If you stand in your kitchen and let a mug of coffee sit on the counter, it cools. The rate at which it cools depends on how much hotter it is than the room. This is Newton’s law of cooling:
That equation says: the rate of temperature change () at any moment depends on how far above room temperature the coffee currently is. It’s a simple statement, and it captures essentially everything you need to know about cooling. Solve the equation and you get the entire cooling curve — how the temperature decreases over time.
This kind of equation — one that relates a quantity to its rate of change — is a differential equation. They are the most important class of equations in applied mathematics. Newton’s law of cooling. Maxwell’s equations for electromagnetism. The Schrödinger equation for quantum mechanics. Einstein’s field equations for general relativity. The Black-Scholes equation for option pricing. The SIR model for epidemics. The Navier-Stokes equations for fluid flow. All of them are differential equations.
This article is a tour of why differential equations are everywhere, what kinds of behavior they can produce, and why solving them remains one of the central challenges in mathematics.
Why nature speaks in derivatives
Most physical laws are statements about local behavior — what happens at a single point in space and time, depending on what’s happening at that point. The rate at which heat flows through a wall depends on the temperature difference across the wall at that location. The rate at which a population grows depends on the population at that moment.
When you write down such a local law mathematically, you almost always get a derivative — the rate of change. And to know what happens over time, you need to integrate the law, which means solving a differential equation.
This isn’t an accident. It’s a consequence of how the universe seems to be built: local rules produce global behavior. From a small set of differential equations, you can derive the trajectory of a planet, the shape of a soap bubble, the spread of an epidemic, the price of a financial derivative. Each is a different problem, but each reduces to the same mathematical task: “given a rule for how things change instant by instant, work out the long-term behavior.”
This is what differential equations do.
The simplest case: exponential growth and decay
Consider the equation
In words: the rate of change of is proportional to itself. The solution is — exponential growth (if ) or decay (if ). This is the simplest non-trivial differential equation, and it appears in:
- Population growth (when resources are unlimited)
- Compound interest
- Radioactive decay
- Bacterial reproduction
- Cooling (after a substitution)
- Capacitor charging in RC circuits
- Drug elimination from the body
- Many neuron models
Read our piece on the number for why exponentials appear so often: any quantity whose growth is proportional to itself produces at its base.
The lesson is general: a single equation describes processes in physics, biology, chemistry, and finance. The mathematics doesn’t care what the variable represents. Once you’ve solved , you’ve solved an enormous family of real-world problems.
Two flavors: ODEs and PDEs
Differential equations come in two main types:
Ordinary differential equations (ODEs) involve derivatives with respect to a single variable, usually time. The unknown is a function . Examples: Newton’s law of cooling, the harmonic oscillator , the SIR model for epidemics.
Partial differential equations (PDEs) involve derivatives with respect to multiple variables — typically time and space. The unknown is a function of several variables. Examples: the heat equation , the wave equation, the Navier-Stokes equations, the Schrödinger equation.
ODEs are dramatically easier than PDEs in general. Many ODEs admit closed-form solutions; most PDEs do not. ODEs describe systems with finitely many degrees of freedom (a planet’s position, a population’s size); PDEs describe continuous fields (temperature distribution, fluid velocity, quantum wave functions).
The transition from ODE to PDE — adding spatial dependence to a temporal equation — is one of the steepest difficulty cliffs in mathematics. Some of the most challenging open problems in mathematics, including the Navier-Stokes problem (one of the Millennium Problems), concern PDEs.
What kinds of behavior do they produce?
A useful way to think about differential equations is in terms of qualitative behavior. Most equations fall into a small number of categories:
Equilibrium: The system reaches a stable state and stays there. A pendulum at rest, a chemical reaction at equilibrium, a population at carrying capacity.
Oscillation: The system swings back and forth periodically. A spring, a clock, an LC circuit, the Lotka-Volterra predator-prey model.
Damped oscillation: Oscillations that decay toward equilibrium. A pendulum with friction, a struck guitar string fading.
Exponential growth or decay: Quantities that scale up or down rapidly.
Wavelike propagation: Disturbances that travel through space. Sound, light, water waves, seismic activity.
Diffusion: Quantities that spread out smoothly. Heat, dye in water, scent through a room.
Chaos: Behavior that is deterministic but unpredictable. The Lorenz system, three-body gravity, weather.
Bifurcation: A small change in parameters causing a qualitative shift in behavior. Used to model regime changes in climate, economics, and ecology.
The remarkable thing is that all these behaviors fall out of equations that are, locally, very simple. The wave equation is a one-line statement; the chaos in the Lorenz system comes from three innocuous-looking ODEs. Local simplicity, global complexity — this is the recurring theme of differential equations.
When can we solve them?
The cruelest fact about differential equations is that most cannot be solved in closed form. Closed-form means: a finite expression in terms of standard functions (exponentials, sines, polynomials, etc.). Some equations admit such solutions; most don’t.
What we can do well:
- Linear ODEs with constant coefficients — fully solvable in terms of exponentials and sinusoids.
- Some nonlinear ODEs with special structure (separable, exact, integrable).
- Linear PDEs on simple domains — solvable via Fourier methods, Green’s functions, separation of variables.
What we can’t do well in closed form:
- Most nonlinear ODEs and systems — for example, the three-body problem.
- Most PDEs with complicated boundary conditions or nonlinearities — fluid flow, weather.
- Stochastic differential equations — those involving random forces.
The response of modern applied mathematics to this hardness has been to develop numerical methods: instead of finding a formula, simulate the equation step by step on a computer. Runge-Kutta methods, finite element methods, spectral methods, and many others let us approximate solutions to enormous accuracy. Modern climate models, structural simulations, and computational fluid dynamics all rely on numerical PDE solvers.
This is a profound shift. For most of mathematical history, the goal was symbolic manipulation — getting a formula. Modern applied math is largely about computation: getting a numerical answer accurate to any desired precision.
A historical pivot
The history of differential equations begins with Newton and Leibniz. Newton used differential equations to derive Kepler’s laws of planetary motion from his law of gravity — perhaps the single most consequential mathematical demonstration in history. Leibniz developed cleaner notation and techniques. Together they invented the calculus that makes differential equations possible.
Euler systematized the field in the 18th century, solving an enormous variety of equations and inventing techniques that are still standard. Lagrange and Hamilton reformulated mechanics in terms of differential equations, opening the door to general relativity and quantum mechanics. Fourier showed how to decompose solutions of PDEs into sums of simpler waves. By the late 19th century, differential equations were the dominant mathematical framework in physics.
The 20th century pushed the field in two directions:
- Pure mathematics: existence and uniqueness theorems (Picard-Lindelöf, Cauchy-Kovalevskaya), study of regularity, abstract spaces of solutions (Sobolev spaces, distributions).
- Computation: numerical methods, special-purpose hardware for simulation, the rise of computational fluid dynamics in the 1960s and on.
Today, differential equations are simultaneously a flourishing area of pure mathematics, the workhorse of applied science and engineering, and the foundation of modern climate, biological, and economic modeling.
Where they appear in modern science
A non-exhaustive list:
Physics. Every fundamental theory is a differential equation. The Schrödinger equation, Maxwell’s equations, Einstein’s field equations, the Standard Model — all are coupled systems of differential equations.
Engineering. Structural analysis, fluid dynamics, electromagnetic design, control systems — all driven by simulation of differential equations.
Biology. Population dynamics, epidemic spread, neural electrical activity (Hodgkin-Huxley equations), gene regulatory networks, cardiac rhythms.
Economics and finance. Black-Scholes for option pricing, macroeconomic models, optimal control of resources.
Climate science. Atmospheric and oceanic circulation, ice sheet dynamics, carbon cycle modeling — all governed by huge coupled PDEs.
Machine learning. Neural networks can be viewed as discretized ODEs (the “Neural ODE” formulation). Diffusion models — a major class of generative AI — are based on stochastic differential equations.
Medicine. Pharmacokinetics (drug absorption and elimination), tumor growth models, modeling of infectious disease spread.
In each case, the equations look different — different variables, different boundary conditions, different nonlinearities. But the framework is the same: write down a local rule for how things change, then solve to get global behavior.
What differential equations teach
The deepest lesson of differential equations is that the world’s behavior is mostly determined by local interactions. You don’t need a global plan; you don’t need to know everything in advance. You need a rule that says, at each instant, what happens next given the current state. From that rule, the entire future of the system unfolds.
This is, in some sense, the mathematics of causality. Each instant produces the next. There is no need for foresight or design. The structure of the equation is the structure of the world’s causal dependencies.
For someone trying to model anything that changes over time, the question to ask is: what’s the rule? What does the rate of change at this point depend on? Answering that question turns a vague description of a phenomenon into a precise mathematical equation.
Once you have the equation, you have a tool. You can solve it (sometimes), simulate it (almost always), analyze its qualitative behavior, study how it responds to perturbations, and extrapolate its predictions. You have, in a meaningful sense, captured the dynamics of the system in mathematical form.
The fact that this approach works as well as it does — that the entire physical universe seems to be governed by relatively few differential equations — is one of the recurring miracles of applied mathematics. It is also, perhaps, the strongest case for the famous remark by Eugene Wigner about “the unreasonable effectiveness of mathematics in the natural sciences.”
Differential equations don’t just describe physics. They are physics. And, increasingly, they are biology, economics, climate, and machine learning. The local rule plus integration is a pattern that mathematics keeps discovering, again and again, in nearly every field where humans look for understanding.
That’s a remarkable thing for a single mathematical framework to do.
Frequently asked
Why do so many real-world processes obey differential equations?
Because most physical laws are local: the rate at which something changes at a point depends only on conditions at that point. Newton's second law (F=ma), heat conduction, fluid flow, population dynamics, chemical reactions — all are statements about rates of change. Translating these statements into mathematics gives a differential equation.
Are all differential equations solvable in closed form?
No. Most aren't. Linear ODEs with constant coefficients have clean solutions in terms of exponentials, but nonlinear systems and most PDEs typically don't. Modern computational mathematics solves them numerically — partial differential equations are simulated on grids, ODEs are integrated step by step. Closed-form solutions are the exception, not the rule.
What's the difference between an ODE and a PDE?
An ordinary differential equation (ODE) involves derivatives with respect to a single variable — usually time. The unknown is a function of one variable. A partial differential equation (PDE) involves derivatives with respect to multiple variables — typically time and space. The unknown is a function of several variables. PDEs are dramatically harder to solve in general.