notes5 - 1 e 1 t v 21 + c 2 e 2 t v 22 : In order to...

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MATH 1005A - Notes 5 Systems of Equations A system of two, linear, homogeneous equations with constant coe±cients has the form x 0 = ax + by y 0 = cx + dy; where a; b; c and d are constants, and x and y are functions of t . The system may be expressed in matrix form as μ x y 0 = μ x 0 y 0 = μ ab cd ¶μ x y : Letting x = μ x y and A = μ ab cd , we obtain the matrix di®erential equation x 0 = A x . Analogous to second-order, linear, homogeneous equations, we seek solutions in the form x = e ¸t v ,where ¸ is a constant and v isavectortobedeterm ined . x = e ¸t v ) x 0 = ¸e ¸t v ; so x 0 = A x , ¸e ¸t v = Ae ¸t v , A v = ¸ v ; i.e., ¸ is an eigenvalue of A with the corresponding eigenvector v . If the matrix A has two independent eigenvectors v 1 and v 2 , corresponding to the eigen- values ¸ 1 and ¸ 2 respectively (distinct or not), then x 1 = e ¸ 1 t v 1 and x 2 = e ¸ 2 t v 2 are two independent solutions of the matrix equation, with the general solution x = c 1 x 1 + c 2 x 2 , where c 1 and c 2 are arbitrary constants. In terms of the original variables, if v 1 = μ v 11 v 21 and μ v 12 v 22 ,then μ x y = c 1 e ¸ 1 t μ v 11 v 21 + c 2 e ¸ 2 t μ v 12 v 22 ) ½ x = c 1 e ¸ 1 t v 11 + c 2 e ¸ 2 t v 12 y = c 1 e
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Unformatted text preview: 1 e 1 t v 21 + c 2 e 2 t v 22 : In order to determine the eigenvalues of a matrix A , set det( I A ) = 0, where I is the identity matrix, and solve for . For each eigenvalue , the corresponding eigenvectors are the nonzero solutions v of the matrix equation ( I A ) v = 0. If the 2 2 matrix A has repeated eigenvalues 1 = 2 = with only one independent eigenvector v , then one solution is given by x = e t v . The determination of a second, inde-pendent solution is beyond the scope of this course. If the 2 2 matrix A has complex eigenvalues, then x 1 = e 1 t v 1 and x 2 = e 2 t v 2 are two complex solutions. The extraction of two, real, independent solutions from the complex ones is beyond the scope of this course....
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This document was uploaded on 04/14/2010.

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