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Unformatted text preview: Math 219, Lecture 2 Ali Devin Sezer September 30, 2010 1 Review of the previous lecture We did three things in this lecture: 1. An informal introduction to differential equations. 2. Introduction of the types of diffential equations that this course will be concerned about 3. Classification of differential equations. 1.1 The subject of this course Let y : R → R , y : t → f ( t ) be a function that is at least n times differen- tiable, in other words d k y dt k exists for k ≤ n . It is very common to denote dy/dt with y ′ , d 2 y dt 2 with y ′′ and so on. For k ≥ 4, this notation becomes cumbersome and y ( k ) is instead used to denote d k y dt k . We will be working on the following types of differential equations: y ( n ) = F ( t, y, y ′ , y ′′ , ··· , y ( n − 1) ) . (1) The goal will be 1. To find all of the functions y that satisfies a given equation of this type, 2. To understand, as much as possible, the properties of the solutions. By a property we mean: is the solution of (1) a monotone function? is it positive for all values of t ? Does it have a limit lim t →∞ y ( t ) as t → ∞ and so on. 1 1.2 The classification of differential equations (1) is a class of differential equations. What are the parameters of this class? The simplest is the parameter n , which denotes the number of derivatives of y that appear in the equation. n is referred to as the order of the differential equation. For each value of n we get a subclass of equations of the class (1)...
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