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The General Second-Order Linear PDE

One template — A u_xx + 2B u_xy + C u_yy + lower-order terms = 0 — secretly contains the heat, wave, and Laplace equations all at once. Meet that template, learn why its highest-derivative part is the only part that matters for type, and read off the single number B^2 - A C that will sort every such equation into three families.

One template that holds the three classics

By now you have carried profiles down rivers with the method of characteristics and tamed first-order equations one carrier curve at a time. Now we step up an order. A huge fraction of the equations physics ever hands you fit a single mould: A u_xx + 2B u_xy + C u_yy + D u_x + E u_y + F u + G = 0. Here u(x,y) is the unknown, and the coefficients A, B, C, D, E, F, G may depend on x and y but not on u or its derivatives — that is what makes it a linear PDE. The whole rung you are starting is the story of how this one template, by the size of its top coefficients alone, splits into three utterly different worlds.

It really does hold the three classics. Set A = 1, C = -1, every other coefficient to zero, and replace y by t: you get u_xx - u_tt = 0, the wave equation in disguise. Set A = 1, C = 1, the rest zero: you get u_xx + u_yy = 0, which is Laplace's equation, Laplacian u = 0. And a slightly different fit, A = 1 with a single first-order time term, gives u_t = k u_xx, the heat equation. Same template, three faces — vibration, equilibrium, diffusion. The job of this rung is to explain why they behave so differently when they look so alike.

A u_xx + 2B u_xy + C u_yy  +  D u_x + E u_y + F u + G = 0

  A=1, C=-1, rest 0   -->  u_xx - u_tt = 0     (wave)
  A=1, C= 1, rest 0   -->  u_xx + u_yy = 0     (Laplace)
  A=1, plus a u_t     -->  u_t = k u_xx        (heat)
Three famous equations, one second-order linear template — separated only by the coefficients of the highest derivatives.

Only the top floor matters: the principal part

Here is the first surprising truth of classification: the lower-order terms D u_x + E u_y + F u + G do not affect the type at all. Only the three highest-derivative terms, A u_xx + 2B u_xy + C u_yy, decide everything. We give that piece a name — the principal part — and treat it as the engine room of the equation. Adding a u_x term or an F u term changes the detailed solution, but it cannot turn a wave equation into a heat equation or a Laplace equation. Type is a property of the top floor alone.

Why should the highest derivatives rule? Intuitively, second derivatives describe the sharpest, fastest-varying part of u — the way the surface bends — and it is exactly this bending that controls how information moves: whether a kink can travel, whether a disturbance stays local or spreads everywhere, whether the future even makes sense to ask about. The lower-order terms nudge and damp the answer, but the architecture of how the equation propagates and where it is well behaved is written entirely in A, B, and C. So from now on, to classify, we squint until only the principal part is in focus.

The number that sorts everything: B^2 - A C

From the principal part A u_xx + 2B u_xy + C u_yy we form a single number, the discriminant: B^2 - A C. Its sign — positive, zero, or negative — sorts the equation into one of three types. If B^2 - A C is negative the equation is elliptic (Laplace's equation is the model). If it is exactly zero the equation is parabolic (the heat equation is the model). If it is positive the equation is hyperbolic (the wave equation is the model). Three signs, three worlds. The next guide is devoted entirely to unpacking this trichotomy, so here we simply meet it and check it on our classics.

Check the three classics and the names earn themselves. For Laplace's equation u_xx + u_yy = 0 we have A = 1, B = 0, C = 1, so B^2 - A C = 0 - 1 = -1 < 0: elliptic, just as promised. For the heat equation, reading u_t = k u_xx in (x,t) with only u_xx present, the principal part has A = k and C = 0 (there is no u_tt term), giving B^2 - A C = 0 - 0 = 0: parabolic. For the wave equation u_tt - c^2 u_xx = 0 the top coefficients have opposite signs, so A C < 0 and B^2 - A C > 0: hyperbolic. The labels are not decoration — they come straight from the conic sections, where B^2 - A C distinguishes an ellipse, a parabola, and a hyperbola from one another by exactly this sign.

  1. Strip away all lower-order terms; keep only A u_xx + 2B u_xy + C u_yy — the principal part.
  2. Read off A (coefficient of u_xx), B (half the coefficient of u_xy), and C (coefficient of u_yy).
  3. Compute the discriminant B^2 - A C.
  4. Negative -> elliptic; zero -> parabolic; positive -> hyperbolic. That single sign fixes the type.

Why the type is worth all this fuss

It would be a thin reward if the type were only a label. It is not — it is a prediction about behaviour. The discriminant secretly counts the characteristic curves of the equation: hyperbolic equations have two real families of them, parabolic have one, and elliptic have none (the curves are imaginary). Those curves are the roads along which information travels, so their number is the difference between a wave that carries crisp signals along sharp fronts, a heat flow that has no fronts at all, and a steady-state field where every point feels every other point at once. Counting characteristics is precisely guide three of this rung.

The type also decides which questions are even fair to ask — the deep theme of guide four. A hyperbolic equation like the wave equation takes initial data and marches forward at a finite speed; that is well-posed. A parabolic equation like the heat equation also runs forward in time, smoothing as it goes, but running it backward is a famously ill-posed disaster: tiny errors explode. And asking an elliptic equation like Laplace's to march forward from Cauchy data is ill-posed too — elliptic problems want boundary values all the way around a closed region, not an initial slice. Same template; completely different rules for what counts as a sensible, well-posed problem.

Honest caveats before you climb on

Three honesties to carry up the ladder. First, when A, B, C depend on x and y, the sign of B^2 - A C can change from place to place, so a single equation can be elliptic in one region and hyperbolic in another. That is a genuine equation of mixed type, and the classic example, the Tricomi equation u_yy = y u_xx, is exactly the transonic-flow model where a flow crosses the speed of sound. Type is local, not always global.

Second, this whole classification is built for linear equations, where the top coefficients are fixed functions of x and y. Once the coefficients depend on u itself — a nonlinear equation — the type can change as the solution evolves, and smooth solutions of conservation laws can even crash into shocks in finite time. So treat B^2 - A C as the master key for the linear world, and expect the nonlinear rungs to qualify it. Third, the recipe here lives in two variables, x and y; in three or more variables you cannot reduce everything to one number, and you instead read the type off the symbol of the operator — a quadratic form whose signs of eigenvalues generalise B^2 - A C. That is the closing guide of this rung.