This preview shows pages 1–2. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
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 ....
View Full
Document
 Spring '11
 Smith
 Linear Systems

Click to edit the document details