tailieunhanh - Handbook of mathematics for engineers and scienteists part 75

Tham khảo tài liệu 'handbook of mathematics for engineers and scienteists part 75', khoa học tự nhiên, toán học phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 486 Ordinary Differential Equations 2 . Suppose p x s x 1 and the function q q x has a continuous derivative. The following asymptotic relations hold for eigenvalues Xn and eigenfunctions yn x as n œ VK l n 12 1 Q Xi x2 o - 2 x2 - x1 n n - 1 n2 n n - 1 x - xi 1 r yn x cos---------------- ----- x1 - x Q x x2 x2 - x1 n n - 1 L x2 - x Q x1 x l sin n n - 1 x - x1 O M J x2 - x1 n2 where Q u v is given by . . Problems with boundary conditions of the third kind. We consider the third boundary value problem for equation subject to condition with a1 a2 1. We assume that p x s x 1 and the function q q x has a continuous derivative. The following asymptotic formulas hold for eigenvalues Xn and eigenfunctions yn x as n - œ Xn ------1 ---p Q x1 x2 - P1 b O 2 x2 - x1 n n - 1 L n2 n n - 1 x - x1 1 r r yn x cos---------- --- j x1 - x Q x x2 Q2 I x2 - x1 n n - 1 t x2 - x Q x1 x - 1 sin 1 O J x2 - x1 n2 where Q u v is defined by . . Problems with mixed boundary conditions. Let us note some special properties of the Sturm-Liouville problem that is the mixed boundary value problem for equation with the boundary conditions yX 0 at x x1 y 0 at x x2. 1 . If q 0 the upper estimate is valid for the least eigenvalue with z z x being any twice-differentiable function that satisfies the conditions z x x1 0 and z x2 0. The equality in is attained if z y1 x where y1 x is the eigenfunction corresponding to the eigenvalue A1. 2 . Suppose p x s x 1 and the function q q x has a continuous derivative. The following asymptotic relations hold for eigenvalues Xn and eigenfunctions yn x as n to v n n 2n 1 2 n Q x1 x2 o02i 2 x2 - x1 n 2n - 1 n2 n 2n - 1 x - x1 2 r yn x cos--2 x2 - x1 ---- n 2n - 1 x1 - x Q x x2 x2 - x Q x1 x sin n 2n 1 x x1 o 2 2 2 J 2 x2 - x1 n2 where Q u v is defined by . . Second-Order Linear Differential Equations 487 . Theorems on Estimates and Zeros of Solutions . Theorems on .