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# set23 - \$ Set 23 Systems of First Order Linear Equations...

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a39 a38 a36 a37 Set 23: Systems of First Order Linear Equations Part 4 Kyle A. Gallivan Department of Mathematics Florida State University Ordinary Differential Equations Fall 2009 1

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a39 a38 a36 a37 Methods for Nonhomogenesous Linear Systems We have methods to solve the homogeneous linear system of ODEs x ( t ) = Ax ( t ) with constant coefficient matrix A . We need methods to generate a particular solution for x ( t ) = Ax ( t ) + g ( t ) 2
a39 a38 a36 a37 Methods Methods we discussed earlier generalize: Method of Undetermined Coefficients Variation of Parameters Laplace Transforms (See textbook pages 438 and 439) Additionally, we can use the structure of A to transform the problem into several scalar problems. 3

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a39 a38 a36 a37 Diagonal A Suppose we have x ( t ) = Ax ( t ) + g ( t ) x 1 ( t ) x 2 ( t ) = α 11 0 0 α 22 x 1 ( t ) x 2 ( t ) + g 1 ( t ) g 2 ( t ) 4
a39 a38 a36 a37 Diagonal A x 1 ( t ) x 2 ( t ) = α 11 0 0 α 22 x 1 ( t ) x 2 ( t ) + g 1 ( t ) g 2 ( t ) x 1 ( t ) = α 11 x 1 ( t ) + g 1 ( t ) x 2 ( t ) = α 22 x 2 ( t ) + g 2 ( t ) Two independent scalar linear constant coefficient nonhomogeneous ODEs. 5

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a39 a38 a36 a37 Diagonal A We know how to solve them using techniques discussed earlier. Specifically, we can find particular solutions to combine with the homogeneous general solutions determined by eigenvalue analysis. This is identical to the combination you get by solving each independently with the general solutions, e.g.: x i ( t ) = c i e α ii t + e α ii t integraldisplay e α ii t g i ( t ) dt, i = 1 , 2 Of course, you can use any method you wish, on the scalar problem and get its essentially equivalent form. Therefore, diagonal structure in A means a particular vector solution is simply a vector of particular solutions from independent scalar linear ODEs.
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set23 - \$ Set 23 Systems of First Order Linear Equations...

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