### Abstract:

In this thesis a Lane-Emden equation of the second-kind is investigated. The equation
is considered with arbitrary parameters with the intention of obtaining a solution
without referring to specific cases. The shape factor is a parameter indicating the type
of vessel relevant to the physical problem considered. There are various forms of the
equation. We will consider two such forms, where the shape factor is specified to be
one and two, which are of some physical significance. One of these equations is derived
from the steady state heat balance equation, and in so doing models a thermal
explosion. The other equation that is of importance is derived from equations of mass
conservation and dynamic equilibrium. This gives a model describing the dimensionless
density distribution in an isothermal gas sphere. This equation when transformed
appropriately may also be used to model Bonnor-Ebert gas spheres or Richardson’s
theory of thermionic currents which is related to the emission of electricity from hot
bodies. Lie’s theory of extended groups is used in order to obtain infinitesimal generators
and in association with Noether’s theorem may be used to find appropriate
first integrals of the equation. Non-local symmetries are used in conjunction with local
symmetries in order to verify already obtained solutions and to obtain new solutions.
For specified values of the shape factor solutions were obtained in this way within an
infinite slab, infinite circular cylinder and sphere. Computational methods, such as finite
differences, are used to obtain new numerical solutions which are useful indicators
for the exactness of the solutions obtained via other means. We were unable to obtain
solutions to certain specific cases because of the nature of the equation in question.
New physical and mathematical insights are revealed through the solutions found and
the comparisons made between them and other already existing solutions.