Hand a mathematician a brand-new PDE and, before attempting to solve a single thing, they read off two labels: its order and whether it is linear. Those two facts alone decide which whole toolbox of techniques even applies — a linear second-order equation and a nonlinear first-order equation are solved by entirely different methods, and mixing them up wastes hours chasing a technique that was never going to work.

We write subscripts for partial derivatives: u_t = \partial u/\partial t, u_{xx} = \partial^2 u/\partial x^2, and so on.

The order is the highest derivative that appears. The heat equation u_t = \alpha\,u_{xx} is second order (because of u_{xx}); the transport equation u_t + c\,u_x = 0 is first order.

Linear, and the operator viewpoint

A PDE is linear if the unknown u and its derivatives appear only to the first power, never multiplied together and never inside a nonlinear function. The cleanest way to say this is with a linear operator L — a rule that eats a function and returns one — such that the equation reads L[u] = g and

For the heat equation, L[u] = u_t - \alpha\,u_{xx} is linear. The equation is homogeneous when g = 0 and inhomogeneous (forced) otherwise.

Worked examples: classifying three equations

Let's put the labels to work on three equations, one at a time.

1. The heat equation, u_t = \alpha\,u_{xx}. The highest derivative is u_{xx}, so it's second order. Every term involving u — namely u_t and u_{xx} — appears to the first power, with no products of u with itself or its derivatives, so it's linear, and since there's no term free of u, it's homogeneous.

2. The wave equation, u_{tt} = c^2 u_{xx}. The highest derivative is now u_{tt} (a second time-derivative), so it is again second order. Exactly as before, u and its derivatives each appear alone and to the first power, so it too is linear and homogeneous. Two very different physical behaviours (smoothing versus oscillating) hiding behind the same two labels — order and linearity classify the algebra of an equation, not what it physically does.

3. A made-up equation, u_t = \alpha\,u_{xx} + u^2. Still second order — but now there's a u^2 term, the unknown multiplied by itself. Try the operator test: if L[u] = u_t - \alpha u_{xx} - u^2, then L[2u] = 2u_t - 2\alpha u_{xx} - 4u^2 \neq 2\,L[u] — doubling the input does not double the output, because the squared term grows four times as fast. This equation is nonlinear. So is the closely related u_t = \alpha\,u_{xx} + (u_x)^2, for exactly the same reason: a derivative of u multiplied by itself is just as fatal to linearity as u multiplied by itself.

Why does this one broken rule matter so much? Because linearity is exactly the property that lets you take two known solutions and add them to build a third — the superposition principle, which powers almost every exact-solution method later in this course. Plug u = u_1 + u_2 into a nonlinear equation with a u^2 term and the cross term 2u_1 u_2 appears out of nowhere and ruins the sum — the addition trick simply stops working the moment linearity breaks.

Order also tells you how many conditions you'll need

Classifying the order isn't just labelling for its own sake — it has an immediate practical payoff. Just as integrating an ordinary second-order equation twice leaves two unknown constants to pin down, differentiating a PDE of order n in a given variable leaves roughly n worth of "free choice" in that variable, which must be fixed by an extra condition.

The heat equation u_t = \alpha\,u_{xx} is first order in t but second order in x: sure enough, it needs exactly one initial condition (the starting temperature everywhere) and two boundary conditions (one fact at each end of the rod). The wave equation u_{tt} = c^2 u_{xx} is second order in t as well, so it needs two initial conditions — not just the string's starting shape, but its starting velocity too. We'll turn this rule of thumb into a precise recipe in boundary and initial conditions, and check when a chosen set of conditions is actually enough — no more, no less — in well-posedness.

Linear versus nonlinear, out in the real world

This isn't just bookkeeping — it sorts physics into two very different difficulty classes. Linear PDEs — the heat equation, the wave equation, Laplace's equation — already model an enormous range of everyday phenomena extremely well: heat spreading through a wall, sound travelling through air, the electric potential around a wire. Because they're linear, we can often write down exact formulas for their solutions, built by superposing simple building blocks.

Nonlinear PDEs are a different world entirely. The Navier–Stokes equations, which describe how a fluid like air or water flows, contain a term where the velocity is multiplied against its own derivative — nonlinear, just like our made-up example above. So does the system of ten coupled PDEs in Einstein's general relativity, which describes how spacetime itself curves. For equations like these, exact pen-and-paper formulas are rare or unknown, and physicists and engineers instead turn to numerical methods — approximating the solution step by step on a computer — to predict the weather, design an aeroplane wing, or simulate colliding black holes.