Discussion
A general solution of the partial differential equation
\[ \begin{aligned} &\frac{\partial^{2}f(x,y)}{\partial x^{2}} \\ &\quad - \frac{\partial^{2}f(x,y)}{\partial y^{2}} = 0 \end{aligned} \]
can be expressed in terms of arbitrary functions u and v as :
Answer: A
The general solution is \(f(x,y) = u(x-y) + v(x+y)\).
The equation is the one-dimensional wave equation with unit speed. In the characteristic variables \(\xi = x - y\) and \(\eta = x + y\) it becomes \(\partial^{2}f/\partial\xi\,\partial\eta = 0\), whose solution is a sum of an arbitrary function of each variable. This is d'Alembert's solution.
Check directly: for \(u(x-y)\), \(f_{xx} = u''\) and \(f_{yy} = (-1)^{2}u'' = u''\), so \(f_{xx} - f_{yy} = 0\).
The form \(u(x+iy) + v(x-iy)\) solves Laplace's equation \(f_{xx} + f_{yy} = 0\) instead. For \(u(2x \pm y)\) or \(u(x \pm 2y)\), \(f_{xx}\) and \(f_{yy}\) differ by a factor of 4, so they solve wave equations with speed 2 or 1/2, not this one.
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