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which describes motions with constant speed c. On considering the dynamics of the
concentration of a pollutant in a stream of water flowing through a thin tube at a constant
speed c, the transport equation can be applied.
Let u(t, x) denote the concentration of the pollutant in gr/cm (unit mass per unit
length) at time t. The solution to the PDE (2.5) can be written as

u(t, x) = f(x − ct) (2.6)

for any arbitrary single-variable function f.
Let us now consider a particular initial condition for u(t, x)

(
x 0 < x < 1
u(0, x) = (2.7)
0 otherwise

According to (2.6), u(0, x) = f(x), which determines the function f. Having found
the function from the initial condition, we can now evaluate the solution u(t, x) of the
transport equation from (2.6). Indeed
(
x − ct 0 < x − ct < 1
u(t, x) = f(x − ct) = (2.8)
0 otherwise

Noticing that the inequalities 0 < x − ct < 1 imply that x is in-between ct and ct + 1,
we can rewrite the above solution as
(
x − ct ct < x < ct + 1
u(t, x) = (2.9)
0 otherwise,

which is exactly the initial function u(0, x), given by (2.7), moved to the right along the
x−axis by ct units. Thus, the initial data u(0, x) travels from left to right with constant
speed c.

2.3 General constant coefficient equations

We can easily adapt the method of characteristics to general constant coefficient linear
first-order equations
au x + bu y + cu = g(x, y) (2.10)

Recall that to find the general solution of this equation it is enough to find the general
solution of the homogeneous equation

au x + bu y + cu = 0, (2.11)

and add to this a particular solution of the nonhomogeneous equation (2.10). Notice that
in the characteristic coordinates (2.4), equation (2.11) will take the form

c
2
2
(a + b )u ξ + cu = 0 or u ξ + u = 0,
2
a + b 2
10