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Topics: Polynomial
Math 251: The Annihilator Method: Using higher order homogeneous equations to solve non-homogeneous equations
The annihilator method is a fast method for solving certain non-homogeneous differential equations. A variation of this method is sometimes called the “method of judicious guessing” or the “method of undetermined coefficients.” In each variation, the work that must be done is the same; the difference is only in the background understanding of why the work is being done. The key idea of the annihilator method is to replace the problem of solving a non-homogeneous equation with the problem of solving a higher order homogeneous equation. The method is discussed in Section 2.11 of Codddington’s book. The more popular alternate approach is discussed in sections 5.4-5.5 of Trench’s book.
So we begin with a brief discussion of higher order linear homogeneous equations with constant coefficients. This is done in Section 2.7 of Coddington, as well as secition 9.2 of
Trench, in more depth and greater detail. Such depth is not necessary for our purposes.
So consider an equation of the form y (n) + an−1 y (n−1) + · · · a1 y + a0 y = 0.
Based on our experience with second order equations, we would naturally try a solution of the form y = erx . If you go through the motions of differentiating and substituting into the equation you will get p(r)erx = 0, where p(r) = rn + an−1 rn−1 + · · · + a1 r + a0 , which is as before called the characteristic polynomial. The difficulty is that now if n > 2, the polynomial is of higher degree than before and such polynomials are hard to factor and find roots. We do not have available the quadratic formula. There are cubic formulas and quartic formulas that are known and used to appear in books, but they are rarely taught any more and no such formulas are available for polynomials of degree 5 or higher. So in practice it can be very hard to find the roots of the characteristic polynomial. Nevertheless, we can at least imagine factoring

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