
Chapter 1 3
Methods for First-Order
Linear PDEs
13.1 Linear PDEs with Two Independent Variables
13.1.1 Special First-Order Linear PDEs with Two Independent
Vari ables
◮ Physical interpretation and the characteristic equation.
Consider a first-order linear homogeneous partial differential equation with two indepen-
dent variables of the special form
f(x, y)
∂w
∂x
+ g(x, y)
∂w
∂y
= 0. (13.1.1.1)
Equation (13.1.1.1) describes a steady-state distribution of the concentration of a sub-
stance in a plane flow (without regard to diffusion). Moreover, it is assumed that the fluid
velocity components along the x- and y-axes are specified by the functions f and g.
The transfo ...