Handbook of mathematics for engineers and scienteists part 132. 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 885 The next value ym is calculated by substituting the initial values into the right-hand side of equation with n 0. We have ym f 0 y0 yi . ym-1 . Then one takes n 1 in . We get ym 1 f 1 yi y2 . ym . Substituting the initial values y1 . ym-1 and the calculated value ym into this relation we find ym 1. Further taking n 2 in and using the initial values y2 . ym-1 and the calculated values ym ym 1 we find ym 2. In a similar way we consecutively find all subsequent values ym 3 ym 4 . The above method of solving difference equations is called the step method. . Linear Difference Equations with a Single Continuous Variable . First-Order Linear Difference Equations . Homogeneous linear difference equations. General properties of solutions. 1 . A first-order homogeneous linear difference equation has the form y x 1 - f x y x 0 where f x is a given continuous real-valued function of real argument and y x is an unknown real-valued function 0 x to. Let y1 y1 x be a nontrivial particular solution of equation y1 x 0. Then the general solution of equation is given by y x 0 x y1 x where 0 x 0 x 1 is an arbitrary 1-periodic function. Example 1. The equation with constant coefficients y x 1 - ay x 0 with a 0 admits a particular solution y0 x ax. Therefore the general solution of equation has the form y x 0 x ax where 0 x is an arbitrary periodic function with unit period. In the special case of a 1 the general solution of equation is an arbitrary 1-periodic function. Remark 1. When using formula for obtaining continuous particular solutions from a given continuous particular solution one should consider not only continuous but also discontinuous or unbounded periodic functions G x . For instance the equation y x 1 y x 0 which determines antiperiodic functions of unit .
