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Unformatted text preview: Supplementary material on finding particular solutions to systems of linear first order equations Diagonalization Method: Consider the system of n linear first order constant coefficient equations ( 29 y Ay g t = + . If A has n linearly independent eigenvectors ( 29 ( 29 ( 29 1 1 , , , n x x x then form the matrix ( 29 ( 29 ( 29 1 1 , , , n X x x x = where each column is one of the eigenvectors. Making the substitution y Xu = we get ( 29 ( 29 ( 29 ( 29 1 1 1 1 2 1 n Xu AXu g t X Xu X AX u X g t u u X g t ---- = + = + = + Hence we now have n uncoupled linear first order equations that can be solved by the methods learn in the first differential equations course. Once we have the us simply take y Xu = to get the particular solution....
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- Spring '08