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PS1_hints - tions This is one type of differential...

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Economic Growth and Development MS&E 249 Professor Olivier de La Grandville Fall 2008 Problem Set 1 Hints 1 Start with equation (6) in Solow paper, and see what equations you have for ˙ r when t < t 1 and t > t 1. You should have two different equations for r ( t ) for when t < t 1, and t > t 1. 2 Using the hint, ˙ x = dx/dt = (1 - α ) r - α dr/dt . Then solve for x ( t ) and change this to r ( t ). 3 For (b) and (c), please write whatever you’d like to say on the topics, but something meaningful :) 4 There can be a few different approaches to this problem. One can be using the fact that the slope of sy should be no less than n . Since we are assuming that y is concave, the minimum slope is approached as r goes to . So we need lim r →∞ sf 0 ( r ) > = n . diff. eqn The following is for those who do not have the background for solving differential equa-
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Unformatted text preview: tions. This is one type of differential equations you’re going to see in class many times. So it’s helpful for you to remember or refer to this. dx dt = ax + b, → dx ax + b = dt, → d ( x + b a ) a ( x + b a ) = dt, ( ∵ dx = d ( x + b a )) → 1 a Z d ( x + b a ) ( x + b a ) = Z dt, → 1 a ln ( x + b a ) = t + C, → ln ( x + b a ) = at + C , → e at + C = x + b a , → x ( t ) = e at + C-b a , → x ( t ) = e C e at-b a , depending on given initial condition , → x ( t ) = ( x + b a ) e at-b a , ( ∵ x = x (0) = e C-b a ) or → x ( t ) = ( x t * + b a ) e a ( t-t * )-b a , ( ∵ x t * = x ( t * ) = e C e at *-b a )...
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