Economics Dynamics Problems 176

Economics Dynamics Problems 176 - are after the general...

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160 Economic Dynamics It follows immediately that the roots are: (1) real and distinct if tr( A ) 2 > 4det( A ) (2) real and equal if tr( A ) 2 = 4det( A ) (3) complex conjugate if tr( A ) 2 < 4det( A ). 4.5 Solutions to the homogeneous differential equation system: real distinct roots Suppose we have an n -dimensional dynamic system ˙x = Ax (4.21) where ˙x = ˙ x 1 ˙ x 2 . . . ˙ x n , A = a 11 a 12 ··· a 1 n . . . . . . . . . . . . a n 1 a n 2 ··· a nn , x = x 1 x 2 . . . x n and suppose u 1 , u 2 , ... , u n are n linearly independent solutions, then a linear combination of these solutions is also a solution. We can therefore express the general solution as the linear combination x = c 1 u 1 + c 2 u 2 + ... + c n u n where c 1 , c 2 , ... , c n are arbitrary constants. In the case of just two variables, we
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Unformatted text preview: are after the general solution x = c 1 u 1 + c 2 u 2 In chapter 2, where we considered a single variable, we had a solution x = ce rt This would suggest that we try the solution 5 x = e λ t v where λ is an unknown constant and v is an unknown vector of constants. If we do this and substitute into the differential equation system we have λ e λ t v = A e λ t v eliminating the term e λ t we have λ v = Av i.e. ( A − λ I ) v = For a nontrivial solution we require that det( A − λ I ) = 5 Here u = e rt v ....
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