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Unformatted text preview: Fall Semester ’05’06 Akila Weerapana LECTURE 15: DIFFERENTIAL EQUATIONS I. INTRODUCTION • The last two lectures covered theory and applications in the area of difference equations. The next two lectures will cover theory and applications in the area of differential equations. You will notice many parallels between the two topics: in fact, differential equations are the continuous time analog of difference equations. In other words, a differential equation describes the behavior of an endogenous variable that is changing in a continuous fashion over time. • The similarities begin with the tools of analysis. The solution to a first order linear differential equation is found in the same fashion as the solution to a first order linear difference equation: by breaking down the solution into a complementary and a particular solution. We will also analyze the dynamics of a differential equation using a phase diagram. • The applications that we will study are also similar to the ones we already discussed in the last two lectures. Since most economic variables grow continuously, the differential equation representation of a model is likely to be more accurate than the difference equation repre sentation of that model. Once again, most of the applications we do will be drawn from the areas of macroeconomics, international economics and finance. II. SOLVING FIRST ORDER LINEAR DIFFERENTIAL EQUATIONS • A first order linear differential equation relates the continuous change in a variable to the levels of that variable and other exogenous variables. In other words a first order differential equation is of the form dy dt = f ( y t , x t ) where x is a vector of exogenous variables. • A convenient shorthand ‘dot’ notation is often used in differential equations. The derivative of a variable y with respect to time is denoted by dy dt = ˙ y . • A solution to a first order differential equation is the time path for the endogenous variable { y } . We can find this time path by calculating the general solution which is the sum of the particular solution, y P and the homogenous solution, y H . • The easiest differential equations to solve are first order linear differential equations of the form ˙ y = αy + β , where the slope and intercept terms are not functions of time....
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 Fall '10
 riley
 @, Akila Weerapana

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