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Economics Dynamics Problems 43

Economics Dynamics Problems 43 - Continuous dynamic systems...

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Continuous dynamic systems 27 In each of the first three examples there is only one variable other than time, namely x . They are therefore referred to as ordinary differential equations . When functions of several variables are involved, such as u and v in example (iv), such equations are referred to as partial differential equations . In this book we shall be concerned only with ordinary differential equations. Ordinary differential equations are classified according to their order. The order of a differential equation is the order of the highest derivative to appear in the equation. In the examples above (i) and (iii) are first-order differential equations, while (ii) is a second-order differential equation. Of particular interest is the linear differential equation , whose general form is a 0 ( t ) d n x dt n + a 1 ( t ) d n 1 x dt n 1 + . . . + a n ( t ) x = g ( t ) (2.2) If a 0 ( t ), a 1 ( t ), . . . , a n ( t ) are absolute constants, and so independent of t , then equation (2.2) is a constant-coefficient n th-order differential equation . Any differential equation not conforming to equation (2.2) is referred to as a
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