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Handbook of mathematics for engineers and scienteists part 73

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Tham khảo tài liệu 'handbook of mathematics for engineers and scienteists part 73', khoa học tự nhiên, toán học phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 472 Ordinary Differential Equations 12.1.10-3. Runge-Kutta method of the fourth-order approximation. This is one of the widely used methods. The unknown values yk are successively found by the formulas yk i yk 6 i 2 2 2 3 4 Ax where 1 Xk yk 2 Xk 2Ax yk 2 iAx 3 Xk 2Ax yk 2 2AX 4 Xk Ax yk 3AX . Remark 1. All methods described in Subsection 12.1.10 are special cases of the Runge-Kutta method a detailed description of this method can be found in the monographs listed at the end of the current chapter . Remark 2. In practice calculations are performed on the basis of any of the above recurrence formulas with two different steps Ax 2Ax and an arbitrarily chosen small Ax. Then one compares the results obtained at common points. If these results coincide within the given order of accuracy one assumes that the chosen step Ax ensures the desired accuracy of calculations. Otherwise the step is halved and the calculations are performed with the steps 2 Ax and 4 Ax after which the results are compared again etc. Quite often one compares the results of calculations with steps varying by ten or more times. 12.2. Second-Order Linear Differential Equations 12.2.1. Formulas for the General Solution. Some Transformations 12.2.1-1. Homogeneous linear equations. Formulas for the general solution. 1 . Consider a second-order homogeneous linear equation in the general form 2 X yXX i X yX o X y 0. 12.2.1.1 The trivial solution y 0 is a particular solution of the homogeneous linear equation. Let yi x y2 x be a fundamental system of solutions nontrivial linearly independent particular solutions of equation 12.2.1.1 . Then the general solution is given by y Ciyi x C2y2 x 12.2.1.2 where C1 and C2 are arbitrary constants. 2 . Let y1 y1 X be any nontrivial particular solution of equation 12.2.1.1 . Then its general solution can be represented as y yi ci C2 J dx where F dx. 12.2.1.3 y2 J J J2 3 . Consider the equation yXx f x y 0 which is written in the canonical form see Paragraph 12.2.1-3 for