Economics Dynamics Problems 181

Economics Dynamics Problems 181 - Systems of rst-order...

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Systems of Frst-order differential equations 165 and det( A λ I ) = λ 2 + 2 λ + 5, which leads to the roots r = 2 + 4 20 2 =− 1 + 2 i and s = 2 4 20 2 1 2 i The associated eigenvectors are ( A λ I ) v r = ± 2 2 i 4 22 2 i ² v r = ± 0 0 ² i.e. (2 + 2 i ) v 1 r + 4 v r 2 = 0 2 v r 1 + (2 2 i ) v r 2 = 0 Let v r 1 = 2, then v r 2 = 2(2 + 2 i ) / 4 = 1 + i . Thus u 1 = e ( 1 + 2 i ) t ± 2 1 + i ² Turning to the second root. With λ = s 1 2 i then ( A λ I ) v s = ± 2 + 2 i 4 + 2 i ²± v s 1 v s 2 ² i.e. ( 2 + 2 i ) v s 1 + 4 v s 2 = 0 2 v s 1 + (2 + 2 i ) v s 2 = 0 Choose v s 1 = 2, then v s 2 ( 2 + 2 i )(2) / 4 = 1 i . Hence the second solution is u 2 = e (1 + 2 i ) t ± 2 1 i ² i.e. v s is the complex conjugate of v r . Hence the general solution is x = c 1 e ( 1 + 2 i ) t ± 2 1 + i ² + c 2 e (1 + 2 i ) t ± 2 1 i ² These are, however, imaginary solutions. To convert them to real solutions we employ two results. One is Euler’s identity (see exercise 10 of chapter 2), i.e.
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This note was uploaded on 12/14/2011 for the course ECO 3023 taught by Professor Dr.gwartney during the Fall '11 term at FSU.

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