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Handbook of mathematics for engineers and scienteists part 135. Tài liệu toán học quốc tế để phục vụ cho các bạn tham khảo, tài liệu bằng tiếng anh rất hữu ích cho mọi người. | 906 Difference Equations and Other Functional Equations Theorem. Any solution of equation 17.2.4.1 in the class of exponentially growing functions of a finite degree a can be represented in the form hk-1 y x E E CksXsefkx Pk x e kx 172.4.5 I4kl s 0 k l where the sum is over all zeroes of the characteristic function 17.2.4.2 in the circle t a Cks are arbitrary constants and Pk x are arbitrary polynomials of degrees nk - 1. Corollary 1. A necessary and sufficient condition for the existence of polynomial solutions of equation 17.2.4.1 is that the characteristic function 17.2.4.2 has zero root 31 0. In this case the coefficients of equation 17.2.4.1 should satisfy the condition am am-1 ai ao 0. Corollary 2. There is only one solution y x 0 in the class 1 a only if a min M 2I 17.2.4-2. Linear nonhomogeneous difference equations. 1 . A linear nonhomogeneous difference equation with constant coefficients in the case of arbitrary differences has the form dmy x hm am-1y x hm-1 a1y x hr a0 y x fo f x 17.2.4.6 where a0am 0 m 1 and hi hj for i j. Let f x be a function of exponential growth of degree a. Then equation 17.2.4.6 always has a solution y x in the class 1 a . The general solution of equation 17.2.4.6 in the class 1 a can be represented as the sum of the general solution 17.2.4.5 of the homogeneous equation 17.2.4.1 and a particular solution y x of the nonhomogeneous equation 17.2.4.6 . 2 . Suppose that the right-hand side of the equation is the polynomial n f x Z bsxs bn 0 n 0. 17.2.4.7 s 0 Then equation 17.2.4.6 has a particular solution of the form n y x xp CsXs Cn 0 17.2.4.8 s 0 where t 0 is a root of the characteristic function 17.2.4.2 the multiplicity of this root being equal to j. The coefficients cn in 17.2.4.8 can be found by the method of indefinite coefficients. 3 . Suppose that the right-hand side of the equation has the form f x epx bsxs bn 0 n 0. 17.2.4.9 s 0 Then equation 17.2.4.6 has a particular solution of the form y x epxxp Csxs cn 0 17.2.4.10 s 0
