the canonical form
When a messy equation can be cleaned up by a clever change of variables until it looks like one of a few standard templates, that tidied-up version is its canonical form. Just as any genuine ellipse can be rotated and stretched until it reads x^2/a^2 + y^2/b^2 = 1, any second-order linear PDE of a given type can be transformed until its principal part matches the model equation for that type. The canonical form is the equation seen in its most honest coordinates.
The recipe is to choose new variables aligned with the characteristics. For a hyperbolic equation, set the new coordinates xi and eta to be constant along the two characteristic families; in those coordinates the cross-derivative is all that survives and the equation reduces to the canonical u_{xi eta} + (lower order) = 0 (or, after a 45-degree turn, the wave form u_{ss} - u_{tt} + ... = 0). For a parabolic equation there is only one characteristic family; aligning one coordinate with it kills two of the three principal terms and leaves the heat-like canonical u_{eta eta} + (lower order) = 0. For an elliptic equation the characteristics are complex, so taking real and imaginary parts produces the Laplace-like canonical u_{xi xi} + u_{eta eta} + (lower order) = 0. Three types, three canonical templates.
Why bother? Because once an equation is in canonical form you may apply everything you already know about the corresponding model — d'Alembert's method for the hyperbolic form, the heat-equation toolkit for the parabolic form, harmonic-function theory for the elliptic form. It is the lever that reduces the whole forest of second-order linear equations to three trees. The honest limitation: the reduction is generally LOCAL when the coefficients vary, and it touches only the principal part — the lower-order terms come along for the ride and may still complicate the actual solving.
Take u_xx - u_yy = 0. Its characteristics are y - x = const and y + x = const, so set xi = y - x, eta = y + x. Substituting and simplifying turns the equation into the canonical hyperbolic form u_{xi eta} = 0, which integrates immediately to u = F(xi) + G(eta) = F(y - x) + G(y + x) — d'Alembert's two travelling waves, read straight off the canonical form.
Aligning coordinates with the characteristics collapses the equation to its canonical template, where the solution often falls out by inspection.
The canonical form normalizes only the principal (highest-order) part; the lower-order terms transform along with it and are not erased, so reaching canonical form is a powerful simplification but rarely the final answer by itself.