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n20 - Atsushi Inoue ECG 765 Mathematical Methods for...

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Atsushi Inoue ECG 765: Mathematical Methods for Economics Fall 2009 Lecture Notes 20 First-Order Differential Equations Differential equations are used to model in continuous time. We will consider first-order differential equations and discuss how to obtain solutions. Outline A. Integration B. First-Order Differential Equations C. First-Order Differential Equations with a Time Varying Term D. Bernoulli Equations E. Application to the Solow-Swan Neoclassical Growth Model A. Integration Mean Value Theorem If f is continuous on [ a, b ], there is c such that f ( c ) = 1 b - a Z b a f ( x ) dx. (1) Fundamental Theorem of Calculus dI ( x ) dx = f ( x ) (2) where I ( x ) = R x a f ( s ) ds and f ( s ) is continuous. Integration by Parts Because ( u · v ) 0 = u 0 · v + u · v, 1
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u · v = Z u 0 · v + Z u · v 0 or equivalently Z u ( x ) v 0 ( x ) dx = u ( x ) v ( x ) - Z u 0 ( x ) v ( x ) dx. B. First-Order Differential Equations Consider first-order linear differential equations of the form dy dt = ay ( t ) + b First consider the homogenous case in which b = 0: 1 y dy dt = a. Integrating both sides, Z 1 y dy dt dt = Z adt. Z 1 y dy dt dt = Z adt. Suppose that y ( t ) > 0 for all t . The left-hand side is Z 1 y dy dt dt = Z 1 y dy = log( y ) + c 1 where as the right-hand side is Z adt = at + c 2 .
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