M334Lec31 - Math 334 Lecture #31 7.5: Homogeneous Linear...

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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 ) = ~ e rt for a constant vector ~ and a constant r ? Substitution of this guess into the system gives r ~ e rt = A ~ e rt A ~ = r ~ ( A- rI ) ~ = ~ . That is, the guess ~x ( t ) = ~ e rt is a solution of ~x = A~x if and only if r is an eigenvalue of A and ~ 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 ) = ~ e 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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M334Lec31 - Math 334 Lecture #31 7.5: Homogeneous Linear...

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