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Unformatted text preview: Inhomogeneous Linear n-th Order Differential Equations; The Method of Undetermined Coefficients We are ready now to discuss n-th 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 n-th 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 n-th 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the operator acting on y on the right hand side is the same as the characteristic polynomial p ( r ), except that r has been replaced by D , we will call that expression p ( D ), write the previous expression as p ( D ) y and write the original equation as p ( D ) y = g. In fact, for any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 m-th 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