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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 12.2.5.8 . 12.2.5-5. Problems with boundary conditions of the third kind. We consider the third boundary value problem for equation 12.2.5.1 subject to condition 12.2.5.2 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 12.2.5.8 . 12.2.5-6. 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 12.2.5.1 with the boundary conditions yX 0 at x x1 y 0 at x x2. 1 . If q 0 the upper estimate 12.2.5.6 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 12.2.5.6 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 12.2.5.8 . 12.2. Second-Order Linear Differential Equations 487 12.2.6. Theorems on Estimates and Zeros of Solutions 12.2.6-1. Theorems on .