tailieunhanh - Handbook of mathematics for engineers and scienteists part 134

Handbook of mathematics for engineers and scienteists part 134. 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. | . Linear Difference Equations with a Single Continuous Variable 899 . Linear nonhomogeneous difference equations with constant coefficients. An mth-order linear nonhomogeneous integer-difference equation with constant coefficients has the form amy x m am-1y x m - 1 a1y x 1 a0 y x f x where a0am 0. The general solution of the nonhomogeneous equation is given by the sum y x u x y x where u x is the general solution of the corresponding homogeneous equation with f 0 and y x is a particular solution of the nonhomogeneous equation . Regarding the solution of the homogeneous equation see Paragraph . Below we consider some methods for the construction of a particular solution of the nonhomogeneous equation . 1 . Table lists the forms of particular solutions corresponding to some special cases of the function on the right-hand side of the linear nonhomogeneous difference equation . 2 . Let P X amXm am-1Xm-1 a1 X a0 be the characteristic polynomial of equation . Consider the function 1 ro 9 X -F E9kXk X X1 k 0 where X1 is the root of the equation P X 0 with the smallest absolute value. Then mi9k k X1 -1. From it follows that s-1 a090 1 a190 a091 0 . as-k-19k 0 s 2 av 0 if v m. k 0 If _ lim f x k 1 k af X1 k ro then the series y x 90f x 91 f x 1 9k f x k is convergent and its sum gives a solution of equation . 3 . Let the right-hand side of equation can be represented by the integral f x y F X Xx-1 dX where the line of integration L does not cross the roots of the characteristic polynomial P X . Direct verification shows that the integral y x i F X X dX Jl p X represents a particular solution of equation . 900 Difference Equations and Other Functional Equations table Forms of particular solutions of the linear nonhomogeneous difference equation with constant coefficients amy x m am-1y x m - 1 aiy x 1 a0y x f x in some special .

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