In this chapter we begin to look at the “Big Three PDEs”—the heat equation (or diﬀusion equation), the wave equation and Laplace’s equation (or the potential equation)—each in two independent variables. Each is a secondorder, linear, homogeneous PDE with constant coeﬃcients. The general such equation is
auxx + buxy + cuyy + dux + fuy + gu = 0, (2.1)
where, again, u = u(x, y) and, of course, a, b, c, d, f and g are constants. We study equation (2.1) in detail in Section 5.4. In particular, there we’ll
classify these equations as in the following deﬁnition and give reasons for such a classiﬁcation.