tailieunhanh - Advanced Mathematical Methods for Scientists and Engineers Episode 4 Part 3

Tham khảo tài liệu 'advanced mathematical methods for scientists and engineers episode 4 part 3', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | Differentiating this relation yields f x as x X. However this is not true since f z x ----------- sin x3 2 cos x3 2 x3 - as x X. x2 The Controlling Factor. The controlling factor is the most rapidly varying factor in an asymptotic relation. Consider a function f x that is asymptotic to x2ex as x goes to infinity. The controlling factor is ex. For a few examples of this xlogx has the controlling factor x as x X. x-2 e1 x has the controlling factor e1 x as x 0. x-1 sinx has the controlling factor sinx as x X. The Leading Behavior. Consider a function that is asymptotic to a sum of terms. f x a0 x a1 x a2 x as x x0. where a0 x a1 x a2 x as x x0. The first term in the sum is the leading order behavior. For a few examples For sinx x x3 6 x5 120 as x 0 the leading order behavior is x. For f x ex 1 1 x 1 x2 as x X the leading order behavior is ex. 1254 Leading Order Behavior of Differential Equations It is often useful to know the leading order behavior of the solutions to a differential equation. If we are considering a regular point or a regular singular point the approach is straight forward. We simply use a Taylor expansion or the Frobenius method. However if we are considering an irregular singular point we will have to be a little more creative. Instead of an all encompassing theory like the Frobenius method which always gives us the solution we will use a heuristic approach that usually gives us the solution. Example Consider the Airy equation y xy. We 1 would like to know how the solutions of this equation behave as x to. First we need to classify the point at infinity. The change of variables x 1 n x n t . A _t21. d2 t4 d2 2t31 y x dx df dx2 dt2 2t dt yields t4u 2t3uz - u 2 u I u - 1 u t5 0. Since the equation for u has an irregular singular point at zero the equation for y has an irregular singular point at infinity. 1Using We may be a bit presumptuous on my part. Even if you don t particularly want to know how the solutions behave I urge you to just