Section 6.1

# Section 6.1 - Higher-Order Linear Differential Equations (...

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Unformatted text preview: Higher-Order Linear Differential Equations ( ) ( 1) 1 1 1 1 order linear differential equation: (1) ( ) ( ) ' ( ) ( ) where , , , , and are continuous functions on some interval . th n n n n n y p x y p x y p x y f x p p p f I--- + + + + = L K ( ) ( 1) 1 1 ( ) ( 1) 1 1 (1) is if ( ) 0 on : ( ) ( ) ( ) ( ) 0. If ( ) is not identically zero on , then (1) is . ( ) ( ) ( ) ( ) ( ). n n n n n n homogeneous f x I H y p x y p x y p x y f x I nonhomogeneous N y p x y p x y p x y f x---- + + + + = + + + + = L L ( ) ( 1) 1 1 1 2 1 2 [ ] ' is a ( ) : [ ] [ ] [ ], [ ] [ ]. Equations (H) and (N) can be written (H) [ ] 0, (N) [ ] ( ) n n n L y y p y p y p y linear differential operator L y y L y L y L cy cL y L y L y f x-- = + + + + + = + = = = L 1 1 ( 1) 1 1 Existence and Uniqueness Theorem: Let be any point on and let , , , be any real numbers. The initial-value problem [ ] ( ); ( ) , '( ) , , ( ) has a unique solution....
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## This note was uploaded on 02/09/2011 for the course MATH 3321 taught by Professor Morgan during the Fall '08 term at University of Houston.

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Section 6.1 - Higher-Order Linear Differential Equations (...

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