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m2k_dfq_nordnh

# m2k_dfq_nordnh - Inhomogeneous Linear n-th Order Dierential...

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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

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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the operator acting on y on the right hand side is the same as the characteristic polynomial p ( r ), except that r has been replaced by D , we will call that expression p ( D ), write the previous expression as p ( D ) y and write the original equation as p ( D ) y = g. In fact, for any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 0 g ( x ) + c 1 g 0 ( x ) + · · · + c m - 1 g ( m - 1) ( x ) .
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m2k_dfq_nordnh - Inhomogeneous Linear n-th Order Dierential...

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