Section 24.7

Series Solutions (Higher Order)

When all else fails, the power series method remains a robust tool for any linear equation.

1

Introduction

For , we assume .

The recurrence relation will usually relate to previous terms.

2

The Method

Steps

  • Compute derivatives up to order .
  • Shift indices.
  • Find recurrence relation .
  • Identify linearly independent solutions corresponding to .
3

Worked Examples

Example 1: 3rd Order Trivial

.

.

All for .

.

Example 2: 3rd Order Airy-like

.

.

Three independent series:
(terms )
(terms )
(terms )

Converges to .

Example 3: Visual

Approximating one of the basis solutions.

4

Practice Quiz

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