For a first-order PDE, the method of characteristics discovers curves (called characteristic curves or just characteristics) along which the PDE becomes an ordinary differential equation (ODE). Once the ODE is found, it can be solved along the characteristic curves and transformed into a solution for the original PDE.
For the sake of motivation, we confine our attention to the case of a function of two independent variables x and y for the moment. Consider a quasilinear PDE of the form
-
(
Suppose that a solution u is known, and consider the surface graph z = u(x,y) in R3. A normal vector to this surface is given by
As a result, equation (1 ) is equivalent to the geometrical statement that the vector field
is tangent to the surface z = u(x,y) at every point. In other words, the graph of the solution must be a union of integral curves of this vector field. These integral curves are called the characteristic curves of the original partial differential equation.
The equations of the characteristic curve may be expressed invariantly by the Charpit-Lagrange equations[1]
or, if a particular parametrization t of the curves is fixed, then these equations may be written as a system of ordinary differential equations for x(t), y(t), z(t):
These are the characteristic equations for the original system.
