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Singularity is almost invariably a clue

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.

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

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

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

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

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

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 me). Here is the link to the first lecture…you can follow the trail on YouTube or use Google to find the others. I have not put these links as they have not uniformly worked on my machine.