Reading an equation before solving it
In the previous guide you met what a partial differential equation actually is: a relation among an unknown function and its partial derivatives with respect to two or more independent variables. That is a big family. Before you can hope to solve a member of it, you want to classify it — the way a botanist names a plant before deciding how to care for it. The three cheapest, most useful labels are its order, its place on the linearity ladder, and its principal part.
These labels are not bureaucracy. They predict behaviour. Order tells you how much data you will need to pin a solution down. Linearity tells you whether the powerful trick of adding solutions together is available. And the principal part — the cluster of highest-order derivatives — secretly decides the equation's whole personality: whether disturbances travel at finite speed or infinite speed, whether sharp corners get smoothed away or preserved, even which problems are sensibly posed at all.
Order: count the deepest derivative
The order of a PDE is simply the highest number of times the unknown is differentiated in any single term. In "u_t = k u_xx", the deepest term is u_xx — two derivatives in x — so this is second order. In the transport equation "u_t + c u_x = 0", every derivative is single, so it is first order. A mixed partial like u_xy still counts as second order, because two differentiations are stacked, just in different variables.
Why does counting matter so much? Because order foreshadows how many side conditions you must supply. A first-order ODE needs one constant; a second-order ODE needs two. PDEs follow the same instinct, but with a twist you should hold onto: a PDE's general solution involves arbitrary functions, not arbitrary constants. The wave equation "u_tt = c^2 u_xx" has general solution u(x,t) = F(x - c t) + G(x + c t) for any two functions F and G. Order roughly governs how many such free functions appear — and therefore how much data is needed to lock them in.
The principal part: highest derivatives rule
Now the subtle one. Inside a second-order equation, gather only the top-order terms — every term carrying exactly two derivatives — and you have the principal part. For a general second-order equation in two variables, write A u_xx + 2B u_xy + C u_yy as the principal part; everything else (the u_x, u_y, u, and any source term) is the lower-order remainder. The principal part is the equation's spine, and it is what we read off when we want to know how the equation behaves.
Here is the punchline that surprises newcomers: the lower-order terms barely affect the deep character of the equation. The qualitative type — elliptic, parabolic, or hyperbolic — is decided entirely by the coefficients A, B, C of the principal part, through the sign of the discriminant B^2 - A C. Negative gives elliptic (Laplace-like, steady states), zero gives parabolic (heat-like, diffusion), positive gives hyperbolic (wave-like, propagation). Add a friction term or a source and the type does not change; the personality lives in the top derivatives.
A u_xx + 2B u_xy + C u_yy + (lower-order terms)
principal part remainder
B^2 - A C < 0 elliptic (Laplace u_xx + u_yy = 0)
B^2 - A C = 0 parabolic (heat u_t = k u_xx)
B^2 - A C > 0 hyperbolic (wave u_tt = c^2 u_xx)If you push further into this ladder you will meet the symbol of a differential operator: replace each derivative by a frequency variable and the principal part becomes the principal symbol, a polynomial whose sign pattern is exactly this classification, generalized cleanly to many variables. For now, just hold the picture: highest derivatives carry the soul of the equation.
The linearity ladder
The second great label sorts PDEs by how the unknown enters. An equation is linear when the unknown and all its derivatives appear only to the first power, multiplied by coefficients that depend on the independent variables alone. The next guide walks the full ladder — linear, semilinear, quasilinear, and fully nonlinear — but here is the quick map so the words feel familiar when you arrive.
- Linear: "u_t = k u_xx". Coefficients depend only on x and t; u and its derivatives appear to the first power. The gold standard, where superposition lives.
- Semilinear: the principal part is still linear with constant or variable coefficients, but a lower-order term is nonlinear, e.g. "u_t = u_xx + u^2". The top derivatives behave; the mess is downstairs.
- Quasilinear: the highest derivatives still appear to the first power, but their coefficients now depend on u or its lower derivatives, e.g. "u_t + u u_x = 0". The equation's geometry depends on its own solution.
- Fully nonlinear: the highest derivatives appear nonlinearly, e.g. "u_x^2 + u_y^2 = 1". Here even the principal part bends to the solution; the toolbox shrinks dramatically.
Be honest about why this ladder matters: most clean theory — exact formulas, Fourier methods, neat existence results — lives at the linear rung. As you climb toward the nonlinear top, beautiful and brutal new effects appear. A smooth solution of a nonlinear conservation law such as "u_t + u u_x = 0" can develop a shock in finite time, after which the gradient blows up and a plain classical solution no longer exists. That is not a flaw in your algebra; it is the physics of the equation, and it is exactly why nonlinearity is its own continent.
Homogeneous, inhomogeneous, and superposition
One more linear-world distinction earns its keep. A linear equation is homogeneous when every term contains the unknown, so the all-zero function u = 0 is automatically a solution. It is inhomogeneous when a standalone source term sits on the right, like "u_t = k u_xx + f(x,t)" with a heater f pumping energy in. Homogeneous means "left alone"; inhomogeneous means "forced".
The reward for staying linear is the superposition principle, the engine of nearly every solution method you will learn: if u1 and u2 each solve a linear homogeneous equation, then so does any combination a u1 + b u2. You can therefore break a hard problem into simple pieces, solve each, and add the answers back. The next guide is devoted to this idea, and it underwrites Fourier series, eigenfunction expansions, and the heat kernel alike. The instant the equation goes nonlinear, superposition fails — adding two solutions of "u_t + u u_x = 0" does not give a third.
A roadmap: the three model equations
Everything above converges on a beautifully small roadmap: almost every second-order linear PDE you meet is, in its principal part, a disguised version of one of three models — one per type. They are the heat equation "u_t = k u_xx" (parabolic), the wave equation "u_tt = c^2 u_xx" (hyperbolic), and Laplace's equation "Laplacian u = u_xx + u_yy = 0" (elliptic), the last built from the Laplacian.
These are not three random examples; they are three archetypes of behaviour. The heat equation has infinite signal speed and smooths data instantly — but is irreversible, so running it backward in time is ill-posed. The wave equation has finite speed and does not smooth, carrying corners forward intact. Laplace's equation has no time at all, describing equilibrium; demanding the wrong kind of data for it, such as Cauchy data, is also ill-posed. Which data makes a problem well-posed is dictated by the type — this is type-dependent well-posedness, and the final guide of this rung makes the trio vivid.
So the loop closes neatly. Read the order, read the principal part, read the discriminant — and you already know which archetype you are facing, which data will be well-posed, and roughly how the solution will behave, all before solving a single thing. That is the quiet power of classification, and it is the lens you will carry through every guide that follows.