M334Lec30 - Math 334 Lecture #30 7.4: Basic Theory of...

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Unformatted text preview: Math 334 Lecture #30 7.4: Basic Theory of Systems of Linear First Order ODEs The Initial Value Problem. The IVP for a system of n first order linear ODEs in n functions x 1 , . . . , x n dependent on t is x 1 = p 11 ( t ) x 1 + . . . + p 1 n ( t ) x n + g 1 ( t ) , . . . . . . . . . . . . . . . x n = p n 1 ( t ) x 1 + . . . + p nn ( t ) x n + g n ( t ) , x 1 ( t ) = x 1 , . . . x n (0) = x n , for n 2 functiions p ij ( t ) and n function g i ( t ). In matrix form, this IVP is ~x = P ( t ) ~x + ~g ( t ) , ~x ( t ) = ~x , where x i is the i th component of ~x , p ij ( t ) is the ij th component of the n n matrix function P ( t ), and g i ( t ) is the i th component of the vector function ~g ( t ). A solution of the IVP is a vector function ~x ( t ) differentiable on an open interval I containing t such that ~x ( t ) = ~x and ~x ( t ) = P ( t ) ~x ( t ) + ~g ( t ) for all t in I. Existence and Uniqueness of Solutions. If the entries of P ( t ) and the components of ~g ( t ) are continuous on an open interval I containing t , then the IVP has a unique solution ~x ( t ) defined on I ....
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This note was uploaded on 03/08/2011 for the course MATH 334 taught by Professor Smith during the Spring '11 term at Vanderbilt.

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M334Lec30 - Math 334 Lecture #30 7.4: Basic Theory of...

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