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lecture_19 (dragged) - MA 36600 LECTURE NOTES: MONDAY,...

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MA 36600 LECTURE NOTES: MONDAY, MARCH 2 Method of Undetermined Coefficients Undetermined Coeﬃcients. Say that we wish to solve a constant coeﬃcient linear second order difer- ential equation in the Form ay °° + by ° + cy = f ( t ) We know how to ±nd the general solution y = y ( t ) once we ±nd homogeneous solutions y 1 = y 1 ( t ) and y 2 = y 2 ( t ), so we explain a method to ±nd a particular solution Y = Y ( t ). ²ollow these three steps: #1. Express the Function on the right-hand side as the sum oF Functions f ( t )= f 1 ( t )+ f 2 ( t )+ ··· + f n ( t ) where each f i ( t ) is the product oF a polynomial, an exponential Function, and a trigonometric Func- tion. That is, say that we can write f i ( t )= d i ° j =0 a ij t j e α i t cos β i t + d i ° j =0 b ij t j e α i t sin β i t For some constants a ij , b ij , α i , and β i . Note that f i ( t ) involves a polynomial oF degree d i . #2. Make a guess that a solution Y i = Y i ( t ) oF the nonhomogeneous equation a ( t ) Y °° i + b ( t ) Y ° i + c ( t ) Y i = f i ( t ) For i =1 , 2 ,...,n ; is in the Form Y
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This note was uploaded on 11/30/2011 for the course MATH 366 taught by Professor Edraygoins during the Spring '09 term at Purdue University-West Lafayette.

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