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Handbook of mathematics for engineers and scienteists part 209. 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. | 1424 Functional Equations 2 . On the interval x e a b there exists a decreasing solution of the original equation involving an arbitrary function .l x _ i x for x e a c y ic 1 x for xe c b where c is an arbitrary point belonging to the interval a b and Q x is an arbitrary continuous decreasing function on a c such that lim Q x b Q c c. x a 0 3 . Solution in parametric form x e 2 . y where 0 t 0 t 1 is an arbitrary periodic function with period 1. 4 . Solution in parametric form x 01 t 02 t sin nt y 0i t - 02 t sin nt where 01 t and 02 t are arbitrary periodic functions with period 1. 5 . The original functional equation has a single increasing solution y x x. 6 . Particular solutions of the equation may be represented in implicit form using the algebraic or transcendental equation x y 0 where x y y x is some symmetric function with two arguments. 8. y y x ay x bx 0. 1 . General solution in parametric form x 0i t Aj 02 t A2 y 0i t A1 1 02 t A2 1 where A1 and A2 are roots of the quadratic equation A2 aA b 0 and 01 01 t and 02 02 t are arbitrary periodic functions with period 1. 2 . For 01 t C1 const and 02 t C2 const we have a particular solution in implicit form A2x - y x _ C Aix - y x 1 In Ai A2 - Ai C1 C2 Ai - A2 taAl 9. y y y x - x 0. A special case of equation T12.1.2.21. 1 . Particular solutions yi x _CE y2 x C - C2 y3 x Cl C x x C2 x where C C1 C2 are arbitrary constants. 2 . Solution in parametric form x 0 1 . y e i l . where 0 t 0 t l is an arbitrary periodic function with period 1. T12.1. Linear Functional Equations in One Independent Variable 1425 T12.1.2-3. Equations involving unknown function with three different arguments. 10. Ay ax By bx y x 0. This functional equation has particular solutions of the form y x Cx where C is an arbitrary constant and 3 is a root of the transcendental equation Aa Bb 1 0. 11. Ay xa By xb y x 0. This functional equation has particular solutions of the form y x C ln x p where C is an arbitrary constant and p is a root of the
