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Advanced Mathematical Methods for Scientists and Engineers Episode 3 Part 9

Tham khảo tài liệu 'advanced mathematical methods for scientists and engineers episode 3 part 9', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | We solve this system with Kramer s rule. y2 C C _ yi C p C -w C 2 C p C -w C Here W x is the Wronskian of y1 x and y2 x . The Green function is yi x y2 i P QW i y2 x yi g p i W e for a x C for C x b. The solution of the Sturm-Liouville problem is y Ị G x C f C dC Result 21.7.2 The problem L y p x y q x y f x subjectto Bi y aiy a W a 0 B2 y Ay b 2y b 0. has the Green function G x C yi x y2 e p e w e y2 x yi e p e W e for a x C for C x b where y1 and y2 are non-trivial homogeneous solutions that satisfy B1 y1 B2 y2 0 and W x is the Wronskian of y1 and y2. 1094 Example 21.7.5 Consider the equation y - y Ỉ x y 0 y i o. A set of solutions to the homogeneous equation is ex e-x . Equivalently one could use the set cosh x sinh x . Note that sinh x satisfies the left boundary condition and sinh x 1 satisfies the right boundary condition. The Wronskian of these two homogeneous solutions is W x sinh x cosh x sinh x 1 cosh x 1 sinh x cosh x 1 cosh x sinh x 1 - sinh 2x 1 sinh 1 - sinh 2x 1 sinh 1 sinh 1 . The Green function for the problem is then sinh x sinh -1 . sinhil sinh x-l sinh Ị sinh l for 0 x for x 1. The solution to the problem is y sinh x 1 sinh 1 sinh f d sinh x sinh 1 sinh 1 f d . 21.7.2 Initial Value Problems Consider L y y p x y q x y f x for a x b .