285lect16 - 1 Lecture 16 1.1 Linear ODE of order n In a...

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Unformatted text preview: 1 Lecture 16 1.1 Linear ODE of order n In a similar way, we can consider a linear ODE of order n : y ( n ) + p 1 ( x ) y ( n & 1) + p 2 ( x ) y ( n & 2) + ::: + p n ( x ) y ( x ) = f ( x ) with the initial conditions: y ( x ) = b ; y ( x ) = b 1 ; :::; y ( n & 1) ( x ) = b n & 1 . We have the following Theorem 1 (Existence and uniqueness of solutions) Consider the ini- tial value problem y ( n ) + p 1 ( x ) y ( n & 1) + p 2 ( x ) y ( n & 2) + ::: + p n ( x ) y ( x ) = f ( x ) ; y ( x ) = b ; y ( x ) = b 1 ; (1) :::; (2) y ( n & 1) ( x ) = b n & 1 . (3) Suppose that the functions p j ( x ) ; j = 1 ; 2 ; :::; n; and f ( x ) are continuous on an open interval I = ( x & " 1 ; x + " 2 ) containing x . Then there is a unique solution y ( x ) ; x & " 1 < x < x + " 2 , of the initial value problem (3). De&nition 2 Functions f 1 ( x ) ; f 2 ( x ) ; :::; f n ( x ) ; a < x < b; are said to be linearly dependent if there are constants & 1 ; & 2 ; :::; & n such that & 2 1 + & 2 2 + ::: + & 2 n 6 = 0 and & 1 f 1 ( x ) + & 2 f 2 ( x ) + ::: + & n f n ( x ) , for every a < x < b . Otherwise functions f 1 ( x ) ; f 2 ( x ) ; :::; f n ( x ) ; a < x < b; are called linearly independent....
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This note was uploaded on 01/26/2012 for the course MATH 285 taught by Professor Staff during the Fall '08 term at University of Illinois, Urbana Champaign.

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285lect16 - 1 Lecture 16 1.1 Linear ODE of order n In a...

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