tailieunhanh - 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 | . Second-Order Nonlinear Differential Equations 493 Further differentiating yields Vxxx fx x y yx fy x y yx yx fy x x y yxyUxX On substituting x xo the initial conditions and the expression of yXX xo of into the right-hand side of equation 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 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 y o yx o 1. Substituting the initial values of the unknown and its derivative into equation yields the initial value of the second derivative yXX o 2. Differentiating equation gives yX x x yy xx y x 2 3yy x. Substituting here the initial values from and we obtain the initial condition for the third derivative y . x o 6. . io Differentiating followed by substituting and we find that yx x xx o 24. On substituting the initial data and into we arrive at the Taylor series expansion of the solution about x o y i x x2 x3 x4 . . i2 This geometric series is convergent only for x i. . 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 so that yk 1 x a0 a1x ak xk. The partial sum 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 .