Section 24.6

Systems of Differential Equations

Any -th order linear equation can be written as a system of first order equations.

1

Introduction

Why do this?

Numerical methods (Runge-Kutta) work best on 1st order systems.
It allows us to use matrix methods (eigenvalues) to solve them.

2

The Method

Define New Variables

Let
Let

Let

Then , , etc.
The final equation comes from the original DE.

3

Worked Examples

Example 1: 2nd Order

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Matrix: .

Example 2: 3rd Order

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Matrix (companion matrix): .

Example 3: Nonhomogeneous

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4

Practice Quiz

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