation may de- pend on constants. 1.4 Initial Conditions To determine a specific solution, one may impose some initial conditions (IC’s): 0-1
y ( x 0 ) = y 0 , y 0 ( x 0 ) = y 0 0 , · · · , y ( n - 1) ( x 0 ) = y ( n - 1) 0 . 1.5 First Order Equation and Direction Field Given y 0 ( x ) = F ( x, y ) , ( x, y ) R . Let R : a x b, c y d be a rectangular domain, and set a = x 0 < x 1 < · · · < x m = b be equally spaced poind in [ a, b ] and c = y 0 < y 1 < · · · < y n = d

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