Section 26.4

Separation of Variables

Assuming turns one PDE into two ODEs.

1

Introduction

If the geometry is simple (rectangle, circle) and the equation is linear, we can separate the dimensions.

2

The Method

Steps

  • Assume .
  • Plug into PDE and divide by .
  • Isolate terms on one side, on the other.
  • Set both equal to a separation constant .
  • Solve the resulting ODEs.
3

Worked Examples

Example 1: Heat Equation Setup

. Assume .

.

Divide by : .

Eq 1: (Spatial BVP).

Eq 2: (Time decay).

Example 2: Wave Equation Setup

. Assume .

.

.

Eq 1: (Spatial BVP).

Eq 2: (Time oscillation).

Example 3: Visual

The profile X(x) maintains shape but scales by T(t).

4

Practice Quiz

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