tailieunhanh - Handbook of mathematics for engineers and scienteists part 74

Tham khảo tài liệu 'handbook of mathematics for engineers and scienteists part 74', khoa học tự nhiên, toán học phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | . Second-Order Linear Differential Equations 479 where f f x k tan Ï. 2m J 3 . Let m be an odd integer. Then f x 1 4 cos -- r Vlf x dx n 1 if x 0 Le J0 4 J 2 k-1 f x -1 4 exp 2 h xf x dx if x 0 f x 1 4 cos -- T Vlf x l dx - n 1 if x 0 Le J0 4 J k f x -1 4 exp -2 j0 y f x dx if x 0 where f f x k sin Ï. 2m 4. Equations not containing y x. Equation coefficients are dependent on e. Consider an equation of the form e2y cx - f x E y 0 on a closed interval a x b under the condition that f 0. Assume that the following asymptotic relation holds f x E fk x Ek E 0. k 0 Then the leading terms of the asymptotic expansions of the fundamental system of solutions of equation are given by the formulas y1 f0-1 4 x exp -E j f0 x dx 11 fx dx 2 O e y2 f -1 4 x exp 1 i y f0 x dx 1 i - L- dx 1 O e . 0 Le J 2 J fx J 5. Equations containing y x. 1 . Consider an equation of the form Ey x 9 x y x f x y 0 on a closed interval 0 x 1. With g x 0 the asymptotic solution of this equation satisfying the boundary conditions y 0 C1 and y 1 C2 can be represented in the form y C1 - kC2 exp -e 1 g 0 x C2 exp i f x dx O e x g x . U1 f x 1 where k exp l0 d . 480 Ordinary Differential Equations 2 . Now let us take a look at an equation of the form e2 yXx eg x y x f x y 0 on a closed interval a x b. Assume D x g x 2 - 4f x 0. Then the leading terms of the asymptotic expansions of the fundamental system of solutions of equation as e - 0 are expressed by n D x l_ 1 4 exn I _i a D x dx_ dx dx 11 O e 1 yi X- x exp 2e J -L x tvx 2 J - D x xJ e J y2 D x 1 4 exp -1 VD x dx - 2 J Dj- dx 1 O . Equations of the general form. The more general equation 2 yXx eg x e yX f x e y 0 is reducible with the aid of the substitution y w exp - f g dx to an equation of the form e1 wXx f - 4 g2 - 2 g x w 0 to which the asymptotic formulas given above in Paragraph are applicable. . Boundary Value Problems . First second third and .