Why just three equations?
You have now met what a PDE is, how order and the principal part work, the ladder from linear to fully nonlinear, and the superposition principle that lets you build complicated linear solutions out of simple ones. This last guide of the rung gives you a map. Almost every second-order linear PDE you meet behaves like one of three model equations, and learning those three deeply is like learning three landmark cities before you tour a whole country.
The three are the heat equation u_t = k u_xx, the wave equation u_tt = c^2 u_xx, and Laplace's equation Laplacian u = u_xx + u_yy = 0. They are the canonical representatives of the three types — parabolic, hyperbolic, and elliptic — that the discriminant B^2 - A C sorted out in an earlier rung. The model-equations roadmap is exactly the promise that understanding these three carries over, by classification, to a vast zoo of cousins.
Heat: the parabolic one that smooths and forgets
The heat equation u_t = k u_xx is the archetype of diffusion. Read it as a rule of motion: u_xx measures how much a point's value sits below or above the average of its neighbors, so wherever a bump pokes up, u_xx is negative and u_t pushes it down; wherever a dip sinks below, u_t lifts it up. Run the rule and every wrinkle gets averaged away — the equation is parabolic, and its job is to flatten differences toward equilibrium.
Two surprises make the heat equation worth respecting. First, it has infinite propagation speed: drop a hot spot at one point and, mathematically, the temperature rises everywhere instantly — see infinite speed. That is a known idealization (real diffusion is not truly instantaneous), but it is exactly what the smooth model predicts. Second, it has a powerful smoothing effect: even jagged, discontinuous initial data becomes perfectly smooth the instant t > 0.
Smoothing has a hard price: the equation forgets. Because sharp features get averaged into mush, you cannot reconstruct the past from the present. Running the backward heat equation — solving for earlier times — is a classic ill-posed problem: tiny measurement errors in the present blow up explosively as you step back. This irreversibility is the famous one-way arrow of time, baked right into the equation.
Wave: the hyperbolic one that travels and remembers
The wave equation u_tt = c^2 u_xx is the opposite temperament. It is hyperbolic, and in one space dimension d'Alembert showed its general solution is breathtakingly simple: any disturbance splits into a right-mover and a left-mover, each gliding at speed c without changing shape. Whatever you plucked into the string sails off intact in both directions.
Written out, d'Alembert's formula is u(x,t) = F(x - c t) + G(x + c t): a right-moving shape F and a left-moving shape G, each a travelling wave of the form u = f(x - c t) that simply slides rigidly along. Contrast this with heat at every turn. The wave equation has finite propagation speed — nothing outruns c, so a signal at one point reaches another only after the travel time, the content of finite speed. And it does not smooth: a kink in the initial data stays a kink forever, riding along the characteristics. Where heat blurs and forgets, the wave equation transmits and remembers, which is why it conserves energy and why sound and light can carry sharp information across a room.
Laplace: the elliptic one that has no time
Laplace's equation Laplacian u = u_xx + u_yy = 0 is the odd one out: it has no time variable at all. It describes equilibrium — the steady state a diffusion settles into once u_t has died to zero, or the shape of a soap film stretched on a wire loop. Its solutions are called harmonic functions, and being elliptic gives them a magical rigidity: the value at any point equals the average of the values on every circle around it.
That averaging rule has immediate teeth. A harmonic function can have no interior peak or valley — its maximum and minimum live only on the boundary, the maximum principle. So the right question for an elliptic equation is a boundary-value problem: prescribe u all the way around the edge of a region and the Dirichlet problem determines u everywhere inside, uniquely and smoothly. There is no "initial time" to march from; there is only the fence around the field.
The type is the punchline: which data is well-posed
Now the roadmap pays off. The single most useful thing a type label tells you is what side conditions make a sensible problem — its type-dependent well-posedness. Each of the big three asks for a different kind of data, and asking the wrong kind gives a problem with no solution, infinitely many, or one that explodes under tiny perturbations.
heat (parabolic): u(x,0) given on a line + boundary values -> well-posed forwards in time wave (hyperbolic): u and u_t given on a line at t=0 (Cauchy) -> well-posed both directions Laplace (elliptic): u (or u_n) given on the whole boundary curve -> well-posed; Cauchy data is NOT
The famous trap is prescribing Cauchy data for Laplace's equation. Hadamard's example shows that two boundary inputs differing by an imperceptible high-frequency ripple produce interior solutions that differ enormously — an ill-posed problem, even though the equation is the gentlest of the three. Symmetrically, the heat equation is well-posed running forwards in time but ill-posed running backwards. The lesson of this rung in one sentence: the equation's type, not your wishes, decides what you are allowed to ask.
Where the map leads next
These three equations are linear, so the superposition principle is in full force: you can add solutions, and the upcoming rungs cash that in — separation of variables, Fourier series, transforms, and Green's functions all build big answers by superposing simple pieces. Keep the three temperaments in mind as you go: smoothing-and-forgetting (heat), travelling-and-remembering (wave), and rigid-equilibrium (Laplace).
Two honest cautions before you climb on. The clean type labels assume linear, roughly constant-coefficient equations; truly nonlinear PDEs break the rules — smooth solutions of a nonlinear conservation law can develop shocks in finite time, and afterward a weak solution is not even unique unless you add an entropy condition to pick the physical one. And not everything is settled: whether 3D solutions of the Navier-Stokes equations stay smooth for all time is a famous open problem. The big three are where you start, not the whole country.