A.1 Deriving Generating Functions via Characteristic Curves

We follow Hildebrand (1976, Chap. 8) in summarizing the method for deriving generating functions defined by quasilinear partial differential equations.

We only consider equations with two independent variables, x and y, and a dependent variable z, of the form

An important special case is

We can interpret this equation geometrically as saying that the vector (P, Q, R) is orthogonal to the gradient VG, i.e., the vector lies in the tangent plane to G(x, y, z) = const.

vector (P, Q, R) is tangent to any curve on the surface passing through at the point. Such curves are called characteristic curves of the differential equation.

A characteristic curve has the same direction as the vector (P, Q, R); hence the tangent vector (dx, dy, dz) is such that

because P = μ dx/ds, Q = μ dy/ds, and R = μ dz/ds, where s represents arc length along the curve. This is equivalent to two ordinary differential equa­tions. When R = 0, we take dz = 0. This and dx/P = dy/Q constitute the two equations.

Let the solutions of the two independent differential equations be

for i = 1, 2, where c₁ and c₂ are two independent constants. These represent two families of surfaces such that the intersections of the two surfaces are the characteristic curves. The two constants are related by

that is,

is an integral surface of the original partial differential equation for any differ­entiable function F.

The function F or f may be chosen to include a specified curve on the integral surface.

In the special case in which P and Q do not depend on z, we can treat

as an ordinary differential equation independent of z. If it can be solved in the form u₁(x, y) = c₁, then we use it to express y, say, in terms of x and c₁. Since we are looking for a second expression u₂(x, y, z) = c₂ such that F (c₁, c₂) = 0, we keep c₁ constant. For example, the second equation becomes

after y is substituted out. When this is solved as