tailieunhanh - Handbook of mathematics for engineers and scienteists part 206

Handbook of mathematics for engineers and scienteists part 206. 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 Second Kind with Constant Limits of Integration 1403 2 . The solution bounded at the endpoint x 1 and unbounded at the endpoint x -1 y x Af x -B f g X f t dt g x 1 x a 1 - x a 3 n 7-1 g t t - x where a is the solution of the trigonometric equation 2 on the interval -1 a 0. 3 . The solution unbounded at the endpoints _ Af w g x f t dt C ntx 1 T 1 -1- y x Af x - n J g t t x Cg x g x 1 x 1 - x where C is an arbitrary constant and a is the solution of the trigonometric equation 2 on the interval -1 a 0. 4. y x - Xfo t-TX x t-2x y t dt f x 0 x 1. Tricomi s equation. Solution . . 1 . 1 ta 1 - x a 1 1 V I C 1 - x g y x 1 . . z t-x-xr-iz d -x r- a arctan An -1 a 1 tan An -2 3 0 n 2 where C is an arbitrary constant. 5. y x A I e-x-ty t dt f x . 7o Solution for X -2 y x f x - exp -V1 2X x -t f t dt V1 2X Jo 1---- 1 i exp -V1 2X x 1 f t dt. V1 2A7 Jo 6. y x - A i e-x-tty t dt 0 A 0. J - Lalesco-Picard equation. Solution f C1 exp xV1 - 2X C2 exp -xV1 - 2X for 0 A - I 1 v 2 y x J C1 C2x for A 2 C1 cos xV2A - 1 C2 sin xV2A - 1 for A i 2 where C1 and C2 are arbitrary constants. 1404 Integral Equations 7. œ e lx-tly t dt f x . œ 1 . Solution for A -1 y x f x - A f exp -V1 2A x -t f t dt. A 2X J- 2 . If A -2 for the equation to be solvable the conditions f x cos ax dx 0 f x sin ax dx 0 J-oo J-oo where a -1 - 2A must be satisfied. In this case the solution has the form a2 1 y x f x ------ sin at f x t dt 2a Jo In the class of solutions not belonging to 2 - the homogeneous equation with f x 0 has a nontrivial solution. In this case the general solution of the corresponding nonhomogeneous equation with A - 2 has the form a2 1 y x C1 sin ax C2 cos ax f x ---- 4a -œ x œ . Î sin a x - t f t dt. J -oo 8. y x A i eA x t y t dt f x . J a 1 . The function y y x obeys the following second-order linear nonhomogeneous ordinary differential equation with constant coefficients y x A 2A - A y f x x A2 f x . 1 The boundary conditions for 1 have the form y x a Ay a f x