ps4-soln - Problem Set 4 Solutions February 3, 2010 Problem...

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Unformatted text preview: Problem Set 4 Solutions February 3, 2010 Problem 1 Compute e At using the Laplace transform method and the eigenvalue/eigenvector method for the following matrix: A = - 2- 2 1- 3- 4 . Using the Laplace transform method, we must compute ( sI- A )- 1 . We have: ( sI- A ) = s + 2 2 s- 1 3 s + 4 . We have that det( sI- A ) = ( s + 2)( s + 1)( s + 3) and hence, ( sI- A )- 1 = 1 ( s + 2)( s + 1)( s + 3) ( s + 1)( s + 3)- 2( s + 4)- 2 ( s + 2)( s + 4) ( s + 2)- 3( s + 2) s ( s + 2) . Next, we take the inverse Laplace transform of each entry. Firstly, we must simplify some of the entries using partial fraction expansions. We have- 2( s + 4) ( s + 2)( s + 1)( s + 3) = 4 s + 2- 3 s + 1- 1 s + 3- 2 ( s + 2)( s + 1)( s + 3) = 2 s + 2- 1 s + 1- 1 s + 3 s + 4 ( s + 1)( s + 3) = 3 / 2 s + 1- 1 / 2 s + 3 1 ( s + 1)( s + 3) = 1 / 2 s + 1- 1 / 2 s + 3- 3 ( s + 1)( s + 3) =- 3 / 2 s + 1 + 3 / 2 s + 3 s ( s + 1)( s + 3) =- 1 / 2 s + 1 + 3 / 2 s + 3 . Lastly, we have e At equals the inverse Laplace transform of the entries of ( sI- A )- 1 . Hence, e At = e- 2 t 4 e- 2 t- 3 e- t- e- 3 t 2 e- 2 t- e- t- e- 3 t 3 2 e- t- 1 2 e- 3 t 1 2 e- t- 1 2 e- 3 t- 3 2 e- t + 3 2 e- 3 t- 1 2 e- t + 3 2 e- 3 t .....
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ps4-soln - Problem Set 4 Solutions February 3, 2010 Problem...

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