Method of Annihilators. In 20 through 22, we consider

Chapter 4, Problem 21

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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.Consider the problem of finding the form of a particular solution Y(t) of(D 2)3(D + 1)Y = 3e2t tet, (i)where the left side of the equation is written in a form corresponding to the factorizationof the characteristic polynomial.(a) Show that D 2 and (D + 1)2, respectively, are annihilators of the terms on the rightside of Eq. (i), and that the combined operator (D 2)(D + 1)2 annihilates both terms onthe right side of Eq. (i) simultaneously.(b) Apply the operator (D 2)(D + 1)2 to Eq. (i) and use the result of toobtain(D 2)4(D + 1)3Y = 0. (ii)Thus Y is a solution of the homogeneous equation (ii). By solving Eq. (ii), show thatY(t) = c1e2t + c2te2t + c3t2e2t + c4t3e2t + c5et + c6tet + c7t2et, (iii)where c1, ... , c7 are constants, as yet undetermined.(c) Observe that e2t, te2t, t2e2t, and et are solutions of the homogeneous equation correspondingto Eq. (i); hence these terms are not useful in solving the nonhomogeneousequation. Therefore, choose c1, c2, c3, and c5 to be zero in Eq. (iii), so thatY(t) = c4t3e2t + c6tet + c7t2et. (iv)This is the form of the particular solution Y of Eq. (i). The values of the coefficients c4, c6,and c7 can be found by substituting from Eq. (iv) in the differential equation (i).Summary. Suppose thatL(D)y = g(t), (v)where L(D) is a linear differential operator with constant coefficients, and g(t) is a sum orproduct of exponential, polynomial, or sinusoidal terms. To find the form of a particularsolution of Eq. (v), you can proceed as follows:(a) Find a differential operator H(D) with constant coefficients that annihilates g(t)that is,an operator such that H(D)g(t) = 0.(b) Apply H(D) to Eq. (v), obtainingH(D)L(D)y = 0, (vi)which is a homogeneous equation of higher order.(c) Solve Eq. (vi).(d) Eliminate from the solution found in step (c) the terms that also appear in the solutionof L(D)y = 0. The remaining terms constitute the correct form of a particular solution ofEq. (v).