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# L91 - Fall 2003 Math 308/501502 9 Matrix Methods for Linear...

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Fall 2003 Math 308/501–502 9 Matrix Methods for Linear Systems 9.1 Introduction (to [Linear] Systems) Wed, 05/Nov c 2003, Art Belmonte Summary General first-order systems A general first-order system of n differential equations in the n unknown functions x 1 , x 2 , . . . , x n , has the normal form x 0 1 = g 1 ( t , x 1 , x 2 , . . . , x n ) x 0 2 = g 2 ( t , x 1 , x 2 , . . . , x n ) · · · x 0 n = g n ( t , x 1 , x 2 , . . . , x n ). In vector form, x 0 = g ( t , x ) , where x 0 = x 0 1 ; x 0 2 ; . . ., ; x 0 n and g ( t , x ) = g 1 ( t , x ) ; g 2 ( t , x ) ; . . ., ; g n ( t , x ) are n × 1 column vectors. The independent variable is t . The dependent variables are x 1 , x 2 , . . . , x n . Or one could say the dependent column vector is x = [ x 1 ; x 2 ; . . . , ; x n ]. In this general system, g may have components that contain nonlinear expressions involving the dependent variables x 1 , x 2 , . . . , x n . Initial value problem This consists of the vector differential equation x 0 = g ( t , x ) together with the vector initial condition x ( t 0 ) = x 0 . Converting higher-order equations to systems Given the general n th order equation y ( n ) = g t , y , y 0 , y 00 , · · · , y ( n - 1 ) , let x 1 = y , x 2 = y 0 , x 3 = y 00 , . . . , x n = y ( n - 1 ) . We thus have x 0 k = y ( k ) = x k + 1 , k = 1 , 2 , . . . n - 1. Moreover, x 0 n = y ( n ) = g t , y , y 0 , y 00 , · · · , y ( n - 1 ) = g ( t , x 1 , x 2 , . . . , x n ).

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L91 - Fall 2003 Math 308/501502 9 Matrix Methods for Linear...

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