Ordinary Differential Equations: Introduction
The language that describes change
Introductory (high school – first-year university)
About This Level
At the introductory level we treat differential equations not as "calculation problems" but as a "language for describing change." The goal is to be able to say "what happens" even without solving them.
Prerequisites
- The basics of differentiation (slope, an intuition for rate of change)
- The basics of integration (area, an intuition for accumulation)
- Foundations of functions and graphs
Table of Contents
1. The Idea of a Rate of Change
What a derivative is.
- Increments, slope, local linearity
- The viewpoint of time evolution
- Separating "state" and "rate of change"
2. Why Differential Equations Appear
Describing natural phenomena.
- Newton's equation of motion
- Population models (Malthus)
- Cooling, decay, growth
3. The Simplest Differential Equations
Getting a feel with basic examples.
- $\dot{x} = c$ (uniform motion)
- $\dot{x} = kx$ (exponential growth / decay)
- The exponential meaning of the solution
4. Solution Curves and Slope Fields
A visual understanding of solutions.
- A solution is a "curve," not a "function"
- The idea of the initial value problem
- Drawing the slope field (vector field)
5. Working with Numerical Solutions
The relationship between computation and theory.
- The idea of Euler's method
- Why it drifts
- Why theory becomes necessary
Key Concepts
Definition of a Differential Equation
An equation containing an unknown function and its derivatives. For example, $\dfrac{dx}{dt} = f(t, x)$ means that the rate of change of the state $x$ at time $t$ is given by $f(t, x)$.
Initial Value Problem
A differential equation together with an initial condition $x(t_0) = x_0$. Fixing the initial value pins down a single solution curve (when uniqueness holds).
Slope Field
A vector of slope $f(t, x)$ drawn at each point $(t, x)$. Solution curves flow along this vector field.
Applications You Can Understand at This Level
Exponential Growth and Decay
The solution $x(t) = x_0 e^{kt}$ of $\dot{x} = kx$ describes population growth, radioactive decay, compound interest, and many other phenomena. $k > 0$ gives exponential growth, $k < 0$ exponential decay.
Newton's Law of Cooling
A body cools in proportion to the difference between its temperature $T$ and the ambient temperature $T_{\text{env}}$: $\dot{T} = -k(T - T_{\text{env}})$. This too has the form of exponential decay.
Terminal Velocity
The speed $v$ of a falling body obeys $\dot{v} = g - kv$ (with air resistance). After a long time it converges to $v \to g/k$ (the terminal velocity).
Logistic Growth
A population model that accounts for the carrying capacity $K$: $\dot{x} = rx(1 - x/K)$. An S-shaped curve converging to $K$.
Simple Harmonic Motion (pendulum, spring)
The motion of a pendulum or spring obeys $\ddot{x} = -\omega^2 x$. This second-order differential equation produces back-and-forth motion rather than growth or decay; it is explored in the reading "Why the World Swings."
Study Tips
- Meaning over computation: focus on "how it behaves" rather than the form of the solution
- Visualization: draw the slope field to grasp the whole picture of the solution curves
- Connect to phenomena: always tie back to concrete examples in physics and biology
- Limits of numerics: be aware that a numerical solution shows only a single trajectory
Readings
Equations That Describe the Future [Reading]
How does the future shape arise from the local rule of "the present rate of change"? A relaxed tour from Newton's wager to determinism and chaos.
Why the World Swings [Reading]
Swings, pendulums, strings, alternating current. Why is the world so fond of swinging? The story of how simple harmonic motion arises from one mechanism — "overshoot and pull-back" — told with Galileo's discovery.
Cooling Coffee and Growing Crowds [Reading]
Cooling coffee, swelling savings, crowds that grow too large. The everyday "grow, shrink, settle" connected in one stroke — from exponential growth and decay to the logistic S-curve, with anecdotes of Malthus and Newton.
Frequently Asked Questions
What is a differential equation? (an explanation a high-schooler can follow)
A differential equation is "an equation containing a rate of change (a derivative)." For example, writing the fact that "a bacterium's growth rate is proportional to the current population" gives $\dfrac{dy}{dt} = ky$. This is a differential equation, and its solution $y = Ce^{kt}$ describes the actual change in population.
What is a slope field (direction field)?
A slope field draws, at each point of the plane, a small arrow with the slope of the solution curves of $\dfrac{dy}{dx} = f(x,y)$. Even when you cannot find a formula for the solution, the slope field lets you read off the rough shape of the solution curves (increasing, decreasing, oscillating, diverging, and so on).
What are differential equations good for?
They are a tool for predicting "what happens next" from "the present state." The motion of objects, the rise and fall of populations or infections, electrical circuits, chemical reactions, economic models — almost anything that changes over time can be written as a single equation and its future behavior read off.