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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 λ λ λ ′ ′ = + ⇒ = + ⋅⋅⋅ ⋅⋅⋅ ′ ⇒ = + ⋅ ⋅ ⋅ ⋅⋅⋅ 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 u’s simply take y Xu = to get the particular solution....
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 Spring '08
 MEI
 Xu, φ, Fundamental matrix

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