ch03_6 - Ch 3.6: Nonhomogeneous Equations; Method of...

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Ch 3.6: Nonhomogeneous Equations; Method of Undetermined Coefficients Recall the nonhomogeneous equation where p , q, g are continuous functions on an open interval I . The associated homogeneous equation is In this section we will learn the method of undetermined coefficients to solve the nonhomogeneous equation, which relies on knowing solutions to homogeneous equation. ) ( ) ( ) ( t g y t q y t p y = + + 0 ) ( ) ( = + + y t q y t p y
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Theorem 3.6.1 If Y 1 , Y 2 are solutions of nonhomogeneous equation then Y 1 - Y 2 is a solution of the homogeneous equation If y 1 , y 2 form a fundamental solution set of homogeneous equation, then there exists constants c 1 , c 2 such that ) ( ) ( ) ( ) ( 2 2 1 1 2 1 t y c t y c t Y t Y + = - ) ( ) ( ) ( t g y t q y t p y = + + 0 ) ( ) ( = + + y t q y t p y
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Theorem 3.6.2 (General Solution) The general solution of nonhomogeneous equation can be written in the form where y 1 , y 2 form a fundamental solution set of homogeneous equation, c 1 , c 2 are arbitrary constants and Y is a specific solution to the nonhomogeneous equation. ) ( ) ( ) ( ) ( 2 2 1 1 t Y t y c t y c t y + + = ) ( ) ( ) ( t g y t q y t p y = + +
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Method of Undetermined Coefficients Recall the nonhomogeneous equation with general solution In this section we use the method of undetermined coefficients to find a particular solution Y to the nonhomogeneous equation, assuming we can find solutions y 1 , y 2 for the homogeneous case. The method of undetermined coefficients is usually limited
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This note was uploaded on 08/13/2009 for the course DIFF 2343632 taught by Professor Csar during the Fall '09 term at Middle East Technical University.

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ch03_6 - Ch 3.6: Nonhomogeneous Equations; Method of...

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