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Unformatted text preview: Inhomogeneous Linear nth Order Differential Equations; The Method of Undetermined Coefficients We are ready now to discuss nth order linear inhomogenous differential equations. In general these take the form d n y dx n + p 1 ( x ) d n 1 y dx n 1 + ... + p n 1 ( x ) dy dx + p n ( x ) y = g ( x ) . To obtain a general solution we obtain the general solution of the corresponding homogeneous equation with g ( x ) = 0 and we add to that a particular solution, i.e., any solution, y p ( x ) of the homogeneous equation. While there are nth order versions of the method of variation of parameters these are very com plex and we will not treat them here. We will concentrate on the constant coefficient case and the method of undetermined coefficients. Differential Operators We consider now the nth order lin ear constant coefficient inhomogeneous equation L y = d n y dx n + p 1 d n 1 y dx n 1 + ... + p n 1 dy dx + p n y = g ( x ) , wherein the p k are real constants. The characteristic polynomial associated with the corresponding homogeneous equation is then p ( r ) r n + p 1 r n 1 + ... + p n 1 r + p n . We have noted that it is also possible to write the left hand side of the differential equation, using the differential operator notation, as L y = D n y + p 1 D n 1 y + ... + p n 1 D y + p n y. 1 Because the operator acting on y on the right hand side is the same as the characteristic polynomial p ( r ), except that r has been replaced by D , we will call that expression p ( D ), write the previous expression as p ( D ) y and write the original equation as p ( D ) y = g. In fact, for any polynomial, of any degree m , in r , q ( r ) = r m + q 1 r m 1 + + q m 1 r + q m , we will define q ( D ) = D m + q 1 D m 1 + ... + q m 1 D + q m I, where I denotes the identity operator such that I y = y . Back to Finite Families We will suppose again that the function g ( x ) appearing as the inhomogeneous term belongs to a finite family under differentiation. We recall that this means for some positive integer m the mth derivative of g ( x ) is a linear combination of g ( x ) itself and its derivatives of order k, k = 1 , 2 , . . . m 1: g ( m ) ( x ) = c g ( x ) + c 1 g ( x ) + + c m 1...
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This note was uploaded on 01/23/2012 for the course MATH 4254 taught by Professor Robinson during the Spring '10 term at Virginia Tech.
 Spring '10
 ROBINSON
 Equations

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