Economics Dynamics Problems 158

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

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Unformatted text preview: CHAPTER 4 Systems of first-order differential equations 4.1 Definitions and autonomous systems (4.1) In many economic problems the models reduce down to two or more systems of differential equations that require to be solved simultaneously. Since most economic 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 ˙ = x = f (x, y, t) dt dy ˙ = y = g(x, y, t) dt Consider the following examples in which x and y are the dependent variables and t is an independent variable: ˙ x = ax − by − cet (i) ˙ y = rx + sy − qet ˙ x = ax − by (ii) ˙ y = rx + sy ˙ x = ax − bxy (iii) ˙ 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 in the system. Examples (ii) and (iii) are autonomous systems since the variable t does not appear explicitly in the system of equations; otherwise a system is said to be nonautonomous, as in the case of example (i). Furthermore, examples (ii) and (iii) are homogeneous because there is no additional constant. Example (i) is nonhomogeneous with a variable term, namely cet . A solution to system (4.1) is a pair of parametric equations x = x(t) and y = y(t) which satisfy the system over some open interval. Since, by definition, they satisfy the differential equation system, then it follows that the solution functions are differentiable functions of t. As with single differential equations, it is often necessary to impose initial conditions on system (4.1), which take the form x0 = x(t0 ) and y0 = y(t0 ) ...
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This note was uploaded on 12/14/2011 for the course ECO 3023 taught by Professor Dr.gwartney during the Fall '11 term at FSU.

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