Differential Equations Workbook For Dummies®

9.5. Answers to Solving Differential Equations with Series Solutions Near Singular Points

Following are the answers to the practice questions presented throughout this chapter. Each one is worked out step by step so that if you messed one up along the way, you can more easily see where you took a wrong turn.

1 What are the singular points of this differential equation?

Solution: x₁ = 0 and x₂ = 0

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 x² = 0, so the singular points are

   x₁ = 0 and x₂ = 0

2 Solve for the singular points of this equation:

Solution: x₁ = 9 and x₂ = 0

1. Covert the differential equation to this form:

   

   where

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

   and

   q(x) = R(x)/P