AMATH_383_Solution_4 - AMATH 383 Solution #4 Winter 2010...

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AMATH 383 Solution #4 Winter 2010 Problem 10.1 (a). The happiness of the husband and wife in marriage is heading towards the equilibrium (x*,y*) , determined by setting f(x*,y*)=0 and g(x*,y*)=0, where f and g are the right-hand side of the differential equations. These two equations yield: 12 00 ,. aa x xy y yx rr   Solving, we get 0 11 0 x yy    x This equilibrium is stable if which is the case. 1 2 0, qr r a a  (b). (i) If , the numerators in the above expressions for the equilibrium solution are positive. The denominators are positive and less than 1. Therefore 0, 0  0 and 0. xx y The long-term happiness of the couple is enhanced by the marriage over their individual states. So this marriage is “regulated”. (ii). If , then The couple is less happy in marriage. So the marriage is “unregulated”. 0, 0  0 and 0. y Problem 10.2 For Eqs. (10.3) and (10.4) governing the marital interaction of a validating couple, the equilibrium is given by (10.5).
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This note was uploaded on 10/21/2011 for the course APPLIED MA AMATH 383 taught by Professor Hongqian during the Spring '10 term at University of Washington.

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AMATH_383_Solution_4 - AMATH 383 Solution #4 Winter 2010...

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