Mathematics

Differential Equations

The language nature uses to describe change — and the toolkit physicists use to read it.

Why Everything Is a Differential Equation

Here is a strange and beautiful fact about the universe: almost every fundamental law in physics is a differential equation. Newton wrote $F = ma$ , which is really $F = m\,\ddot{x}$ — a second-order ODE relating position to its second time derivative. Maxwell wrote four differential equations. Schrödinger wrote one. Einstein wrote ten, coupled, nonlinear partial differential equations.

The reason is deep. Differential equations describe how a system changes given its current state. Nature seems to be fundamentally local — what happens here and now depends on what is here and now, not on some global overview of the whole universe. Differential equations are the mathematical expression of that locality.

So understanding differential equations is not just a technical skill. It is learning to read the language in which the laws of physics are written.

Ordinary Differential Equations

An ODE involves a function of one variable and its derivatives. The order is the highest derivative that appears. Linear ODEs are ones where the unknown function and its derivatives appear only linearly — no products, no powers.

The key insight for linear ODEs with constant coefficients is that exponential functions are eigenfunctions of differentiation: $\frac{d}{dt}e^{rt} = r e^{rt}$ . So if you guess a solution $y = e^{rt}$ , the ODE turns into an algebraic equation for $r$ — the characteristic equation.

Characteristic Equation

a_n r^n + a_{n-1} r^{n-1} + \cdots + a_1 r + a_0 = 0

  Root type          Solution form           Physical meaning
  ───────────        ─────────────           ────────────────
  Real r             e^{rt}                  Exponential growth or decay
  Complex r = α±iβ  e^{αt}(cos βt + sin βt)  Damped oscillation
  Repeated root r    t^k e^{rt}              Resonance / polynomial growth

Nature of solutions depends on roots of the characteristic equation

The simple harmonic oscillator — a mass on a spring, a pendulum near equilibrium, an LC circuit, a quantum particle in a parabolic potential — all reduce to the same equation: $\ddot{x} + \omega^2 x = 0$ , with characteristic roots $r = \pm i\omega$ and solution $x(t) = A\cos\omega t + B\sin\omega t$ . The universality of this equation is not a coincidence. It is the generic behaviour of any system near a stable equilibrium.

Partial Differential Equations

PDEs involve functions of multiple variables. The fundamental equations of physics are mostly PDEs, because physical fields — temperature, pressure, electric potential — vary continuously in space and time.

The three archetypal PDEs each describe a qualitatively different kind of physics:

Wave Equation

\frac{\partial^2 u}{\partial t^2} = c^2 \nabla^2 u

Describes propagation without dissipation — sound, light, vibrating strings. Information travels at finite speed c.

Heat / Diffusion Equation

\frac{\partial u}{\partial t} = \alpha \nabla^2 u

Describes irreversible spreading — heat conduction, Brownian motion, the diffusion of probability. Solutions smooth out immediately.

Laplace's Equation

\nabla^2 u = 0

Describes steady-state configurations — electrostatic potential with no charges, steady temperature with no sources. Solutions are as smooth as possible (harmonic).

Boundary Conditions and Eigenvalue Problems

Specifying a PDE is only half the problem. You also need boundary conditions — information about the solution at the edges of the domain. Different physical situations call for different conditions:

  Type        Condition              Physical Example
  ────────    ───────────────        ──────────────────────────────
  Dirichlet   u = f  on boundary     Temperature fixed at wall
  Neumann     ∂u/∂n = g on boundary  Insulated wall (no heat flux)
  Robin       au + b∂u/∂n = h        Convective heat transfer

Three common boundary condition types

When you separate variables in a PDE on a bounded domain, you typically encounter a Sturm-Liouville eigenvalue problem. These are beautiful: they guarantee a discrete spectrum of eigenvalues and a complete orthogonal set of eigenfunctions — exactly the structure quantum mechanics needs for quantisation.

Sturm-Liouville Problem

\frac{d}{dx}\!\left[p(x)\frac{dy}{dx}\right] + q(x)y + \lambda w(x) y = 0

The vibrating string, the quantum particle in a box, and the hydrogen atom's angular part are all Sturm-Liouville problems. Their eigenfunctions — sinusoids, Legendre polynomials, spherical harmonics — are the basis functions that recur endlessly across physics.

Green's Functions — The Impulse Response

One of the most powerful ideas in mathematical physics is the Green's function. The idea is simple: if you understand how a system responds to a single point impulse, you understand how it responds to anything — because any source can be decomposed into a superposition of impulses.

Formally, the Green's function $G(x, x')$ satisfies the equation with a delta function source at $x'$ :

Green's Function

\mathcal{L}\, G(x,x') = \delta(x - x')

Once you have $G$ , the solution for any source $f$ is just a convolution:

Solution by Convolution

u(x) = \int G(x, x')\, f(x')\, dx'

In electrostatics, $G$ is the potential due to a point charge. In quantum mechanics, it is the propagator — the amplitude for a particle to go from one place to another. In signal processing, it is the impulse response of a filter. Different names, same idea.