Method of Annihilators. In 20 through 22, we consider

Chapter 4, Problem 20

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Method of Annihilators. In 20 through 22, we consider another way of arriving atthe proper form of Y(t) for use in the method of undetermined coefficients. The procedureis based on the observation that exponential, polynomial, or sinusoidal terms (or sums andproducts of such terms) can be viewed as solutions of certain linear homogeneous differentialequations with constant coefficients. It is convenient to use the symbol D for d/dt. Then, forexample, et is a solution of (D + 1)y = 0; the differential operator D + 1 is said to annihilate,or to be an annihilator of, et. In the same way, D2 + 4 is an annihilator of sin 2t or cos 2t,(D 3)2 = D2 6D + 9 is an annihilator of e3t or te3t, and so forth.Show that linear differential operators with constant coefficients obey the commutativelaw. That is, show that(D a)(D b)f = (D b)(D a)ffor any twice-differentiable function f and any constants a and b. The result extends atonce to any finite number of factors.