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Edith Mayer 2022-05-17 Answered
Let
[ 2 1 0 0 2 1 0 0 2 ]
and
| x ( t ) | = ( x 1 2 ( t ) + x 2 2 ( t ) + x 3 2 ( t ) ) 1 / 2
Then any solution of the first order system of the ordinary differential equation
{ x ( t ) = A x ( t ) x ( 0 ) = x 0
satisfies
1. lim t | x ( t ) | = 0
2. lim t | x ( t ) | =
3. lim t | x ( t ) | = 2
4. lim t | x ( t ) | = 12
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Answers (1)

lizparker6q8h9o
Answered 2022-05-18 Author has 12 answers
Since −2 is the only eigenvalue of A, the general solution of x ( t ) = A x ( t ) has the form
x ( t ) = e 2 t ( c 1 v 1 ( t ) + c 2 v 2 ( t ) + c 3 v 3 ( t )
where c 1 , c 2 and c 3 are constants and v j ( t ) is a vector in R 3 whose coordinates are polynomials with grade j 1.
Hence | x ( t ) | 0 for t .
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