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The Discriminant: Elliptic, Parabolic, Hyperbolic

One number, B^2 - A C, computed from the second-derivative coefficients, sorts every second-order linear PDE into three families. Meet the discriminant, the three model equations it picks out, and the first hint of why that single label decides everything.

One number that reads off the whole equation

In the previous guide you met the general second-order linear PDE in two variables and learned that only its top derivatives — the principal part — set its character. Write that leading part as A u_xx + 2 B u_xy + C u_yy, with everything else (first derivatives, the u term, the right-hand side) collected into lower-order junk that does not affect the type. The question of this whole rung is: which of three deeply different worlds does this equation live in? Astonishingly, the answer is decided by a single number built from just A, B, and C.

That number is the discriminant, B^2 - A C. Its sign — and only its sign — names the equation. If B^2 - A C is negative, the equation is elliptic; if it is exactly zero, the equation is parabolic; if it is positive, the equation is hyperbolic. The size of the number does not matter, only whether it sits below, on, or above zero. That is the great trichotomy this rung is named for, and the entire subject is organized around it.

The three model equations it picks out

The names elliptic, parabolic, hyperbolic are not chosen at random — and not, despite a tempting story, because the solutions are shaped like those conic sections. They come from the algebra: replace u_xx, u_xy, u_yy by the symbols X^2, X Y, Y^2 and you get A X^2 + 2 B X Y + C Y^2, the very expression whose conic curve A X^2 + 2 B X Y + C Y^2 = const is an ellipse, a parabola, or a hyperbola exactly according to the sign of B^2 - A C. The classification is conic-section algebra wearing a PDE costume. Each type has one cleanest representative, the model equation you should keep in mind whenever you hear the name.

A u_xx + 2 B u_xy + C u_yy + (lower order) = 0

    B^2 - A C < 0   ELLIPTIC     model:  u_xx + u_yy = 0   (Laplace)
    B^2 - A C = 0   PARABOLIC    model:  u_t = k u_xx     (heat)
    B^2 - A C > 0   HYPERBOLIC   model:  u_tt = c^2 u_xx  (wave)
One sign, one type, one model equation. Memorize this little table and you carry the whole rung in your pocket.

Run the table on each model. Laplace's equation u_xx + u_yy = 0 has A = 1, B = 0, C = 1, so B^2 - A C = -1 < 0: elliptic, the equation of steady equilibrium. The heat equation u_t = k u_xx, read with (x, t) as the two variables, has a u_xx term but no u_tt term at all, so A = k, B = 0, C = 0 and B^2 - A C = 0: parabolic, the equation of irreversible diffusion. The wave equation u_tt = c^2 u_xx rearranges to c^2 u_xx - u_tt = 0, giving A = c^2, C = -1 with the opposite sign, so B^2 - A C = c^2 > 0: hyperbolic, the equation of finite-speed signals. Three signs, three behaviours, three of the most important equations ever written.

Where the names come from: counting characteristic directions

There is a second, more dynamic way to read the discriminant that prepares the next guide. Look for the special directions along which the principal part degenerates — the characteristic directions. Trying a solution that varies only in the direction dy/dx = m leads to the quadratic A m^2 - 2 B m + C = 0 (signs depend on convention, but the discriminant is what matters). Solving for the slopes m gives m = (B ± sqrt(B^2 - A C)) / A. The number of real solutions of this little quadratic is governed by — you guessed it — the sign of B^2 - A C.

  1. If B^2 - A C > 0 (hyperbolic), the square root is real and nonzero, so there are TWO distinct real characteristic slopes — two real families of curves along which signals travel. This is why hyperbolic equations propagate.
  2. If B^2 - A C = 0 (parabolic), the square root vanishes and the two slopes merge into ONE repeated real characteristic — a single carrier direction, the borderline case sitting between the other two.
  3. If B^2 - A C < 0 (elliptic), the square root is imaginary, so there are NO real characteristics at all — no real curves carry information, which is exactly why elliptic equations describe instantaneous, everywhere-coupled equilibrium with no propagation.

So the discriminant is literally counting real characteristic families: two, one, or none. Hyperbolic equations are the ones with real tracks for signals to run along — the wave equation factors into a left-mover and a right-mover, each riding its own characteristic, just as you saw the transport equation do one rung below. Elliptic equations have none, so a disturbance anywhere is felt everywhere at once. Parabolic sits exactly on the knife-edge with a single repeated track. The next guide turns these characteristic curves into new coordinates and uses them to fold each type into its canonical form.

How to classify any equation — including ones that change type

Here is the whole procedure in miniature, the move you will make hundreds of times. Take, say, u_xx + 4 u_xy + u_yy = 0. Read off A = 1, and from the cross term 4 u_xy = 2 B u_xy so B = 2, and C = 1. Then B^2 - A C = 4 - 1 = 3 > 0: hyperbolic. Or take u_xx + 2 u_xy + u_yy = 0: now B = 1 and B^2 - A C = 1 - 1 = 0: parabolic. The recipe never changes — identify A, B, C from the second-order terms only, compute B^2 - A C, read the sign. The lower-order terms are along for the ride and never alter the type.

A crucial honesty: when A, B, C are not constants but functions of x and y, the discriminant B^2 - A C is itself a function of position, so the same equation can be elliptic in one region and hyperbolic in another. These are equations of mixed type, and the most famous is the Tricomi equation y u_xx + u_yy = 0: here A = y, B = 0, C = 1, so B^2 - A C = -y. It is elliptic where y > 0, hyperbolic where y < 0, and parabolic right along the line y = 0 — a single equation that is all three types at once, which is exactly why transonic airflow (subsonic and supersonic regions side by side) is so subtle.

Why this one label decides everything

It would be a poor classification if it only sorted equations by algebra. The reason the trichotomy earns its central place is that the type controls behaviour — what the solutions look like and, above all, what data makes the problem solvable. This is type-dependent well-posedness, the heart of the next two guides, and it is worth previewing now so the discriminant feels like more than bookkeeping. Each type wants a different kind of data, and giving it the wrong kind is not merely harder — it is mathematically broken.

A few honest previews of how different these worlds are. Elliptic problems (Laplace) want boundary data all the way around a region and give a single smooth equilibrium; but prescribing Cauchy data — value and normal derivative on an open arc — for Laplace's equation is famously ill-posed, with solutions that explode under the tiniest wobble in the data. Parabolic problems (heat) run forward in time and smooth any rough start instantly, but running the heat equation backward in time is ill-posed and effectively impossible to do stably. Hyperbolic problems (wave) propagate at finite speed along characteristics, preserve sharp features rather than smoothing them, and happily accept Cauchy data — initial position and velocity. Same trichotomy, three completely different fates.

So the discriminant is not a label you pin on for tidiness; it is a forecast. Compute B^2 - A C and you already know, before solving anything, whether to expect smoothing or sharp fronts, infinite or finite speed, boundary data or initial data, equilibrium or evolution. That predictive power is why this single number sits at the gateway of the whole subject — and why the remaining guides in this rung unpack, one by one, exactly how each type lives out the destiny its discriminant assigned.