Section 22.4
Systems of Differential Equations
Any -th order differential equation can be rewritten as a system of first-order equations. This is how computers solve them.
1
Introduction
A system of first-order linear differential equations looks like:
where is a vector of functions and is a matrix.
2
Converting DEs to Systems
The Recipe
To convert :
- Define new variables:
- Write derivatives: , ...
- Last equation comes from the original DE.
3
Worked Examples
Example 1: Second Order Conversion
Convert into a system.
1. Define Variables:
.
.
2. Differentiate:
.
.
3. Matrix Form:
.
Example 2: Physical System
Write the system for two tanks mixing. Tank A flows to B at 3L/min. Input to A is 3L/min..
1. Matrix Form:
Just read the coefficients.
.
This is already a "coupled" linear system.
Example 3: Third Order
Convert .
1. Variables:
.
2. Equations:
.
.
.
3. Matrix Form:
.
4
Practice Quiz
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