Unformatted text preview: â€“ HW10 â€“ gilbert â€“ (56540)
Find the solution of the diï¬€erential equation
du
= A u ( t) ,
dt when A is a 2 Ã— 2 matrix with eigenvalues 2,
and corresponding eigenvectors
1
,
1 v1 = 1
2 1. xk = 2e2t v1 + 4et/2 v2 correct
2. xk = 4e2t v1 âˆ’ 4et/2 v2
2t t/2 3. xk = 4e v1 + 4e Au(t) = 2e2t Av1 + 4et/2 Av2 so
u(t) = 2e2t v1 + 4et/2 v2
solves the diï¬€erential equation.
005 10.0 points Find the solution of the diï¬€erential equation
du
= A u ( t) ,
dt v2 A=
5. xk = 2e2t v1 + 2et/2 v2 Explanation:
Since v1 , v2 are eigenvectors corresponding
to distinct eigenvalues of A, they form an
eigenbasis for R2 . Thus 4e3t âˆ’ 10eâˆ’t
âˆ’4e3t + 2eâˆ’t 2. u(t) = 4eâˆ’3t âˆ’ 10et
âˆ’4eâˆ’3t + 2et 3. u(t) = âˆ’4eâˆ’3t + 10et
4e3t âˆ’ 2et 4. u(t) = âˆ’4e3t + 10eâˆ’t
4e3t âˆ’ 2eâˆ’t u(0) = c1 v1 + c2 v2 .
To compute c1 , c2 we apply row reduction to
the augmented matrix
u(0) ] =
âˆ¼ 1
0 1
1 âˆ’1
1 âˆ’2
6 02
,
14 correct Explanation:
Since
det[A âˆ’ Î»I ] = âˆ’2 âˆ’ Î»
1 âˆ’5
4âˆ’Î» = 5 âˆ’ (2 + Î»)(4 âˆ...
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This document was uploaded on 03/16/2014 for the course M 340L at University of Texas.
 Spring '08
 PAVLOVIC
 Matrices

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