Economics Dynamics Problems 42

Economics Dynamics Problems 42 - CHAPTER 2 Continuous...

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Unformatted text preview: CHAPTER 2 Continuous dynamic systems 2.1 Some definitions A differential equation is an equation relating: (a) (b) (c) (d) the derivatives of an unknown function, the function itself, the variables in terms of which the function is defined, and constants. More briefly, a differential equation is an equation that relates an unknown function and any of its derivatives. Thus dy + 3xy = ex dx is a differential equation. In general dy = f (x, y) dx is a general form of a differential equation. In this chapter we shall consider continuous dynamic systems of a single variable. In other words, we assume a variable x is a continuous function of time, t. A differential equation for a dynamic equation is a relationship between a function of time and its derivatives. One typical general form of a differential equation is dx = f (t, x) dt (2.1) Examples of differential equations are: (i) dx + 3x = 4 + e−t dt (ii) d2 x dx + 4t − 3(1 − t2 )x = 0 dt2 dt (iii) dx = kx dt (iv) ∂u ∂v + + 4u = 0 ∂t dt ...
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