Ordinary Differential Equations
Modeling change — slope fields, phase portraits, and physical systems
1. Building Intuition
A differential equation describes how things change. Instead of telling you where something is, it tells you which direction it's heading and how fast. Population growth, radioactive decay, planetary orbits, electrical circuits — all governed by ODEs.
A slope field visualizes this beautifully: at every point in the plane, draw a tiny arrow showing the direction dy/dx points. Solutions are curves that follow these arrows — like leaves floating in a stream.
Slope field for dy/dx = -x/y. The arrows show the direction of solutions at every point. Solution curves (circles) emerge by following the arrows — each starting point produces a different trajectory, but they all obey the same law.
2. The Mathematics
An ODE relates a function to its derivatives. The order is the highest derivative that appears:
For constant-coefficient linear ODEs, we use the characteristic equation. Assume y = erx and substitute to get a polynomial in r:
The roots r determine the solution form: two real roots give exponentials, complex roots give oscillations, and repeated roots give t·ert terms. For numerical solutions when analytical methods fail, Euler's method gives a simple approximation:
3. Applications
Four types of equilibrium for 2D linear systems. Each trajectory shows how the state evolves from different starting points. The eigenvalues of the coefficient matrix determine the type.
Population Dynamics
Logistic growth, predator-prey (Lotka-Volterra), epidemiology (SIR model). ODEs model how populations evolve.
Mechanical Vibrations
Springs, pendulums, bridges, buildings. Second-order ODEs with mass, damping, and stiffness.
Electrical Circuits
RC, RL, and RLC circuits are all ODEs. Kirchhoff's laws produce first and second-order equations.
Chemical Kinetics
Reaction rates as first-order ODEs. Concentration over time, half-life, and equilibrium.
4. Worked Examples
Example 1: Logistic Population Growth
A population P grows proportionally to its size but is limited by carrying capacity K:
This is separable. Integrating yields the S-shaped logistic curve:
As t → ∞, P(t) → K. The inflection point (fastest growth) occurs at P = K/2.
Example 2: Damped Harmonic Oscillator
The discriminant Δ = b² - 4mk determines the regime:
- Underdamped (Δ < 0): Oscillates with decaying amplitude — complex roots
- Critically damped (Δ = 0): Fastest return to rest without oscillation — repeated root
- Overdamped (Δ > 0): Slow exponential return — two real negative roots
Example 3: RLC Circuit
An RLC series circuit with R = 4Ω, L = 1H, C = 1/4 F, and q(0) = 1C, I(0) = 0:
Characteristic equation: r² + 4r + 4 = (r + 2)² = 0. Repeated root r = -2 (critically damped):