From one variable to several
Climbing up from ODEs, you already have the central instinct: a differential equation does not hand you a function, it hands you a rule the function's derivatives must obey, and your job is to find every function that obeys it. The leap to a partial differential equation is to let the unknown depend on more than one thing at once. Instead of u(t), a temperature that changes only in time, picture u(x, t): a temperature that lives at every point x of a metal bar and also drifts as the clock t runs. Now the equation can relate how u changes in space to how it changes in time — and that is exactly what a partial differential equation does.
Two roles need naming. The things u depends on — here x and t — are the independent variables, the coordinates you are free to slide around. The quantity u itself is the dependent variable, the unknown function you are solving for. With several independent variables, a single derivative is no longer enough; we differentiate with respect to one variable at a time, holding the others fixed. That is the partial derivative you met in multivariable calculus, and we write it compactly: u_t for the time derivative, u_x for the x-derivative, u_xx for differentiating in x twice, u_xy for once in x then once in y.
Reading order and the leading part
Two quick numbers tell you most of what a PDE is. The first is its order: the highest number of derivatives stacked on the unknown anywhere in the equation. The heat equation u_t = k u_xx is second order, because u_xx is two x-derivatives and that beats the single u_t. A transport equation u_t + c u_x = 0 is first order. Order matters because, loosely, it counts how much data you must supply to pin a solution down — just as a second-order ODE needed two conditions, a second-order PDE in time will want both an initial position and an initial velocity.
The second is subtler and more powerful: the principal part, the bundle of highest-order terms with everything lower thrown away. For u_xx + u_yy + 3 u_x - u = 0 the principal part is just u_xx + u_yy; the 3 u_x and the -u are lower-order noise for this purpose. Why care about only the top terms? Because they govern the deepest behavior — how fast information travels, whether solutions stay smooth, what kind of data is allowed. The principal part is the part that, in the next rung, gets distilled into the discriminant B^2 - A C and sorts every second-order equation into one of three families.
The linearity ladder
How a PDE depends on the unknown and its derivatives splits the whole subject into a ladder of difficulty. At the friendly bottom sits the linear PDE: u and all its derivatives appear only to the first power, never multiplied by each other, and their coefficients depend at most on the independent variables. u_t = k u_xx is linear; so is u_xx + u_yy = 0. Linearity is the property that makes everything in this rung possible, because — as the fourth guide will celebrate — linear equations let you add solutions together.
Climb one step and nonlinearity creeps in, but politely. A semilinear PDE keeps the principal part linear and only lets the lower-order terms go nonlinear, like u_t - u_xx = u^2. A quasilinear PDE lets the unknown into the coefficients of the highest derivatives, but those derivatives still appear only to the first power, like u_t + u u_x = 0 — the inviscid Burgers equation, where the solution sets its own speed. At the wild top, a fully nonlinear PDE has the highest derivatives entering nonlinearly, as in u_x^2 + u_y^2 = 1. Each rung up costs you tools and buys you difficulty.
Forcing, balance, and adding solutions
Inside the linear world there is one more split worth naming early. A linear PDE is homogeneous when every term contains the unknown, so the all-zero function u = 0 is automatically a solution; it is inhomogeneous when there is a leftover term with no u in it at all. That extra term is the source term — a heater pumping warmth into the bar, a force shaking the string, a charge density driving a potential. The distinction is exactly the homogeneous versus inhomogeneous one you already met for ODEs, and it works the same way: the source is what makes the equation describe a system being driven rather than left alone.
Homogeneity and linearity together unlock the single most important tool of this whole rung: the superposition principle. If u and v both solve a homogeneous linear PDE, then so does any combination a u + b v — the equation simply cannot tell the difference. This is why you can build a complicated temperature profile by adding up simple ones, why a vibrating string is a sum of pure tones, why a Fourier series solves so much. The entire machinery of separation of variables, eigenfunction expansions, and Fourier methods that fills the coming rungs rests on the superposition principle — and it is precisely what nonlinear equations refuse to grant.
There is one honest subtlety to carry forward. A general solution of a PDE involves arbitrary functions, not just the arbitrary constants you got used to with ODEs. The little equation u_t + c u_x = 0, for instance, is solved by u(x, t) = f(x - c t) for any differentiable f — a whole function's worth of freedom. So a PDE's general solution describes an enormous family, and what finally pins down the one solution you want is data: initial conditions and boundary conditions. Deciding which data makes the problem well-posed — and which leaves it ill-posed — is itself governed by the equation's type, the theme of the very next rung.
Three equations that map the whole field
You do not have to face the infinite zoo of PDEs all at once. Almost the entire subject is organized around three model equations, each a pure representative of one fundamental kind of behavior. Think of them as the three landmarks on the roadmap: master these and you have a compass for everything else, because a new equation usually behaves like whichever of the three it most resembles.
HEAT (parabolic): u_t = k u_xx smooths; infinite speed; one-way in time WAVE (hyperbolic): u_tt = c^2 u_xx no smoothing; finite speed c; reversible LAPLACE (elliptic): u_xx + u_yy = 0 no time; equilibrium; perfectly smooth
Each tells a different story, and the contrasts are honest, not slogans. The heat equation u_t = k u_xx describes spreading and diffusion: it instantly smooths out any roughness in the data, but it does so with infinite propagation speed (a mathematical idealization — a spike at one point nudges the temperature everywhere at once), and it runs one way in time, so the backward heat equation is ill-posed. The wave equation u_tt = c^2 u_xx describes vibration and signals: it does not smooth — a kink in the initial shape travels along untouched — and it carries information at a finite speed c, neatly via u(x, t) = f(x - c t) + g(x + c t), a right-mover plus a left-mover.
The third, Laplace's equation Laplacian u = 0 (in two dimensions, u_xx + u_yy = 0), has no time variable at all: it describes a system in perfect equilibrium — a steady temperature, an electrostatic potential, a soap film pulled taut. Its solutions are extraordinarily smooth, and the right data is values on the boundary, not an initial condition. Trying to prescribe Cauchy data (value and normal derivative on a curve) for Laplace's equation is famously ill-posed. Those three — diffusion, propagation, equilibrium — are the parabolic, hyperbolic, and elliptic prototypes, and sorting any equation into its type is exactly where the next rung begins.