To show how one may reformulate a differential equation as Pfaff's problem for a set of differential forms, consider the wave equation $${\partial^2 u \over \partial t^2} = {\partial^2 u \over \partial x^2} + {\partial^2 u \over \partial y^2}$$

The first step is to rewrite the equation as a system of first-order equations $${\partial a \over \partial t} - {\partial b \over \partial x} - {\partial c \over \partial y} = 0$$ $${\partial u \over \partial t} - a = 0$$ $${\partial u \over \partial x} - b = 0$$ $${\partial u \over \partial y} - c = 0$$

To translate these equations into the language of differential forms, we shall use the fact that $$du = {\partial u \over \partial t} \, dt + {\partial u \over \partial x} \, dx + {\partial u \over \partial y} \, dy$$ from which it follows that $$du \wedge dx \wedge dy = {\partial u \over \partial t} \, dt \wedge dx \wedge dy$$ $$du \wedge dy \wedge dt = {\partial u \over \partial x} \, dt \wedge dx \wedge dy$$ $$du \wedge dt \wedge dx = {\partial u \over \partial y} \, dt \wedge dx \wedge dy$$ We can do likewise with $a$ or $b$ or $c$ in the place of $u$ ; there is no point in repeating the formulas for each of these variables.

Multiplying the differential equations through by the form $dt \wedge dx \wedge dy$ and using the above identities to eliminate partial derivatives, we obtain the following system of differential forms: $$da \wedge dx \wedge dy - db \wedge dy \wedge dt - dc \wedge dt \wedge dx$$ $$du \wedge dx \wedge dy - a \, dt \wedge dx \wedge dy$$ $$du \wedge dy \wedge dt - b \, dt \wedge dx \wedge dy$$ $$du \wedge dt \wedge dx - c \, dt \wedge dx \wedge dy$$ From the way these forms were constructed, it is clear that a three dimensional surface in the seven dimensional space with coordinates $x,y,t,a,b,c,u$ which solves Pfaff's problem and can be parameterized by $x,y,t$ corresponds to the graph of a solution to the system of differential equations, and hence to a solution of the wave equation.

Note: These considerations are purely local. The global topology of the seven-dimensional space will depend on the domain on which the original wave equation was formulated and on the boundary conditions.