that are free to be determined, the idea being that in most cases the number of independent choices that can be made is the excess of the latter over the former.
In the context of partial differential equations, constraint counting is a crude but often useful way of counting the number of free functions needed to specify a solution to a partial differential equation.
satisfies the wave equation, we can rewrite where in the first equality, we appealed to the fact that partial derivatives commute.
To answer this in the important special case of a linear partial differential equation, Einstein asked: how many of the partial derivatives of a solution can be linearly independent?
Specifically, the power series counting the variety of arbitrary functions of three variables (no constraints) is but the power series counting those in the solution space of some second order p.d.e.
for an arbitrary function of n variables is where the coefficients of the infinite power series of the generating function are constructed using an appropriate infinite sequence of binomial coefficients, and the power series for a function required to satisfy a linear m-th order equation is Next, which can be interpreted to predict that a solution to a second order linear p.d.e.
To verify this prediction, recall the solution of the initial value problem Applying the Laplace transform
to the two spatial variables gives or Applying the inverse Laplace transform gives Applying the inverse Fourier transform gives where Here, p,q are arbitrary (sufficiently smooth) functions of two variables, so (due their modest time dependence) the integrals P,Q also count as "freely chosen" functions of two variables; as promised, one of them is differentiated once before adding to the other to express the general solution of the initial value problem for the two dimensional wave equation.
In the case of a nonlinear equation, it will only rarely be possible to obtain the general solution in closed form.
However, if the equation is quasilinear (linear in the highest order derivatives), then we can still obtain approximate information similar to the above: specifying a member of the solution space will be "modulo nonlinear quibbles" equivalent to specifying a certain number of functions in a smaller number of variables.