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Tham khảo tài liệu 'handbook ofintegral equations phần 3', ngoại ngữ, ngữ pháp tiếng anh phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 30. y x A Ị x2 - t2 eXlx t y t dt f x . The substitution u x e Xxy x leads to an equation of the form 2.1.11 u x A i x2 - t2 u t dt f x e Xx. J0 31. y x aỊ x - t nex -x -tty t dt f x n 1 2 . Solution y x f x Ị R x - t f t dt R x r eXxy exp ơk x ơk cos pk x - Pk sin Pkx n 1 k 0 where ơk An n cosf 2nk n 1 . . . __L 2nk X pk An n 1 sin n 1 2nk n . 2nk nA ơk An n 1 cos 2 pk An n 1 sin n 1 n 1 for A 0 for A 0. 32. fx exp X x -1 y x b ----7 y t dt f x . Ja x x - t Solution y x eXx IF x nb2 y exp nb2 x - t F t dt j where F x e-Xxf x - bi e Xft dt. J a y x t 33. y x aỊ x - t tkexl x-t y t dt f x . The substitution u x e-Xxy x leads to an equation of the form 2.1.49 u x A i x - t tku t dt f x e-Xx. a 34. y x xk - tk ex x-t 1 y t dt f x . The substitution u x e-Xxy x leads to an equation of the form 2.1.52 u x aỊ xk - tk u t dt f x e-Xx. yx e x-t 35. y x - X A- y t dt f x Jo x - t a Solution y x f x y R x - t f t dt 0 a 1. - x Ar 1 - a xP-- n where R x Ax - 2 r ------------A-. n i xr n 1 - a 1998 by CRC Press LLC 36. y x A Ị exp X x2 -t2 y t dt f x . Solution y x f x - A i exp A x2 -t2 - A x -t f t dt. J a 37. y x A i exp Ax2 pt2 y t dt f x . J a In the case p -A see equation 2.2.36. This is a special case of equation 2.9.2 with g x -Aexp Ax2 and h t exp pt2 . 38. y x A í exp -XVt - x y t dt f x . J x This is a special case of equation 2.9.62 with K x A exp -Aự-x . 39. y x aÍ exp X xM -1 y t dt f x 1 0. a This is a special case of equation 2.9.2 with g x -A exp AxA and h t exp -At . Solution y x f x - A j exp X xM - tM - A x -t f t dt. 40. y x kj exp -X y t dt g x . This is a special case of equation 2.9.71 with f z ke 2. N For a polynomial right-hand side g x 52 Anxn a solution is given by n 0 N y x 12 n 0 An_ xn 1 kBn n Bn xn r - e- Ế n AFk r- k 0 2.3. Equations Whose Kernels Contain Hyperbolic Functions 2.3-1. Kernels Containing Hyperbolic Cosine x 1. y x - Al cosh Xx y t dt f x . a This is a special case of equation 2.9.2 with g x A cosh Xx and h t 1. Solution y x f x aỊ
