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Unformatted text preview: 2 = 3 , A3 I = 6 61010 1 1 , and v 2 = (1 , 1) . Writing the eigenvectors as columns, P = 31 5 1 will have P1 AP = 1 3 = D . Now the method of Section 6.2 is to solve the new system of DE y = P1 AP y, which is the uncoupled system y 1 =y 1 y 2 = 3 y 2 , and has solution z 1 = c 1 et , z 2 = c 2 e 3 t so the original system had solution y = P z, which we write as y 1 y 2 = 31 5 1 c 1 et c 2 e 3 t = c 1 et 3 1 + c 2 e 3 t 1 1 . Notice that this solution is a linear combination of two LI solutions of the form e j t v j , where v j is an eigenvector with eigenvalue j . For nondiagonalizable systems, we recall that the method of the Supplement Part B is (except for the linear algebra, nding the matrix P so P1 AP = J ) described in our text, 6.3, as in Example 2....
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 Fall '08
 zhang
 Math

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