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 put this into a more symmetrical form. Suppose that G(x, y, z) = c defines a solution implicitly, i.e., this equation determines z as a function of x and y that satisfies the partial differential equation. Assume that ∂G/∂z ≠ 0.

Then,

and

Substituting these into the original equation, we arrive at

We can interpret this equation geometrically as saying that the vector (P, Q, R) is orthogonal to the gradient ∇ G, i.e., the vector lies in the tangent plane to G(x, y, z) = const. At any point on the solution (integral) surface, the 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.