Section 23.3

Series Solutions to DEs

When we can't find a closed form solution, we assume .

1

Introduction

This method works for linear DEs near an Ordinary Point.

An ordinary point is where and coefficients are analytic. Usually .

2

The Method of Frobenius (Technically just Series)

The Recipe

  1. Assume .
  2. Compute and .
  3. Plug into the DE.
  4. Shift indices so all terms have .
  5. Combine into a single sum: .
  6. Set the coefficient bracket to 0 to find the Recurrence Relation.
3

Worked Examples

Example 1: First Order

Solve .

.

Shift first sum (): .

Relation: .

Pattern: .

Result: .

Example 2: Airy's Equation

Solve .

.

Shift indices to match powers. This one has a "gap" of 3.

Relation: .

This yields special functions (Airy functions).

Example 3: Visualizing Convergence

More terms better approximation further from .

4

Practice Quiz

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