Math 54 - Quiz 10 Ans, Spring 2006

Math 54 - Quiz 10 Ans, Spring 2006 - 2 1 = 1 3 Then...

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Math 54, Quiz 10 Solutions Sections 202/203 GSI Carter 1. Give the general solution to the following system of differential equations. x 0 = - 2 - 1 0 0 - 1 - 1 1 1 0 x The characteristic polynomial of the matrix is ± ± ± ± ± ± - 2 - λ - 1 0 0 - 1 - λ - 1 1 1 - λ ± ± ± ± ± ± = - ( λ + 2) [( λ )( λ + 1) + 1] + 1 = - ( λ + 1) 3 . Then we have that A + I = - 1 - 1 0 0 0 - 1 1 1 1 which has a one-dimensional null space. Then our guess can be any vector not in the null space of ( A + I ) 2 = 1 1 1 - 1 - 1 - 1 0 0 0 such as v 1 = (0 , 0 , 1) T . Then let v 2 = ( A + I ) v 1 = (0 , - 1 , 1) T and v 3 = ( A + I ) v 2 = (1 , - 1 , 0) T . Then the general solution takes the form x ( t ) = c 1 exp( At ) v 1 + c 2 exp( At ) v 2 + c 3 exp( At ) v 3 = e - t ² c 1 ³ I + t ( A + I ) + t 2 2 ( A + I ) 2 ´ v 1 + c 2 ( I + t ( A + I )) v 2 + c 3 v 3 µ = e - t ² c 1 ³ v 1 + t v 2 + t 2 2 v 3 ´ + c 2 ( v 2 + t v 3 ) + c 3 v 3 µ = e - t c 1 0 0 1 + t 0 - 1 1 + t 2 2 1 - 1 0 + c 2 0 - 1 1 + t 1 - 1 0 + c 3 1 - 1 0 . 2. Give the general solution to the following system of differential equations. x 0 = 0 1 1 - 1 - 2 - 1 0 0 - 1 x The characteristic polynomial of the matrix is ± ± ± ± ± ± - λ 1 1 - 1 - 2 - λ - 1 0 0 - 1 - λ ± ± ± ± ± ± = - ( λ + 1) [
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Unformatted text preview: λ ( λ + 2) + 1] =-( λ + 1) 3 . Then we have that A + I =   1 1 1-1-1-1   1 which has a null space spanned by v 1 = (1 ,-1 , 0) T and v 2 = (1 , ,-1) T . We can let v 3 be any vector not in the null space, such as (0 , , 1) T . Note that ( A + I ) v 3 = v 1 . Then the general solutions takes the form x ( t ) = c 1 exp( At ) v 1 + c 2 exp( At ) v 2 + c 3 exp( At ) v 3 = e-t [ c 1 v 1 + c 2 v 2 + c 3 ( I + t ( A + I )) v 3 ] = e-t [ c 1 v 1 + c 2 v 2 + c 3 ( v 3 + t v 1 )] = e-t   c 1   1-1   + c 2   1-1   + c 3     1   + t   1-1       . 2...
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This note was uploaded on 02/20/2010 for the course AS a taught by Professor 11 during the Spring '10 term at École Normale Supérieure.

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Math 54 - Quiz 10 Ans, Spring 2006 - 2 1 = 1 3 Then...

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