3.6we - 3.6: NONHOMOGENEOUS EQUATIONS AND METHOD OF...

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METHOD OF UNDETERMINED COEFFICIENTS KIAM HEONG KWA We would like to construct the general solution of the nonhomoge- neous equation (1) ay 00 + by 0 + cy = g ( t ) , where a,b,c R with a 6 = 0 and g ( t ) is a smooth function, i.e., a function that is differentiable as many times as we want, on an open interval I with the property that only a finitely many number of linearly independent terms arise as a result of repeated differentiation. This means that there is an N N and N constants c 0 ,c 1 , ··· ,c N - 1 R such that (2) ( c 0 ,c 1 ,c 2 , ··· ,c N - 1 ) 6 = (0 , 0 , 0 , ··· , 0) and (3) c 0 g ( t ) + c 1 g 0 ( t ) + c 2 g 00 ( t ) + ··· + c N - 1 g ( N - 1) ( t ) = 0 for all t I . It happens that such a function g ( t ) can be expressed as a finite sum of the following types of functions: (4) P ( t ) , P ( t ) e αt , P ( t ) e αt cos βt, and P ( t ) e αt sin βt. Here P ( t ) denotes a generic polynomial with real coefficients, while α,β R are constants. To calculate the general solution of (1), we first show that it suffices to have computed a particular solution of the same equation. This is the content of the following theorem. Theorem 1. 1 Let Y ( t ) be any solution of the nonhomogeneous equa- tion (5) y 00 + p ( t ) y 0 + q ( t ) y = g ( t ) , Date : January 22, 2011. 1
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3.6we - 3.6: NONHOMOGENEOUS EQUATIONS AND METHOD OF...

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