a linear first-order PDE

Among first-order equations, the friendliest are the linear ones — the equations where the unknown u and its first derivatives appear only by themselves, each multiplied by a coefficient that does not depend on u. They are friendly because their characteristics can be drawn before the solution is known, and because solutions add: combine two solutions and you get another.

In two variables the general linear first-order PDE is a(x,y) u_x + b(x,y) u_y + d(x,y) u = e(x,y). The coefficients a, b, d, e are functions of position only, never of u. Two important sub-cases sit inside it. If d and e are zero you have the homogeneous transport-type equation a u_x + b u_y = 0, and u is constant along characteristics. If only e is allowed to be non-zero and the equation has no u term, a u_x + b u_y = c(x,y) with c independent of u, the characteristics are still set purely by a and b — a notch simpler than the semilinear case, where the right-hand side may depend on u. Because the characteristic directions dx = a ds, dy = b ds do not involve u, you can lay down the whole web of characteristics first, then carry the data along them.

Linearity is what makes the superposition principle work: the sum of two solutions, or a constant times a solution, solves the same homogeneous equation, so you can build complicated solutions from simple building blocks. This is the gentle end of the subject — no shocks, no gradient blow-up purely from the equation's structure — and it is the right place to learn how characteristics carry data before meeting the nonlinear surprises.