QABE_Notes18

QABE_Notes18 - QABE Lecture 18 Differential Equations...

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Unformatted text preview: QABE Lecture 18 Differential Equations & Growth I School of Economics, UNSW 2011 Contents 1 Introduction 1 2 Differential Equations 2 2.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2.2 Solution Technique . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.3 General and Definite Solutions . . . . . . . . . . . . . . . . . . . . . . . . 4 3 Topics in Growth 6 3.1 Exponential Growth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1 Introduction Last time we introduced the concept of the anti-derivative , otherwise known as the integral . This opened up to us the area of business dynamics , that is, where we are likely to start with an observation concerning the way a variable changes over time, and then move to an expression for the value of the variable at a chosen point in time. In this lecture, we extend these ideas to nibble off a small piece of a very large area in economic, engineering and physical mathematics that of differential equations . At this level, we will only consider the first class of such equations, those of first- order, and first-degree. There are, however, many more types of differential equations, some linear, some non-linear. Again, like we found with our integration rules, some very smart people have put together tables of differential equations. Thus, we only need to recognise the type of equation and then apply the correct rule from the table. Well investigate the derivation of a couple of these in this lecture but most of these derivations are beyond the scope of our course. So where do differential equations get applied in business? The answer is almost everywhere. The class we look at towards the end of this lecture that of exponential growth are so common in economic activity, that they are often referred to as natural growth equations. This partly has to do with the way that biological populations tend to grow (hence the natural bit), but also is about the commonness of this kind of growth. It comes about where the growth results in more entities (people, bacteria, firms etc.) that can accomplish further growth! That is, each cycle of growth, produces more ability for the population to grow; it builds on itself. Not surprisingly, this kindmore ability for the population to grow; it builds on itself....
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QABE_Notes18 - QABE Lecture 18 Differential Equations...

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