Models in Differential Equations

  1. By adding factors one-by-one, we may understand the effects of each factor and thereby gain the ability to adjust our model to better simulate reality.
    Example 1. [Arn1, 1.6][Arn1, 1.7] ([Arn1, Fig. 9][Arn1, Fig.11 (p.23, l.!11)]).
    Example 2. [Arn1, 1.6][Arn1, 1.8][Arn1, 1.9][Arn1, 1.10] ([Arn1, Fig.13 (Notice the point of divergence [p.25, l.19])][Arn1, Fig.14][Arn1, Fig.15]).

  2. The process of mathematical modeling [Edw, p.4, l.-7-p.5, l.-9].

  3. We must be flexible when effectively establishing a model. Our goal is to use a simple model to solve sophisticated problems. We should not use a sophisticated model to solve simple problems.
        [Wat1, 20.222] uses csc2 z to establish a model so that we may follow the pattern to derive the differential equation given in [Wat1, p.437, l.2]. In [Wat1, p.439, l.9-l.10], Watson proves c = 0 by using the power series of f(z) and f '(z). He sticks with a strictly axiomatic approach by considering f(z) as only a series. It is unnecessary to use this complicated method to establish a simple model. We can let z = p/2 to prove c = 0. In addition, [Wat1, p.436, l.-1] has nothing to do with the above complicated method. Thus, Watson's complicated method involving negative powers only creates a huge obstacle for us to establish a simple model.

  4. It would be difficult to understand that the development given in [Wat1, 20.6] is based on standardization if we fail to recognize the primitive model upon which the development is based.
        Since dz/d(z) = (43(z) - g2(z) - g3)-1/2 and dz/d sin z = (1 - sin2 z)-1/2 are similar, z = {f(t)}-1/2 dt [Wat1, p.453, l.5] can be considered the corresponding counterpart of z = (2at - t2)-1/2 dt [Edw1, back cover, formula 103].