Differential Equations Workbook For Dummies®

9.1. Finding Singular Points

NOTE

Singular points occur when a coefficient in a particular differential equation becomes unbounded.

For example, in this differential equation

where

p(x) = Q(x)/P(x)

and

q(x) = R(x)/P(x)

the singular points occur where Q(x)/P(x) and/or R(x)/P(x) become unbounded.

In the following problems, you practice finding singular points in differential equations. But first, a quick example.

NOTE

EXAMPLE

Q. What are the singular points of this differential equation?

A. x₁ = 2 and x₂ = −2

1. First, put the equation into the following form:

   

   where

   p(x) = Q(x)/P(x)

   and

   q(x) = R(x)/P(x)

   Doing so gives you

   

2. Therefore

   

   and

   

3. Looks like p(x) and q(x) both become unbounded when 4 − x² = 0, so the singular points are

   x₁ = 2 and x₂ = −2

1. What are the singular points of this differential equation?

2. Solve for the singular points of this equation:

3. What are the singular ...