Posted by ubpdqn on July 16, 2012

The post title is a quote from “The Adventures of Sherlock Holmes: The Boscombe Valley Mystery” (1892). I am working through the wonderful book: Advanced Mathematical Methods for Scientists and Engineers: Asymptotic Methods and Perturbation Theory by Professors Bender and Orszag. I will write about this book in due course.

The book chapters start with quotes from Sherlock Holmes. In the chapter on local analysis of non-linear differential equations there are two examples (nice separable equations) illustrating fixed and spontaneous singularities.

$latex y’=\frac{y}{1-x}$ (linear differential equation fixed singularity at x=1)

$latex y’=y^2$ (non-linear differential equation with spontaneous or movable singularity)

The solutions for the initial conditions $latex y(0)=a$ are respectively:

$latex y(x)=\frac{a}{1-x}$

$latex y(x)=\frac{a}{1-a x}$

In Spontaneous and Fixed Singularities I graphically illustrate this.

Professor Bender’s video lectures on Mathematical Physics are excellent and with the book have made the veil of perturbation and asymptotic theory start to fall (for…

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