Differential Equations Run the Universe

The laws of physics are not lists of facts about where things are. They are statements about how things change. Newton's second law says force equals mass times acceleration — and acceleration is the second derivative of position. Maxwell's equations for electromagnetism describe how electric and magnetic fields change in response to charges and each other. Schrödinger's equation in quantum mechanics describes how probability amplitudes evolve over time.

Every fundamental law of physics is a differential equation. Not "can be written as" one — is one. The universe, at its most basic level, appears to be governed by rules about rates of change.

Newton's laws: where it started

Newton invented calculus specifically to handle the equations he was writing. His second law: $F = ma$.

Acceleration is $a = \frac{d^2x}{dt^2}$. Force depends on position (for a spring: $F = -kx$). So Newton's law becomes:

$m\frac{d^2x}{dt^2} = -kx$

This is a differential equation. Its solutions are $x(t) = A\cos(\omega t) + B\sin(\omega t)$ where $\omega = \sqrt{k/m}$ — sinusoidal oscillations. This is why springs, pendulums, LC circuits, and sound waves all behave the same way: they all satisfy the same differential equation.

Exponential growth and decay

The simplest differential equation: $\frac{dy}{dt} = ky$.

Solution: $y(t) = y_0 e^{kt}$.

This one equation models:

• Bacterial growth ($k > 0$): populations doubling at a constant rate
• Radioactive decay ($k < 0$): nuclei decaying at a rate proportional to the number remaining
• Compound interest: money growing proportional to current balance
• Newton's law of cooling: temperature difference decaying proportional to current difference

The reason the same equation describes such different phenomena: they all share the same mathematical structure. "Rate of change is proportional to current value" uniquely determines the exponential function.

Logistic growth: the real world

Pure exponential growth is unrealistic for populations — resources run out. The logistic equation adds a carrying capacity $K$:

$\frac{dP}{dt} = rP\left(1 - \frac{P}{K}\right)$

When $P \ll K$: growth is nearly exponential (resources aren't limiting yet). When $P \to K$: growth slows to zero (the population stabilises). The solution is an S-shaped curve called the logistic function.

This model describes population ecology, the spread of epidemics (with $P$ as infected individuals), the adoption of new technologies, and the spread of information on social networks. All the same equation, very different interpretations.

The wave equation

Sound, light, water ripples, vibrating strings — all governed by the wave equation:

$\frac{\partial^2 u}{\partial t^2} = c^2 \frac{\partial^2 u}{\partial x^2}$

Here $u(x,t)$ is the displacement (of air, of the string, of the electromagnetic field) at position $x$ and time $t$, and $c$ is the wave speed.

The wave equation is a partial differential equation (PDE): it involves partial derivatives with respect to multiple variables. Its solutions include $u(x,t) = f(x - ct)$ (any wave shape travelling to the right at speed $c$) and $u(x,t) = g(x + ct)$ (travelling left). General solutions are superpositions of these.

Why differential equations work so well

The effectiveness of differential equations at describing the physical world is, as physicist Eugene Wigner put it, "unreasonably effective."

Part of the explanation: nature is local. What happens at a point depends on what's happening at nearby points right now — not on the entire history of the universe all at once. Differential equations are exactly the tool for capturing local, instantaneous relationships. The derivative encodes "what's happening right now," and differential equations encode "how what's happening right now determines what happens next."

The other part: we largely define "physical laws" to be things that can be expressed as differential equations. If a relationship between measurable quantities can't be written as one, we tend to keep looking until we find a more fundamental description that can.