order and linearity of a PDE
Before you can hope to solve a PDE, you sort it by two coarse labels that already tell you most of what to expect: its order and whether it is linear. The order is simply the highest derivative that appears. The linearity asks whether the unknown function and its derivatives enter only in a simple, additive way, or whether they are multiplied together, squared, or fed into nonlinear functions. These two questions are the first thing a working mathematician asks about a new equation.
The order is the largest total number of derivatives taken in any single term: the heat, wave, and Laplace equations are all second order because their highest term is a second derivative. A PDE is linear when the unknown u and all its derivatives appear only to the first power and are never multiplied by each other — schematically, a sum of (coefficient) times (some derivative of u), where the coefficients depend at most on the independent variables, plus a term with no u at all (the source). If those coefficients do not involve u it is linear; if the coefficients depend on u or its lower derivatives it is quasilinear; if the highest derivatives themselves enter nonlinearly it is fully nonlinear. A linear PDE is homogeneous if the source term is zero and inhomogeneous otherwise.
This classification matters because linearity is what makes a problem tractable. For a linear PDE the superposition principle holds: any sum of solutions is again a solution, so you can build complicated answers by adding simple ones — this is exactly what separation of variables and Fourier expansions exploit. Nonlinear PDEs, by contrast, can form shocks, break superposition, and generally resist closed-form solution; the famous open problems (Navier-Stokes regularity, for instance) live there. Knowing the order tells you how many boundary and initial conditions you will need.
partial u / partial t = k partial^2 u / partial x^2 is second order, linear, homogeneous. But Burgers' equation partial u / partial t + u partial u / partial x = nu partial^2 u / partial x^2 is second order and nonlinear, because the term u times partial u / partial x multiplies the unknown by its own derivative.
Same order, opposite tractability: the linear one yields to Fourier methods, the nonlinear one can develop shock waves.
Linear does not mean easy and nonlinear does not always mean impossible, but the dividing line is real: superposition is available only for linear equations, and almost every elementary solution method silently assumes it.