tailieunhanh - Handbook of mathematics for engineers and scienteists part 204

Handbook of mathematics for engineers and scienteists part 204. 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. | . Linear Equations of the First Kind with Variable Limit of Integration 1389 26. i cos A x - i b y t dt f x f a 0. J a For b 0 see equation . For b -1 see equation . 1 . Solution for b b 1 0 y x y X sin k x - t f t dt where k xJ. b 1 k b 1 2 Ja V b 1 2 . Solution for b b 1 0 y x f - X 2 sinh k x - t f t dt where k X . b 1 k b 1 2 a b 1 27. i sin A x - t y t dt f x f a f a 0. a Solution y x XfXx x Xf x . 28. i sin AVx - t y t dt f x f a 0. a 2 d2 fx cosh A x -1 Solution y x r-------------- --------f t dt. nX dx2 J a y x -1 29. i Jo X x - t y t dt f x . a Here Jv z is the Bessel function of the first kind and f a 0. Solution 2 2 y x vf T7 X2 i x -1 J1 X x -t f t dt. X dx2 a 30. i Jo AVx -1 y t dt f x . a Here Jv z is the Bessel function of the first kind and f a 0. Solution d2 x __ y x T0 X x - t f t dt. 31. i I0 A x - t y t dt f x . a Here Iv z is the modified Bessel function of the first kind and f a 0. Solution 2 2 y x dx2 X2 x -1 I1 X x -1 f t dt. 1390 Integral Equations 32. i I0 XVx -1 y t dt f x . J a Here Iv z is the modified Bessel function of the first kind and f a 0. d2 i-x _ Solution y x .2 Jo X x - f t dt. dx J a rx 33. J g x - g t y t dt f x . It is assumed that f a fx a 0 and fx g x const. Solution y x d dx g x 34. y g x - g t b y t dt f x For b 0 see equation . Solution for b 0 f a 0. y x bfx x - 12gx x j exp g t bg x ft t dt 35. y g x h t y t dt f x f a 0. For h t -g t see equation . Solution d r x rx fi t dt 1 r rx h t t dt yX dx _g x h x J a t X CX J a g t h t 36. i K x - t y t dt f x . a 1 . Let K 0 1 and f a 0. Differentiating the equation with respect to x yields a Volterra equation of the second kind y x K x x - t y t dt f x x . a The solution of this equation can be represented in the form rx y x fx x J R x - t ft t dt. 1 Here the resolvent R x is related to the kernel K x of the original equation by R x L-1 1 pK p K p L K x -1 . Linear Equations of the Second Kind with Variable Limit of .