Economics Dynamics Problems 45

Economics Dynamics Problems 45 - curve’s general shape...

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Continuous dynamic systems 29 while substituting for p we obtain ap bp 2 = a 2 p 0 bp 0 + ( a bp 0 ) e at b ± ap 0 bp 0 + ( a bp 0 ) e at ² 2 = a 2 p 0 ( a bp 0 ) e at [ bp 0 + ( a bp 0 ) e at ] 2 which is identically true for all values of t . Equation x ( t ) = ce kt is an explicit solution to example (iii) because we can solve directly x ( t ) as a function of t . On occasions it is not possible to solve x ( t ) directly in terms of t , and solutions arise in the implicit form F ( x , t ) = 0 (2.4) Solutions of this type are referred to as implicit solutions . A graphical solution to a Frst-order differential equation is a curve whose slope at any point is the value of the derivative at that point as given by the differential equation. The graph may be known precisely, in which case it is a quantitative graphical representation. On the other hand, the graph may be imprecise, as far as the numerical values are concerned; yet we have some knowledge of the solution
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Unformatted text preview: curve’s general shape and features. This is a graph giving a qualitative solution. The graph of a solution, whether quantitative or qualitative, can supply con-siderable information about the nature of the solution. ±or example, maxima and minima or other turning points, when the solution is zero, when the solution is increasing and when decreasing, etc. Consider, for example, dx / dt = t 2 whose solution is x ( t ) = t 3 3 + c where c is the constant of integration. There are a whole series of solution curves depending on the value of c . ±our such curves are illustrated in Fgure 2.1, with solutions x ( t ) = t 3 3 + 8 , x ( t ) = t 3 3 + 2 , x ( t ) = t 3 3 , x ( t ) = t 3 3 − 3 Figure 2.1....
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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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