tailieunhanh - Handbook of mathematics for engineers and scienteists part 211

Handbook of mathematics for engineers and scienteists part 211. 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. | 1438 Functional Equations 16. F x y do x y di x . y 6n-i x 0. Notation 0k x 0 x T where k 0 1 . n - 1. The functions G x are assumed to be periodic with period T . d x d x T . Furthermore the left-hand side of the equation is assumed to satisfy the condition F x . F x T . . In the original equation let us substitute x sequentially by x T with k 0 1 . n-1 to obtain the following system the original equation is given first F x yo y1 . yn-1 0 F x nt y1 y2 . yo o F x t yn-1 yo . yn-2 o where the notation yk y 0k x is used for brevity. Eliminating y1 y2 . yn-1 from the system of nonlinear algebraic or transcendental equations 1 one finds the solutions of the original functional equation in implicit form x yo o where yo y d x . . Functional Equations in Several Independent Variables . Linear Functional Equations . Equations involving functions with a single argument. 1- f x y f x f y - Cauchy s equation. Solution f x Cx where C is an arbitrary constant. 2 f x y f x f y Jensen s equation. Solution f x C1X C2 where C1 and C2 are arbitrary constants. 3. af x bf y f ax by c. 1 . Solution c Ax T UT if a b -1 Ax B if a b - 1 and c where A and B are arbitrary constants. 2 . If a b - 1 and c then there is no solution. . Functional Equations in Several Independent Variables 1439 4. Af aix biy ci Bf a2x b2y c2 Cf a3x b3y c3 D. All continuous solutions of this equation have the form f x ax 3 where the constants a and 3 are determined by substituting this expression into the original equation. 5. f x y f x - y 2f x 2f y . Solution f x Cx2 where C is an arbitrary constant. 6. f x y f x eay. Solution f x Ceax where C is an arbitrary constant. 7. f x y f x - y 2f x cosh y. Solution f x Ciex where C1 and C2 are arbitrary constants. 8. f x y f x - y 2f x cosh ay 2f y . Solution f x C 2 - cosh ax where C is an arbitrary constant. 9. f x y f x - y 2f x cos y. Solution f x Ci cos x C2 sin x where C1 and C2 are arbitrary constants. 10. f x y f x - y 2f x cos ay .