linear
,
first

order partial differential equations
237
instantaneous rate of change of
f
in the direction of the vector
v
. If we choose
the unit vector
v
=(
1, 0
)
, which points in the direction of the positive
x
axis,
then the directional derivative is simply
∇
f
(
x
,
y
)
•
v
=
±
∂
f
∂
x
,
∂
f
∂
y
²
•
(
1, 0
)=
∂
f
∂
x
,
the partial derivative of
f
with respect to
x
. Similarly, the directional derivative in
the direction of
v
0, 1
)
is given by
∂
f
/
∂
y
. We now use the notion of directional
derivatives to give a quick, clean solution of a special case of Equation (
9
.
1
).
A linear, homogeneous, constantcoefFcient equation.
Consider the
pde
α
u
t
+
β
u
x
=
0,
(
9
.
2
)
where
α
and
β
are nonzero constants. Our goal is to determine all functions
u
(
x
,
t
)
that satisfy this equation. Observe that since the gradient of
u
is given by
∇
u
u
x
,
u
t
)
, the
pde
(
9
.
2
) is equivalent to the equation
∇
u
(
x
,
t
)
•
(
β
,
α
0.
In other words, the solutions of the
pde
are precisely those functions
u
(
x
,
t
)
whose
directional derivative in the direction of the vector
(
β
,
α
)
is
0
. Geometrically, this
implies that
u
(
x
,
t
)
must remain
constant
as we move along any line parallel to
the vector
(
β
,
α
)
. These lines have slope
α
/
β
in the
xt
plane, and the general
equation for such lines is
t
α
/
β
)
x
+
constant. Equivalently,
u
(
x
,
t
)
must
remain constant along any line of the form
α
x

β
t
=
C
, where
C
is an arbitrary
constant. Since
u
remains constant along each such line,
u
depends only on
the value of the constant
C
. In other words,
u
depends only upon the quantity
α
x

β
t
, but otherwise has no restrictions at all. It follows that the general
solution of Equation (
9
.
2
) is
u
(
x
,
t
f
(
α
x

β
t
)
,(
9
.
3
)
where
f
is any (differentiable) function of a single variable.