  The solution in hyperspherical coordinates for $N$ dimensions is given for a
general class of partial differential equations of mathematical physics
including the Laplace, wave, heat and Helmholtz, Schr\"{o}dinger, Klein-Gordon
and telegraph equations and their combinations. The starting point is the
Laplacian operator specified by the scale factors of hyperspherical
coordinates. The general equation of mathematical physics is solved by
separation of variables leading to the dependencies: (i) on time by the usual
exponential function; (ii) on longitude by the usual sinusoidal function; (iii)
on radius by Bessel functions of order generally distinct from cylindrical or
spherical Bessel functions; (iv) on one latitude by associated Legendre
functions; (v) on the remaining latitudes by an extension, namely the
hyperspherical associated Legendre functions. The original associated Legendre
functions are a particular case of the Gaussian hypergeometric functions, and
the hyperspherical associated Legendre functions are also a more general
particular case of the Gaussian hypergeometric functions so that it is not
necessary to consider extended Gaussian hypergeometric functions.
