the model equations of each type

Imagine learning birds by first memorizing three perfect specimens — a sparrow, a hawk, an owl — so that every new bird can be placed by which one it most resembles. The theory of second-order PDEs is taught the same way. There are three model equations, one for each type, and almost any second-order linear equation you meet can be reduced (by a change of variables) to look locally like one of them. Master these three and you have a map of the whole territory.

The three archetypes are: Laplace's equation u_xx + u_yy = 0, the elliptic model — the equation of equilibrium, of a steady temperature, of a soap film, of an electrostatic potential. The heat (diffusion) equation u_t = k u_xx, the parabolic model — the equation of irreversible smoothing and spreading. The wave equation u_tt = c^2 u_xx, the hyperbolic model — the equation of vibration and propagation at finite speed. They are not arbitrary examples; the classification theorem says that the canonical form of any elliptic, parabolic, or hyperbolic equation IS (to leading order) Laplace's, the heat, or the wave equation respectively, so studying these three is studying every equation of that type.

Each model carries a complete personality that its whole type inherits. Laplace: boundary data on a closed region, no time, infinitely smooth solutions, the mean-value property. Heat: initial data plus boundary data, infinite propagation speed, instant smoothing, irreversibility. Wave: initial displacement and velocity, finite speed, a domain of dependence, no smoothing (singularities travel). The honest caveat is that lower-order terms and variable coefficients add detail the bare models do not show, and away from constant coefficients an equation can even change type from place to place — but the three models remain the indispensable mental anchors.