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Unformatted text preview: Math 334 Lecture #31 § 7.5: Homogeneous Linear Systems with Constant Coefficients Finding Solutions. For an n × n constant matrix A , what are n linearly independent solutions of system ~x = A~x ? Recall that solutions of homogeneous linear ODEs involve exponentials: are there any solutions of the system of the form ~x ( t ) = ~ ξe rt for a constant vector ~ ξ and a constant r ? Substitution of this “guess” into the system gives r ~ ξe rt = A ~ ξe rt ⇔ A ~ ξ = r ~ ξ ⇔ ( A rI ) ~ ξ = ~ . That is, the guess ~x ( t ) = ~ ξe rt is a solution of ~x = A~x if and only if r is an eigenvalue of A and ~ ξ is an eigenvector of A corresponding to r . Example. Find all solutions of ~x = A~x where A = 1 1 4 1 of the form ~x ( t ) = ~ ξe rt . The eigenvalues of the coefficient matrix A are the solutions of det( A rI ) = 0 ⇒ det 1 r 1 4 1 r = 0 ⇒ (1 r ) 2 4 = 0 ⇒ r 2 2 r + 1 4 = 0 ⇒ r 2 2 r 3 = 0 ⇒ r = 2 ± √ 4 + 12 2 = 1 ± 2 = 1 , 3 ....
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 Spring '11
 Smith
 Linear Algebra, Linear Systems, Vector Space, trivial solution, asymptotically stable node, R. 2e3t −2e−t

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