tailieunhanh - Handbook of mathematics for engineers and scienteists part 210

Tham khảo tài liệu 'handbook of mathematics for engineers and scienteists part 210', khoa học tự nhiên, toán học phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | . Nonlinear Functional Equations in One Independent Variable 1431 . Functional equations involving y x and y ax . 10. y 2x - ay2 x 0. A special case of equation . Particular solution y x - eCx a where C is an arbitrary constant. 11. y 2x -2y2 x a 0. A special case of equation . Particular solutions with a 1 y x 0 y x cos Cx y x cosh Cx where C is an arbitrary constant. 12. y x y ax f x . A special case of equation . . Functional equations involving y x and y a x . 13. y x y a x b2. Solution y x b exp x a x where x z - z x is any antisymmetric function with two arguments. On setting x z C ln x - ln z one arrives at particular solutions of the form y ba C x2C where C is an arbitrary constant. 14. y x y a x f 2 x . The function f x must satisfy the condition f x f a x . For definiteness we take f x f a x . Solution y x f x exp x a x where x z - z x is any antisymmetric function with two arguments. On setting x z C ln x - ln z one arrives at particular solutions of the form y a C x2C f x where C is an arbitrary constant. 15. y2 x Ay x y a x By2 a x Cy x Dy a x f x . A special case of equation . Solution in parametric form w is a parameter y2 Ayw Bw2 Cy Dw f x w2 Ayw By2 Cw Dy f a x . Eliminating w gives the solutions in implicit form. 1432 Functional Equations . Other functional equations with quadratic nonlinearity. 16. y x2 - ay2 x 0. Solution y x - xC a where C is an arbitrary constant. In addition y x 0 is also a continuous solution. 17. y x y xa f x a 0. A special case of equation . 18. y x y a A2. V 1 bx z Solutions y x A exp i x a - x 1 bx where x z - z x is any antisymmetric function with two arguments. a - x 19. y x y T ta f x a_x - J. For definite-t z a - x x ness we take f x f ----- . 1 bx Solutions y x f x exp x a-x 1 bx where x z - z x is any antisymmetric function with two arguments. 20. y2 x Ay x y By x f x . V1 bx z A special case of equation . 21. y x y Va2 - x2 b2