Designs in Differential Equations

  1. The Euler-Maclaurin Summation Formula and Bernoulli Polynomials

  2. Successive approximations for a second-order ODE
    Goal: [Wat1, p.195, l.-7]
    Design: [Wat1, p.195, l.6]
    Principle based on: successive approximations
    key idea: (d/dz)ò [b, z] (z-z) f(z)dz = [b, z] f(z)dz.

  3. The reason for distinguishing regular singular points from irregular singular points [Wat1, p.198, l.15-l.19]
        If we want to appreciate the distinction between regular and irregular singular points, we must create an example, say
    2x3y" + (1+x)y'+3xy = 0, whose indicial equation gives only one root for a.

  4. Integral transforms [Guo, §2.12]
    Goal: Solve ODE's by integrals [Guo, p.79, (9)].
    Design: [Guo, p.79, l.11-p.80, l.10]
    Principle based on: [Guo, p.79, (6)]
    Key idea: [Guo, p.80, l.14-l.19]
    Examples: [Guo, p.80, l.-9-l.-1]

  5. A contour design for the case given in [Guo, p.82, (ii)]
    Goal: Solve ODE's using the Laplace transform. More specifically, the contour must satisfy the condition given in [Guo, p.82, (7)].
    Design: [Guo, p.83, Fig. 3]
    Principle based on: a1 is a branch point of the Riemann surface for w = log (z-a1). If a point P goes along the contour circling a1 counterclockwise once and then clockwise once, P will remain on the same sheet of the Riemann surface. Consequently, the starting point and the ending point of the contour given in [Guo, p.83, Fig. 3] will have the same value for log (z-a1).
    Key ideas: It is unnecessary to consider the Riemann surface for w = (z-a1)l1 (z-a2)l2 [Guo, p.83, l.2-l.3]. All we have to do is paste the contour given in [Guo, p.83, Fig. 3] on the Riemann surface for w = log (z-a1). After considering (z-a1)l1, we paste the contour on the Riemann surface for w = log (z-a2).
    Remark. Instead of using the hint given in [Guo, p.83, l.5], I use [Ru2, p.27 Theorem 1.34] to justify the differentiation under the integral sign given in [Guo, p.83, (12)].

  6. Differentiability of angle functions [O'N, p.50, Exercise 12]

  7. The possible radical change of a curve at the point whose curvature is zero [O'N, p.76, Exercise 19]