Final Answer:
The solutions to the given differential equation (16.5) are
and

Step-by-step explanation:
The differential equation (16.5) is given by
The general solution to this differential equation is
where
are the Bessel functions of the first and second kind, respectively, and
are constants.
In this particular case, the presence of Bessel functions
in the solution indicates that the problem involves cylindrical symmetry or circular geometry. Bessel functions commonly appear in problems where the geometry is not Cartesian but rather cylindrical or spherical. The term
in the argument of the Bessel functions is characteristic of these specialized functions.
The Bessel functions
are solutions to Bessel's differential equation and are widely used in physics and engineering, particularly in problems with circular symmetry such as heat conduction, wave propagation, and electromagnetic fields. Therefore, the final answer
is a valid representation of the solutions to the given differential equation, capturing the circular nature of the underlying geometry.