UndeterminedCoefficients[2]

UndeterminedCoefficients[2] - equations are inconsistent,...

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The method of undetermined coefficients: "inspired guessing" for the particular solution of inhomogeneous linear constant coefficient ordinary differential equations of arbitrary order. Steps: (1) Determine the homogeneous solution. (2) If g(x) has several additive terms, decompose into individual terms by superposition and find a Yix) corresponding to each term separately. (3) Find each term of g(x) in the table below and examine the form of the trial Yix). (4) Choose the smallest boost power ~'s" that lifts every term of the gues~s in the, table for Yp(x) above every term of the homogeneous solution. (5) Plug the trial solution into the differential equation and determine the coefficients by equating like terms on both sides of t~e equation. Fot n arbitrary coefficients, exp~ct to find n equations. If you have the wrong number of equations, or if any of the
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Unformatted text preview: equations are inconsistent, or if th~ only solutions are trivial, you have either guessed incorrectly or made an error in differentiation or in algebra. The good news: no integration tricks needed in this method. "-, :1 Ii ;X,') g(x) , Yp( X) r i I, i, P ( )" b m + b m-'l,+ + b m X : OX iX ' . .. m . p '(x)eax , m i \ l"' !, ,sin X P m'( X ) iax cos ,~ xS(Aoxm + +Am) + A ) eax m . ' XS ((Aoxm + . .. + Am) eax cas x +(Boxm + .' . +B~)ea~ sinx) .. . Nates. l:ere f is the smallest ilonnegative i~!eger for which every term in Yp ( x) difers from very term in the complementar functionyc(x). Equivalently, for the three cases, s is, . tne numper, of, times 0 'is a ,.root of, the characteristic ~quation, a is a root of. the characteristic equation, and a t i is a root Qf the characteristic equation, respectively....
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This note was uploaded on 11/02/2009 for the course APMA 2102 taught by Professor Keyes during the Spring '08 term at Columbia.

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