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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 , , etc. We also identify vectors to column vectors , i.e., x = x 1 x 2 . . . x n , = 1 2 . . . 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 ], [ 1 , 2 , , 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 , ( t ) = [ 1 ( t ) , 2 ( t ) , , 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...
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