# Applications of First-Order ODEs

Canonical: https://manyspace.aporia.systems/topics/applications-of-first-order-odes-mth-ode-187
Updated: 2026-10-13

Mathematics · Differential Equations (ODE) · MTH-ODE-187

- Difficulty: 3 / 5
- Importance: 5 / 5
- Study time: 3–6 h

mixing, cooling, population, radioactive decay, circuits

## Why it matters

The mathematical language of change in time: physics laws, engineering systems, populations, finance and control are mostly ODEs.

## Where it is used

Control systems, circuits, mechanical vibrations, radioactive decay, epidemic models, economic dynamics.

## How you know you have learned it

Complete 3 independent examples correctly + explain common failure modes.

## What you need first

- [Integration by Parts](https://manyspace.aporia.systems/topics/integration-by-parts-mth-calc2-140) (required)
- [Gaussian Elimination](https://manyspace.aporia.systems/topics/gaussian-elimination-mth-linalg-155) (helpful)

## Resources

- Numerical Analysis (Burden & Faires) — Burden & Faires
- Elementary Differential Equations (Boyce & DiPrima) — Boyce & DiPrima
- Elementary Differential Equations (Boyce & DiPrima) — Boyce & DiPrima