7.1we - 7.1 AN INTRODUCTION TO SYSTEMS OF FIRST ORDER...

This preview shows pages 1–3. Sign up to view the full content.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: 7.1: AN INTRODUCTION TO SYSTEMS OF FIRST ORDER LINEAR EQUATIONS KIAM HEONG KWA The Notation For notational convenience, we denote vectors by boldfaced lower- case Latin letters x , y , etc. or lower-case Greek letters ξ , η , etc. We also identify vectors to column vectors , i.e., x = x 1 x 2 . . . x n , ξ = ξ 1 ξ 2 . . . ξ n , etc., whenever it is desired to list their components. For compactness, we usually write these column vectors as the transpose of the row vectors [ x 1 ,x 2 , ··· ,x n ], [ ξ 1 ,ξ 2 , ··· ,ξ n ], etc. For instance, if x is the transpose of [ x 1 ,x 2 , ··· ,x n ], then we write x = x 1 x 2 . . . x n = [ x 1 ,x 2 , ··· ,x n ] T . In this case, we also say that the the row vector [ x 1 ,x 2 , ··· ,x n ] is the transpose of x and write [ x 1 ,x 2 , ··· ,x n ] = x T = x 1 x 2 . . . x n T . Likewise, we denote vector-valued functions by x ( t ) = [ x 1 ( t ) ,x 2 ( t ) , ··· ,x n ( t )] T , ξ ( t ) = [ ξ 1 ( t ) ,ξ 2 ( t ) , ··· ,ξ n ( t )] T , etc. Date : February 24, 2011. 1 2 KIAM HEONG KWA The derivative of the vector-valued function x ( t ) is the function d x dt = dx 1 dt , dx 2 dt , ··· , dx n dt T . This is usually expressed as x ( t ) = [ x 1 ( t ) ,x 2 ( t ) , ··· ,x n ( t )] T , where the prime denotes differentiation with respect to the indepen- dent variable t . 1. Systems of Differential Equations With this notation, systems of n first order equations such as x 1 = F 1 ( t,x 1 ,x 2 , ··· ,x n ) , (1.1) x 2 = F 2 ( t,x 1 ,x 2 , ··· ,x n ) , . . . x n = F n ( t,x 1 ,x 2 , ··· ,x n ) , that arise naturally in problems involving several dependent variables x 1 ,x 2 , ··· ,x n , all of which are functions of the same single independent variable t , can be written compactly in the form (1.2) x = F ( t, x ) , where (1.3) F ( t, x ) = F 1 ( t,x 1 ,x 2...
View Full Document

{[ snackBarMessage ]}

Page1 / 9

7.1we - 7.1 AN INTRODUCTION TO SYSTEMS OF FIRST ORDER...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online