Regular Singular Points

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• July 7th 2007, 12:48 PM
fudawala
Regular Singular Points
The problem is taken from "Elementary Differential Equations and Boundary Value Problems," 8th Edition. By Boyce. Section 5.4 (Page 271), problem 1. The problem states to find all singular points of the given equation and determine whether each one is regular or irregular. How would I approach this problem?

The problem is x*y'' + (1 - x)*y' + x*y = 0
• July 7th 2007, 12:55 PM
Jhevon
Quote:

Originally Posted by fudawala
The problem is taken from "Elementary Differential Equations and Boundary Value Problems," 8th Edition. By Boyce. Section 5.4 (Page 271), problem 1. The problem states to find all singular points of the given equation and determine whether each one is regular or irregular. How would I approach this problem?

What was the problem?

A differential equation of the form: $P(x)y'' + Q(x)y' + R(x)y = 0$ is said to have a "regular singular point," $x_0$, if:

$\lim_{x \to x_0} (x - x_0) \frac {Q(x)}{P(x)} < \infty$

AND

$\lim_{x \to x_0} (x - x_0)^2 \frac {R(x)}{P(x)} < \infty$

That is, if both the above limits are finite for the point $x_0$, then $x_0$ is said to be a regular singular point

Now try the problem and post the solution if you wish, so we can check it

Note: We say $x_0$ is a singular point if $\frac {Q(x)}{P(x)}$ and/or $\frac {R(x)}{P(x)}$ are discontinuous functions at $x_0$. We classify the singular point as "regular" or "irregular" by the method above