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Economics Dynamics Problems 158

Economics Dynamics Problems 158 - CHAPTER 4 Systems of...

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CHAPTER 4 Systems of first-order differential equations 4.1 Definitions and autonomous systems In many economic problems the models reduce down to two or more systems of differential equations that require to be solved simultaneously. Since most eco- nomic models reduce down to two such equations, and since only two variables can easily be drawn, we shall concentrate very much on a system of two equations. In general, a system of two ordinary first-order differential equations takes the form dx dt = ˙ x = f ( x , y , t ) dy dt = ˙ y = g ( x , y , t ) (4.1) Consider the following examples in which x and y are the dependent variables and t is an independent variable: ( i ) ˙ x = ax by ce t ˙ y = rx + sy qe t ( ii ) ˙ x = ax by ˙ y = rx + sy ( iii ) ˙ x = ax bxy ˙ y = rx sxy Examples (i) and (ii) are linear systems of first-order differential equations because they involve the dependent variables x and y in a linear fashion. Example (iii), on the other hand, is a nonlinear system of first-order differential equations because of the term xy occurring on the right-hand side of both equations
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