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Handbook of mathematics for engineers and scienteists part 76

Tham khảo tài liệu 'handbook of mathematics for engineers and scienteists part 76', khoa học tự nhiên, toán học phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 12.3. Second-Order Nonlinear Differential Equations 493 Further differentiating 12.3.3.1 yields Vxxx fx x y yx fy x y yx yx fy x x y yxyUxX 12.3.3.5 On substituting x xo the initial conditions 12.3.3.2 and the expression of yXX xo of 12.3.3.4 into the right-hand side of equation 12.3.3.5 we calculate the value of the third derivative y Xx x xo fx xo yo yi fy xo yo yi yi f xo yo yi fyx xo yo yi . The subsequent derivatives of the unknown are determined likewise. The thus obtained solution 12.3.3.3 can only be used in a small neighborhood of the point x xo . Example 1. Consider the following Cauchy problem for a second-order nonlinear equation yXx yy x y3 12.3.3.6 y o yx o 1. 12.3.3.7 Substituting the initial values of the unknown and its derivative 12.3.3.7 into equation 12.3.3.6 yields the initial value of the second derivative yXX o 2. 12.3.3.8 Differentiating equation 12.3.3.6 gives yX x x yy xx y x 2 3yy x. 12.3.3.9 Substituting here the initial values from 12.3.3.7 and 12.3.3.8 we obtain the initial condition for the third derivative y . x o 6. 12.3.3. io Differentiating 12.3.3.9 followed by substituting 12.3.3.7 12.3.3.8 and 12.3.3.10 we find that yx x xx o 24. 12.3.3.II On substituting the initial data 12.3.3.7 12.3.3.8 12.3.3.10 and 12.3.3.11 into 12.3.3.3 we arrive at the Taylor series expansion of the solution about x o y i x x2 x3 x4 . 12.3.3. i2 This geometric series is convergent only for x i. 12.3.3-2. Pade approximants. Suppose the k 1 leading coefficients in the Taylor series expansion of a solution to a differential equation about the point x 0 are obtained by the method presented in Paragraph 12.3.3-1 so that yk 1 x a0 a1x ak xk. 12.3.3.13 The partial sum 12.3.3.13 pretty well approximates the solution at small x but is poor for intermediate and large values of x since the series can be slowly convergent or even divergent. This is also related to the fact that yk œ as x œ while the exact solution can well be bounded. In many cases instead of the .