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y
v

x

Figure 2.1: Characteristic lines of (2.2)

u(x, y) = f(c) = f(bx − ay). Since c is arbitrary, we have formula (2.3) for all values of
x and y. In xyu-space the solution defines a surface that is made up of parallel horizontal
straight lines like a sheet of corrugated iron.
Coordinate approach. To have an ODE, we need to eliminate one of the partial
derivatives in the equation. Let us change the coordinates

ξ = ax + by
(2.4)
η = bx − ay
and find partial derivatives

∂ξ ∂η
u x = u ξ + u η = au ξ + bu η
∂x ∂x
∂ξ ∂η
u y = u ξ + u η = bu ξ − au η
∂y ∂y
Thus,
2
2
au x + bu y = a(au ξ + bu η ) + b(bu ξ − au η ) = (a + b )u ξ = 0.
2
2
Now, since a + b 6= 0, then, as we anticipated,
u ξ = 0,

which is an ODE. We can solve this last equation just as we did in the case of equation
(2.1), arriving at the solution
u(ξ, η) = f(η).

Changing back to the original coordinates gives

u(x, y) = f(bx − ay).

This method of reducing the PDE to an ODE is called the method of characteristics
(or geometric method), and the coordinates (ξ, η) given by formulas (2.4) are called
characteristic coordinates.

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