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M334Lec30

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

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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 0 ) = x 0 1 , . . . x n (0) = x 0 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 0 ) = x 0 , 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 0 such that x ( t 0 ) = x 0 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 0 , then the IVP has a unique solution x ( t ) defined on I . As before, this theorem does not tell us how to find the unique solution, but when the continuity conditions are satisfied, it does tell us that there is a unique solution to find.

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

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