Common Models for Differential Equations Explained

What Differential Equations Actually Are

A differential equation is simply an equation that contains a function and its derivatives. That's it. Nothing fancy. The derivative tells you how something changes, and the equation sets up the relationship between the change and the thing itself.

These equations show up everywhere in physics, engineering, biology, and economics. If something changes over time or space, there's probably a differential equation behind it.

The order of a differential equation is determined by the highest derivative present. A first-order equation involves only the first derivative. A second-order equation involves up to the second derivative. You get the pattern.

Why Models Matter

You don't solve differential equations for the sake of solving them. You solve them because they model real systems. The model captures the rules governing the system, and solving the equation tells you how the system behaves.

A good model lets you predict future states, understand stability, and test scenarios without running experiments.

Common First-Order Models

Exponential Growth and Decay

The simplest differential equation model:

dN/dt = rN

Where N is the quantity and r is the growth rate. If r is positive, you get exponential growth. If r is negative, you get exponential decay.

The solution is N(t) = N₀e^(rt). This shows up constantly—radioactive decay, population growth, compound interest, cooling coffee.

The catch: exponential models break down when resources become limited. A bacteria colony doesn't grow forever in a petri dish.

Logistic Growth Model

This fixes the exponential model's flaw:

dN/dt = rN(1 - N/K)

K is the carrying capacity—the maximum the system can support. When N is small, growth is nearly exponential. As N approaches K, growth slows and eventually stops.

This is the standard model for population dynamics, resource competition, and technology adoption curves.

Logistic Equation with Harvesting

Add a constant removal rate:

dN/dt = rN(1 - N/K) - h

Fisheries use this. Set the harvest rate h too high, and the population collapses. There's a maximum sustainable yield around h = rK/4.

Second-Order Models

Simple Harmonic Oscillator

The backbone of classical mechanics:

d²x/dt² + ω²x = 0

This describes systems that oscillate without damping—springs, pendulums, LC circuits. The solution is x(t) = A cos(ωt) + B sin(ωt).

Add a damping term and you get the damped harmonic oscillator:

d²x/dt² + 2ζω₀ dx/dt + ω₀²x = 0

ζ is the damping ratio. ζ < 1 means underdamped (oscillates before settling). ζ = 1 is critically damped. ζ > 1 is overdamped.

Forced Oscillations and Resonance

Add an external driving force:

d²x/dt² + 2ζω₀ dx/dt + ω₀²x = F₀ cos(ωt)

When the driving frequency ω matches the natural frequency ω₀, you get resonance. This can be useful (MRI machines) or destructive (bridge collapse).

Partial Differential Equation Models

ODEs handle single-variable systems. PDEs handle systems with multiple independent variables—usually space and time.

Heat Equation (Diffusion)

∂u/∂t = α ∇²u

u is temperature, α is thermal diffusivity, ∇² is the Laplacian operator. This describes how heat spreads through a material over time.

The same equation describes chemical diffusion, population spreading, and option pricing (Black-Scholes is derived from it).

Wave Equation

∂²u/∂t² = c² ∇²u

c is wave speed. This describes sound waves, light waves, water waves, and vibrations in solids. Unlike heat diffusion, waves bounce back.

Black-Scholes Equation

Used in finance for option pricing:

∂V/∂t + ½σ²S² ∂²V/∂S² + rS ∂V/∂S - rV = 0

S is the stock price, σ is volatility, r is the risk-free rate. This transformed PDE into a diffusion equation through a change of variables.

Systems of Differential Equations

Real systems often have multiple interacting quantities. You need a system of equations.

Lotka-Volterra (Predator-Prey)

dx/dt = αx - βxy (prey)
dy/dt = δxy - γy (predators)

x is prey population, y is predator population. αx is prey growth, βxy is predation, δxy is predator growth from eating prey, γy is predator death.

This produces cyclical population dynamics. The cycles aren't stable—external perturbations can throw them off.

Competitive Exclusion Model

Two species competing for the same resource:

dx/dt = r₁x(K₁ - x - α₁₂y)/K₁
dy/dt = r₂y(K₂ - y - α₂₁x)/K₂

α₁₂ is how much species 2 interferes with species 1. The outcome depends on competition coefficients and growth rates. Usually, one species wins.

SIR Model (Epidemiology)

dS/dt = -βSI
dI/dt = βSI - γI
dR/dt = γI

S is susceptible, I is infected, R is recovered. β is transmission rate, γ is recovery rate. The basic reproduction number R₀ = β/γ determines whether an epidemic spreads or dies out.

Comparing Common Models

Getting Started: How to Approach a Differential Equation

Most differential equations you encounter fall into a few categories. Here's how to tackle them:

Step 1: Identify the Type

Is it ODE or PDE? What's the order? Are there multiple equations? Classify before you try solving.

Step 2: Check for Standard Forms

• Separable equations: dN/dt = f(N)g(t) → integrate both sides
• Linear first-order: dN/dt + P(t)N = Q(t) → use integrating factor
• Constant coefficients: try exponential solutions
• Exact equations: check if ∂M/∂y = ∂N/∂x

Step 3: Look for Conservation Laws

Energy, momentum, mass—these give constraints that simplify problems. If something is conserved, you might not need to solve the full differential equation.

Step 4: Use Numerical Methods When Analytic Solutions Fail

Most real-world differential equations don't have closed-form solutions. That's fine. Euler's method is simple, Runge-Kutta is better, and specialized solvers exist for stiff systems.

Step 5: Validate Your Solution

Check initial conditions. Verify dimensional consistency. Does the behavior make sense at limits?

Common Pitfalls

• Assuming exponential growth applies when resources are limited
• Ignoring boundary conditions in PDEs
• Not checking stability of equilibrium points
• Overlooking dimensional analysis
• Using continuous models where discrete models make more sense

When to Use Which Model

Exponential models work for early-stage phenomena with abundant resources. Logistic models handle saturation effects. Harmonic oscillator models handle oscillatory systems. PDEs are necessary when spatial variation matters.

If you're modeling a real system, start simple. Add complexity only when the data demands it. A complicated model that doesn't fit is worse than a simple model that captures the essentials.