Handbook of mathematics for engineers and scienteists part 133. 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. | 892 Difference Equations and Other Functional Equations 3 . For equations with rational right-hand side y x 1 - ay x x c where c 0 and a 0 1 a particular solution is written in the form of the integral 1 c-1 y x by ---------dX. 4 . For the equation with exponential right-hand side n y x 1 - ay x bk Xkx k 0 there is a particular solution of the form y x bk X x eXk - a k 0 if a eXm bm xeXm x-1 V eXk x eXk - a k 0 ktm if a eXm where m 0 1 . n. 5 . For the equation with sinusoidal right-hand side y x 1 - ay x b sin 3x there is a particular solution of the form y x 2 1 ---------- cos 3 - a sin 3x - sin 3 cos 3x . 6 . For the equation with cosine in the right-hand side y x 1 - ay x b cos 3x there is a particular solution of the form y x ------- cos 3 - a cos 3x sin 3 sin 3x l. a2 1 - 2a cos 3 . Nonhomogeneous linear equations with the right-hand side of general form. 1 . Consider equation whose right-hand side is an analytic function that admits the expansion in power series œ g x E akxk k 0 which converges on the entire complex plane it follows that lim ak X k 0 . If k œ Tim k ak 1 k 2n k œ then the solution of equation can be represented as a convergent series œ y x S k 1 Bk 1 x k 0 where Bk x are Bernoulli polynomials see Paragraph Item 1 . Below we state a more general result that requires no conditions of the type . . Linear Difference Equations with a Single Continuous Variable 893 2 . Suppose that the right-hand side of equation is an analytic function admitting expansion by power series convergent on the entire complex plane such functions are called entire functions . In this case the solution of equation can be represented in the form tt y x 2 akfik x k 0 where fik x are Hurwitz functions defined by n k f etx -1 dt A e2isnx -1 ßk X 77 TTfcTT k 77 7 7 2ni Jm i et - 1 tk 1 2isn s 1 s k s u Pk 1 x k x Pn 1 x is a polynomial of degree n 1 k x is an entire 1-periodic
