tailieunhanh - HANDBOOK OFINTEGRAL EQUATIONS phần 5

Tham khảo tài liệu 'handbook ofintegral equations phần 5', ngoại ngữ, ngữ pháp tiếng anh phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 2 . For A 2A A -k2 0 the general solution of equation 1 is given by y x C1 cosh kx C2 sinh kx f x --Ỵ sinh k x -1 f t dt 3 k J a where C1 and C2 are arbitrary constants. For A 2A A k2 0 the general solution of equation 1 is given by 2AA x y x C1 cos kx C2 sin kx f x ---J sin k x -1 f t dt. 4 For A 2A the general solution of equation 1 is given by y x C1 C2x f x 4A2 y x - t f t dt. 5 The constants C1 and C2 in solutions 3 - 5 are determined by conditions 2 . 30. y x a t sin A x - t y t dt f x . This is a special case of equation with g t At. The solution of the integral equation can be written via the Bessel functions or modified Bessel functions of order 1 3. 31. y x A Ị sin3 A x - t y t dt f x . Using the formula sin3 3 -1 sin 33 3 sin 3 we arrive at an equation of the form with n 2 y x i -4 A sin 3A x -1 4 A sin A x -t y t dt f x . a 32. y x Ak sin Afc x -1 y t dt f x -tt a b tt. 1 . Let us remove the modulus in the kth summand of the integrand Ik x sin Ak x - t y t dt sin Ak x - t y t dt sin Ak t - x y t dt. 1 J a J a J x Differentiating 1 with respect to x twice yields Ik Ak cos Ak x - t y t dt - Ak cos Ak t - x y t dt a x 2 . x Ị. b I k 2Aky x - Aị sin Ak x - t y t dt - Aị sin Ak t - x y t dt ax where the primes denote the derivatives with respect to x. By comparing formulas 1 and 2 we find the relation between I k and Ik Ik 2Ak y x - Ak Ik Ik Ik x . 3 1998 by CRC Press LLC 2 . With the aid of 1 the integral equation can be rewritten in the form y x 2 AkIk f x . 4 k 1 Differentiating 4 with respect to x twice and taking into account 3 we find that yXx x ny x - 2 Ak kjk fXX x ơn 2 Ak k. 5 k 1 k 1 Eliminating the integral In from 4 and 5 yields n-1 yxx x ơn An y x 2 Ak n - xk ik fXx x x2n f x . 6 k 1 Differentiating 6 with respect to x twice and eliminating In-1 from the resulting equation with the aid of 6 we obtain a similar equation whose left-hand side is a second-order linear differential operator acting on y with constant coefficients plus .