Classification — elliptic, parabolic, hyperbolic (discriminant test)

80/20 core: A linear 2nd-order PDE in two variables is classified by the sign of the discriminant $B^2 - 4AC$ . Negative → elliptic, zero → parabolic, positive → hyperbolic. This single sign tells you the physics, the boundary conditions you need, and the solution method.

The setup

Definition General linear 2nd-order PDE in two variables

$A\,u_{xx} + B\,u_{xy} + C\,u_{yy} + D\,u_x + E\,u_y + F\,u + G = 0$ A PDE is linear when the coefficients $A,B,C,D,E,F$ (and $G$ ) depend only on the independent variables $x,y$ — never on $u$ or its derivatives. It is quasilinear when the coefficients of the highest-order derivatives are allowed to depend on $u$ and its lower-order derivatives (but the highest derivatives still appear linearly). The discriminant test below applies to linear and quasilinear equations alike — but for the latter the type can depend on the solution itself. Only the highest-order (principal) part $A u_{xx}+B u_{xy}+C u_{yy}$ decides the class.

WHERE the discriminant comes from (derivation, not a dump)

We want to know: along which curves can a solution have a "kink" (discontinuity in 2nd derivatives)? Those curves are called characteristics. Their existence is the deep meaning of the classification.

Step 1 — Look for characteristic curves $y = y(x)$ . Why? Because a PDE behaves like an ODE along special directions where the second derivatives are not fully determined by the data. Finding those directions tells us the type.

Suppose a curve $\phi(x,y)=\text{const}$ is a characteristic. Substitute $\xi=\phi$ as a new coordinate. The principal part transforms; the coefficient of $u_{\xi\xi}$ becomes $Q(\phi_x,\phi_y) = A\,\phi_x^2 + B\,\phi_x\phi_y + C\,\phi_y^2.$ Why this expression? Because under a change of variables the chain rule sends $u_{xx}\to \phi_x^2 u_{\xi\xi}+\dots$ , $u_{xy}\to \phi_x\phi_y u_{\xi\xi}+\dots$ , $u_{yy}\to\phi_y^2 u_{\xi\xi}+\dots$ . Collecting $u_{\xi\xi}$ gives exactly $Q$ .

Step 2 — A characteristic is where this coefficient vanishes: $Q=0$ . Along $\phi=\text{const}$ we have $d\phi=\phi_x\,dx+\phi_y\,dy=0$ , so the slope is $\frac{dy}{dx} = -\frac{\phi_x}{\phi_y}.$ Divide $Q=0$ by $\phi_y^2$ and let $m=\phi_x/\phi_y$ : $A\,m^2 + B\,m + C = 0 \quad\Rightarrow\quad m = \frac{-B \pm \sqrt{B^2-4AC}}{2A}.$

The names echo conic sections $Ax^2+Bxy+Cy^2=\text{const}$ , classified by the same $B^2-4AC$ .

Figure — Classification — elliptic, parabolic, hyperbolic (discriminant test)

Worked examples

Boundary/initial conditions each type needs (the practical payoff)

Type	Typical PDE	Domain	Correct data
Elliptic	Laplace	closed region	values on whole boundary (Dirichlet/Neumann)
Parabolic	Heat	open in $t$	initial $u(x,0)$ + side conditions, march forward in $t$
Hyperbolic	Wave	open in $t$	initial $u$ and $u_t$ , march forward in $t$

Common mistakes (steel-manned)

Active recall

Recall Cover and answer first

What three quantities go into $\Delta$ ? (coeffs of $u_{xx},u_{xy},u_{yy}$ )
Sign rule? (− elliptic, 0 parabolic, + hyperbolic)
How many real characteristics for each? (0, 1, 2)
Which physical equation for each? (Laplace, Heat, Wave)
Linear vs quasilinear? (linear: coeffs depend only on $x,y$ ; quasilinear: top-derivative coeffs may depend on $u$ and lower derivatives)

Recall Feynman: explain to a 12-year-old

Imagine dropping a pebble in three different worlds. In the wave world, ripples shoot out along straight lines at a fixed speed — you can see exactly where the ripple front is. That's hyperbolic (two clear lines). In the heat world, a drop of warm dye slowly blurs out, no sharp edge, just smearing as time goes on. That's parabolic (one smearing direction). In the calm-pond world, the water is already perfectly still and every part of the surface gently balances every other part — no fronts, no smearing, just smooth equilibrium. That's elliptic. The little number $B^2-4AC$ is a magic detector that tells you which world your equation lives in: negative = calm pond, zero = blurring dye, positive = shooting ripples.

Connections

Wave Equation — d'Alembert solution (hyperbolic, uses characteristics $x\pm ct$ )
Heat Equation — separation of variables (parabolic)
Laplace Equation — harmonic functions (elliptic, maximum principle)
Method of Characteristics (built directly on the slopes derived here)
Linear vs Quasilinear PDEs (coefficient dependence and what it allows)
Conic Sections (same $B^2-4AC$ test!)
Well-posedness and Boundary Conditions

What is the discriminant used to classify a 2nd-order PDE?

\Delta = B^2 - 4AC

, using coefficients of

u_{xx},u_{xy},u_{yy}

Sign of

\Delta

for elliptic, parabolic, hyperbolic?

\Delta<0

elliptic,

\Delta=0

parabolic,

\Delta>0

hyperbolic.

Which standard PDE is hyperbolic and why?

Wave equation

u_{tt}=c^2u_{xx}

;

\Delta=4c^2>0

, two real characteristics

x\pm ct

Which standard PDE is parabolic and why?

Heat equation

u_t=k u_{xx}

;

C=0

(no

u_{tt}

), so

\Delta=0

Which standard PDE is elliptic and why?

Laplace

u_{xx}+u_{yy}=0

;

\Delta=-4<0

, no real characteristics.

Number of real characteristics for each type?

Hyperbolic 2, parabolic 1 (repeated), elliptic 0 (complex).

Why do only second-order terms classify the PDE?

They control characteristics/propagation; lower-order terms only shift and damp, not change type.

What is the characteristic slope formula?

dy/dx = (B \mp \sqrt{B^2-4AC})/(2A)

Classify Tricomi

y u_{xx}+u_{yy}=0

\Delta=-4y

: elliptic (

y>0

), parabolic (

y=0

), hyperbolic (

y<0

) — mixed type.

Why does the heat equation have

C=0

u_t

is first order; there is no

u_{tt}

term, so the coefficient of

u_{yy}

is zero.

What data does an elliptic PDE need?

Values on the entire closed boundary (Dirichlet/Neumann).

What data does a hyperbolic PDE need?

Initial

u

AND

u_t

, then march forward in time.

Definition of a LINEAR 2nd-order PDE?

All coefficients (of every derivative, including the highest) depend only on the independent variables

x,y

— not on

u

or its derivatives.

Definition of a QUASILINEAR PDE?

Coefficients of the highest-order derivatives may depend on

u

and its lower-order derivatives, but the highest derivatives still appear linearly.

Concept Map

Hinglish (regional understanding)

Intuition Hinglish mein samjho

Dekho, second-order PDE ko classify karna bahut simple hai agar tum sirf ek number pe dhyaan do: discriminant $\Delta = B^2 - 4AC$ . Yahan $A, B, C$ sirf second-order terms $u_{xx}, u_{xy}, u_{yy}$ ke coefficients hain — first-order $u_x, u_y$ wale terms ko bilkul ignore karo, woh classification mein aate hi nahi. Bas sign dekho: $\Delta<0$ matlab elliptic (Laplace), $\Delta=0$ matlab parabolic (Heat), $\Delta>0$ matlab hyperbolic (Wave).

Ek important baat: linear aur quasilinear ek nahi hote. Linear PDE mein saare coefficients (highest derivatives ke bhi) sirf independent variables $x,y$ pe depend karte hain, $u$ ya uske derivatives pe nahi. Quasilinear mein highest-order derivatives ke coefficients $u$ aur uske lower-order derivatives pe depend kar sakte hain (par highest derivatives phir bhi linearly aate hain). Isliye quasilinear case mein type khud solution pe bhi depend kar sakta hai.

Ye discriminant aata kahan se hai? Hum poochte hain ki solution kis curve ke along "kink" rakh sakta hai — usko characteristic bolte hain. Calculation karne par ek quadratic milta hai $Am^2+Bm+C=0$ , aur uske real roots ki ginti $\Delta$ ke sign se decide hoti hai. Do real roots = hyperbolic, ek root = parabolic, koi real root nahi = elliptic.

Physics-wise: Wave mein signal fixed speed se sharp line ke along jaata hai (hyperbolic), Heat mein dye dheere-dheere blur hota hai (parabolic), aur Laplace mein paani bilkul shaant, har point har boundary point ko feel karta hai (elliptic). Common galti: Heat equation mein $C=1$ maan lena — galat! $u_t$ first order hai, $u_{tt}$ hai hi nahi, to $C=0$ aur $\Delta=0$ . Aur Tricomi jaise equations mein coefficients $x,y$ pe depend karte hain, to har point pe $\Delta$ evaluate karo.

Classification — elliptic, parabolic, hyperbolic (discriminant test)

The setup

WHERE the discriminant comes from (derivation, not a dump)

Worked examples

Boundary/initial conditions each type needs (the practical payoff)

Common mistakes (steel-manned)

Active recall

Connections

Concept Map

Hinglish (regional understanding)

Go deeper — visual, from zero

Test yourself — Partial Differential Equations

Connections