204PS62007 - ues of the matrix in the rst order equation....

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Problem Set 6 Economics 204 - August 2007 Due Monday, 20 August in Evgeny Yakovlev’s mailbox in 612 Evans by 9am 1. (a) Show that the convex hull of a compact set in R n is compact. Hint :U s e Caratheodory’s Theorem, Theorem 1.10 on page 233 of de la Fuente. (b) Is the convex hull of a closed set in R n necessarily closed? 2. Let f : R n R n be a C 1 function. De±ne F : R n × R n R n by F ( x, ω )= f ( x )+ ω. Show that there is a set Ω 0 R n , of Lebesgue measure zero such that if ω/ Ω 0 ,then for each x 0 satisfying F ( x 0 0 ) = 0 there is an open set U containing x 0 ,anopenset V containing ω 0 ,anda C 1 function h : V U such that that for all ω V , x = h ( w ) is the unique element of U satisfying F ( x, ω ) = 0. (Hint: use the Transversality Theorem.) 3. Consider the second order linear di²erential equation given by y 0 = y y 0 . (a) Show how this equation can be rewritten as frst order linear di²erential equation of two variables y 1 and y 2 . (b) Describe the solutions of the ±rst order system (verbally) by ±nding the eigenval-
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Unformatted text preview: ues of the matrix in the rst order equation. (c) In a phase diagram, show the behavior of the system using the previous analysis and by solving for y 1 ( t ) = 0 and y 2 ( t ) = 0. (d) Give the solution of the system when y ( t ) = 0 and y ( t ) = 1. 4. Consider the initial value problem y 1 ( t ) y 2 ( t ) = 5 y 1 ( t ) 13 y 2 ( t ) y 3 1 ( t ) y 1 ( t ) y 2 2 ( t ) 13 y 1 ( t ) 5 y 2 ( t ) y 2 1 ( t ) y 2 ( t ) y 3 2 ( t ) , y 1 (0) = 3 , y 2 (0) = 3 (1) (a) Find the solution of the linearized initial value problem around the stationary point . (b) Show that the solution of the original nonlinear initial value problem converges to as t . 1...
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This note was uploaded on 08/01/2008 for the course ECON 204 taught by Professor Anderson during the Fall '08 term at University of California, Berkeley.

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