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Economics Dynamics Problems 214

# Economics Dynamics Problems 214 - 198 Economic Dynamics(i...

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198 Economic Dynamics (i) Show that the characteristic roots of the system are r = 5 and s = − 1. (ii) Derive the eigenvectors associated with the eigenvalues obtained in (i). (iii) Show that the solution values are: x ( t ) = c 1 e 5 t + c 2 e t y ( t ) = c 1 e 5 t c 2 e t and verify that c 1 e 5 t 1 1 , c 2 e t 1 1 are linearly independent. (iv) Given x (0) = 1 and y (0) = 0 , show that x ( t ) = 1 2 e 5 t + 1 2 e t y ( t ) = 1 2 e 5 t 1 2 e t 6. For the dynamic system ˙ x = x + 3 y ˙ y = 5 x + 3 y Show: (i) that the two eigenvalues are r = 6 and s = − 2 (ii) that the two eigenvectors are v r = 1 5 / 3 and v s = 1 1 (iii) and that the general solution satisfying x (0) = 1 and y (0) = 3 is x ( t ) = 3 2 e 6 t
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