tailieunhanh - Handbook of mathematics for engineers and scienteists part 135

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 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 I4kl s 0 k l where the sum is over all zeroes of the characteristic function 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 is that the characteristic function has zero root 31 0. In this case the coefficients of equation 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 . 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 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 always has a solution y x in the class 1 a . The general solution of equation in the class 1 a can be represented as the sum of the general solution of the homogeneous equation and a particular solution y x of the nonhomogeneous equation . 2 . Suppose that the right-hand side of the equation is the polynomial n f x Z bsxs bn 0 n 0. s 0 Then equation has a particular solution of the form n y x xp CsXs Cn 0 s 0 where t 0 is a root of the characteristic function the multiplicity of this root being equal to j. The coefficients cn in 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. s 0 Then equation has a particular solution of the form y x epxxp Csxs cn 0 s 0