What a PDE Is & Its Three Types

From one independent variable to several

Every differential equation you have solved so far had a single independent variable — usually time t or position x — and the unknown was a function of that one thing, y(t) or y(x). The whole previous rung lived there. But the real world rarely cooperates so neatly. The temperature in a metal bar is not just a function of where you are; it is a function of where AND when, u(x, t). The displacement of a drumhead depends on two space coordinates and time, u(x, y, t). The moment the unknown depends on more than one independent variable, its derivatives split apart into partial derivatives — du/dx holding t fixed, du/dt holding x fixed — and an equation relating them is a partial differential equation, a PDE.

This is not a small step; it changes the character of the game completely. When you solved an ODE, the general solution carried a few arbitrary constants, and a handful of initial conditions pinned them down. A PDE's general solution instead carries arbitrary FUNCTIONS, and it takes whole curves or surfaces of data — the temperature along the entire bar at the starting instant, the value held fixed all around the edge of a plate — to single out one answer. The bookkeeping of which data make a problem well posed is genuinely harder, and the type of equation, which we are about to define, is what decides what data are even allowed.

Order and linearity: reading a PDE's fingerprint

Before classifying, learn to read two basic features of any PDE — its order and linearity. The order is simply the highest derivative that appears, exactly as for ODEs. The heat equation du/dt = k d^2u/dx^2 is second order because of that d^2u/dx^2; so is the wave equation d^2u/dt^2 = c^2 d^2u/dx^2, and Laplace's equation d^2u/dx^2 + d^2u/dy^2 = 0. This is no accident: the overwhelming majority of the PDEs that physics hands us are second order, because the laws of nature are mostly written in terms of how fast a rate of change is itself changing — accelerations and curvatures, which are second derivatives.

Linearity is the property that makes the whole subject tractable. A PDE is linear if the unknown u and all its derivatives appear only to the first power, never multiplied by each other, never wrapped inside a sine or a square. The three equations above all pass: u and its derivatives sit there plainly, each to the first power. The reward is exactly the superposition principle you leaned on for linear ODEs — if u_1 and u_2 both solve a linear homogeneous PDE, so does any combination c_1 u_1 + c_2 u_2. That is the licence to build a complicated solution by adding up simple ones, and it is the foundation of separation of variables, the method this rung is built to teach. Contrast something like u du/dx (a term in fluid flow): u times its own derivative is nonlinear, superposition dies, and shock waves and turbulence become possible.

The discriminant: one number that sorts everything

Now the central act. Write the general second-order linear PDE in two variables in its standard shape: A u_xx + 2B u_xy + C u_yy + (lower-order terms) = 0, where u_xx means d^2u/dx^2, u_xy means the mixed d^2u/dx dy, and A, B, C are the coefficients of the second-derivative terms — the highest-order part, which dominates how solutions behave. Strip away everything but those three numbers and form a single combination, the discriminant B^2 - AC. That one number, computed from nothing but the leading coefficients, is what the classification of second-order linear PDEs hangs on entirely.

Standard form:   A u_xx + 2B u_xy + C u_yy + (lower order) = 0
Discriminant:    D = B^2 - AC      (built from the leading coefficients only)

  D < 0   ELLIPTIC     equilibrium / steady state   e.g. Laplace  u_xx + u_yy = 0
  D = 0   PARABOLIC    diffusion / smoothing         e.g. heat     u_t = k u_xx
  D > 0   HYPERBOLIC   waves / signals at finite speed e.g. wave    u_tt = c^2 u_xx

Laplace:  A=1, B=0, C=1  ->  D = 0 - 1 = -1  < 0   elliptic
Heat:     A=k, B=0, C=0  ->  D = 0 - 0 =  0  = 0   parabolic   (here y plays the role of t)
Wave:     A=c^2,B=0,C=-1 ->  D = 0 - (-c^2) = c^2 > 0 hyperbolic (here y plays the role of t)

The three personalities of the great three

The label is not a filing convention — it is a prophecy about how the solution lives. An [[elliptic-equation|elliptic]] equation, the prototype being Laplace's equation d^2u/dx^2 + d^2u/dy^2 = 0, describes a system in equilibrium, with no time at all: a stretched soap film resting on a bent wire loop, the steady temperature of a plate whose edges are held fixed, an electrostatic potential. Disturb the boundary anywhere and the whole interior feels it instantly. Its solutions are extraordinarily smooth (they are infinitely differentiable inside), and they obey a maximum principle — the hottest and coldest points sit on the boundary, never in the calm interior.

A [[parabolic-equation|parabolic]] equation, the prototype being the heat equation du/dt = k d^2u/dx^2, describes irreversible smoothing in time. Start a bar with a sharp spike of heat and watch: the spike instantly slumps into a rounded bump, then flattens and spreads, forever blurring toward uniform warmth. Two honest features mark this type. First, it smooths violently — even a jagged, corner-ridden initial profile becomes perfectly smooth the very next instant. Second, it is irreversible: run the clock backward and the equation becomes wildly unstable, which is the mathematics behind the plain fact that you cannot un-stir cream from coffee. Information leaks away and cannot be recovered.

A [[hyperbolic-equation|hyperbolic]] equation, the prototype being the wave equation d^2u/dt^2 = c^2 d^2u/dx^2, describes signals travelling at a finite speed c, and it behaves nothing like the other two. It does not smooth — a sharp kink in a plucked string stays sharp and simply travels, a corner riding along the string undiminished. It is reversible — film a vibration and run it backward and it still solves the equation. And it has a strict speed limit: a disturbance at one point cannot be felt at a distant point until enough time has passed for the wave to arrive. This finite speed of propagation (the elliptic and parabolic types both spread influence infinitely fast, at least formally) is the hyperbolic signature, and it is why this is the equation of light, sound, and signals.

Honest fine print, and the road ahead

A few honesties keep this from becoming a slogan you misapply. The clean three-way split is exactly for second-order LINEAR equations in two independent variables; the discriminant test is the two-variable case. With more variables the leading part is a quadratic form whose matrix of coefficients has a signature, and you classify by the signs of its eigenvalues — elliptic when they all agree, hyperbolic when one disagrees, parabolic in the degenerate borderline — but the spirit is identical. And the type can vary from place to place: an equation can be elliptic in one region and hyperbolic in another (transonic gas flow famously switches at the speed of sound), with all the interesting physics living right on the boundary between them.

1. Count the independent variables and find the highest derivative — that tells you it is a PDE and gives its order (almost always two for the physics here).
2. Check linearity: do u and its derivatives appear only to the first power, never multiplied together? If yes, superposition is available and the methods of this rung apply.
3. Read off A, B, C from the second-derivative terms and compute the discriminant B^2 - AC; its sign names the type — elliptic (< 0), parabolic (= 0), or hyperbolic (> 0).
4. Use the type to expect the right behaviour and supply the right data: equilibrium with a closed boundary (elliptic), smoothing from an initial state (parabolic), or finite-speed signals from an initial state (hyperbolic).

That is the lay of the land for this whole rung. You can now look at an unfamiliar PDE, name its order, test its linearity, compute one discriminant, and predict in broad strokes how its solutions will live and what data will pin them down. The next guides take the three model equations one at a time and actually solve them — by separation of variables, which turns each PDE into a family of the ODEs you already mastered, and which works precisely because these equations are linear. The classification you have just learned is the map; separation of variables is the vehicle that gets you across it.