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Unformatted text preview: Math 219, Lecture 8 Ali Devin Sezer October 21, 2010 Contents 1 Second order differential equations 1 2 First order, second order, what is the difference? 2 2.1 First order equations in R 2 . . . . . . . . . . . . . . . . . . . 2 2.2 The general first order differential equation in R 2 . . . . . . . 3 3 R 2 as a vector space 4 3.1 Linear functions from R 2 to R 2 . . . . . . . . . . . . . . . . . 5 4 Linear first order differential equation in R 2 7 5 How to solve first order differential equations in R 2 7 6 How to go from second order to first order 7 1 Second order differential equations Today we will begin studying second order linear differential equations; the third chapter of your textbook is on this subject. These are equations of the form: y ′′ ( t ) = f ( y ( t ) , y ′ ( t ) , t ) (1) where f : R 3 → R is a continuous function. To remind you, first order differential equations are equations of the form y ′ ( t ) = g ( y ( t ) , t ) , where g : R 2 → R . 1 2 First order, second order, what is the difference? An entire chapter of your book is devoted to first order equations and another chapter to second order equations. If you look at the contents of these chapters they very look different. This may give you the impression that indeed first and second order equations are very different in their natures and they require very different methods for their solution. This would be not an entirely correct impression. Second order equations can in fact be written as first order equations. To understand this, we need to learn a bit about first order differential equations in R 2 . In the next section this is what we will try to do. 2.1 First order equations in R 2 Remark 1. A member of R 2 is a vector with two components which are both real numbers, i.e., members of R . There is a variety of notation to denote a member of R 2 : ( x 1 , x 2 ) , parenleftbigg x 1 x 2 parenrightbigg , x , [ x 1 x 2 ] , or just x. Other notation is also possible. Instead of the letter x another letter may be used. If one writes α ∈ R 2 α should be thought of as a vector with two real 1 components. The equations we have studied so far have been of the form y ′ ( t ) = f ( t, y ( t )) (2) and we looked for functions y from an interval [ t , T ] to R satisfying this equation. Definition 2.1. A function from any interval I ⊂ R to any set S is called a curve in S ....
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