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Chapter 2

First-order linear equations

In the previous lecture we studied how some simple PDEs can be reduced to ODEs, and
subsequently solved using ODE methods. For example, the equation

u x = 0 (2.1)

has ”constant in x” as its general solution, and hence u depends only on y, thus

u(x, y) = f(y)

is the general solution, with f(y) an arbitrary function of a single variable.
In this lecture we will see that any linear first order PDE can be reduced to an ordinary
differential equation, which will then allow as to solve it with already familiar methods
from ODEs.
Let us start with a simple example. Consider the following constant coefficient PDE

au x + bu y = 0 (2.2)

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2
Here a and b are constants, such that a + b 6= 0, i.e. at least one of the coefficients is
nonzero (otherwise this would not be a differential equation).

2.1 The method of characteristics

Geometric approach. The quantity au x + bu y is the directional derivative of u in the
direction of the vector V = (a, b) = ai + bj. It must always be zero. This means that
u(x, y) must be constant in the direction of V. The vector (b, −a) is orthogonal to V.
The lines parallel to V (see Figure 2.1) have the equations bx−ay = constant. (They are
called the characteristic lines.) The solution is constants on each such line. Therefore,
u(x, y) depends on bx − ay only. Thus the solution is

u(x, y) = f(bx − ay), (2.3)

where f is any function of one variable. Let’s explain this conclusion more explicitly. On
the line bx − ay = c, the solution u has a constant value. Call this value f(c). Then

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