Spherical waves in odd-dimensional space

The general solution is given of the $\left ( {2N + 1} \right )$-dimensional wave equation with spherical symmetry, ${u_{tt}} - {u_{xx}} - \frac {{2N}}{x}{u_x} = 0$, in terms of two arbitrary functions and their first $N$ derivatives. Simple transformations then yield the general solutions to the Euler-Poisson-Darboux equation, ${u_{xy}} + \frac {N}{{\left ( {x + y} \right )}}\left ( {{u_x} + {u_y}} \right ) = 0$, for integer $N$, and also the one-dimensional wave equation, ${u_{tt}} - {c^2}{u_{xx}} = 0$, for certain variable wave speeds $c\left ( x \right )$.