204A-noteDiffEq - Note on Linear Differential Equations...

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1 Note on Linear Differential Equations Econ 204A - Prof. Bohn 1 We will have to work with differential equations throughout this course. Differential equations – and their discrete-time analogs: difference equations – are economically interesting because they link levels to changes; we are often interested in linking the current situation or status of an economy to changes that we are trying to predict or understand. This note is about linear differential equations, linear relationships between a variable and its time- derivative. The general specification is (1) dy ( t ) dt = ! ( t ) " y ( t ) + x ( t ) The variable y = y ( t ) is a function of time, to be determined. The derivative dy/dt is also a function of time, variously denoted ) ( ) ( ' ) ( ) ( t y t y t t y dt dy dt d ! = = = . The terms γ (t) and x(t) are known functions of time, called the coefficients or forcing variables. Solving a differential equation means writing y(t) as function of time that does not involve the derivative. Categorizations: 1. Fixed and variable coefficients : The solutions simplify if γ and x are constants, also called fixed coefficients. Caution : Writers often suppress time-dependence when working with differential equations. That is, equation (1) is often written more compactly as (1’) x y y + ! = " ! Readers are expected to determine from the context if γ and/or x are constant or if they should be treated as variables. 2. Homogenous and non-homogenous equations : A differential equation is homogenous if there is no additive part, i.e., if 0 ) ( ! t x for all t. Otherwise it is non-homogenous. Useful fact: The solution to a non-homogenous equation is always the solution to the homogeneous part—omitting the x-part—plus a function of time. In some applications, the non-homogenous part is not economically interesting and one can simply examine the homogenous part. 3. General and special solutions : Equation (1) is typically solved by a parametric class of functions, which is called the general solution to (1). To pin down a unique y-function—a specific solution —we 1 Disclaimer and request: The Note is meant as a summary and reference, not a self-contained text. You should also read Barro/Sala-i-Martin’s appendix and (if you need it) consult a suitable math-for- economists text (e.g., Chiang). For the benefit of other students, please let me know if sections are unclear or you find glitches.
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2 need additional pieces of information called boundary conditions . For linear differential equations, the general solution is indexed by a single parameter (denoted A below). A single boundary condition is sufficient to determine this parameter. For this note, I will assume that the boundary condition is the y-value at a particular time t=t
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This note was uploaded on 12/26/2011 for the course ECON 240a taught by Professor Staff during the Fall '08 term at UCSB.

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204A-noteDiffEq - Note on Linear Differential Equations...

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