285lect17 - 1 Lecture 17 De&nition 1 A set of solutions...

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Unformatted text preview: 1 Lecture 17 De&nition 1 A set of solutions y 1 ( x ) ;y 2 ( x ) ;:::;y n ( x ) ;a < x < b , of the linear ODE of order n y ( n ) + p 1 ( x ) y ( n & 1) + p 2 ( x ) y ( n & 2) + ::: + p n ( x ) y ( x ) = 0 is called the fundamental set of solutions if W ( y 1 ;y 2 ;:::;y n ) 6 = 0 at least at one point x satisfying a < x < b: Theorem 2 IF y 1 ( x ) ;y 2 ( x ) ;:::;y n ( x ) ;a < x < b , is a fundamental set of solutions of the linear ODE of order n y ( n ) + p 1 ( x ) y ( n & 1) + p 2 ( x ) y ( n & 2) + ::: + p n ( x ) y ( x ) = 0 ; then, every solution y ( x ) can be written as a linear combination of solutions y 1 ( x ) , y 2 ( x ) ;::;y n ( x ) : y ( x ) = C 1 y 1 ( x ) + C 2 y 2 ( x ) + ::: + C n y n ( x ) ;a < x < b . If y 1 ( x ) ;y 2 ( x ) ;:::;y n ( x ) ;a < x < b; is a fundamental set of solutions, then y ( x ) = C 1 y 1 ( x ) + C 2 y 2 ( x ) + ::: + C n y n ( x ) ;a < x < b; is called general solution of the linear ODE of order n ....
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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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285lect17 - 1 Lecture 17 De&nition 1 A set of solutions...

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