## Computer analysis of equations using Mathematica.

##### Abstract

In this thesis we analyse particular differential equations that arise in physical situations.
This is achieved with the aid of the computer software package called
Mathematica. We first describe the basic features of Mathematica highlighting its
capabilities in performing calculations in mathematics. Then we consider a first order
Newtonian equation representing the trajectory of a particle around a spherical
object. Mathematica is used to solve the Newtonian equation both analytically and
numerically. Graphical plots of the trajectories of the planetary bodies Mercury,
Earth and Jupiter are presented. We attempt a similar analysis for the corresponding
relativistic equation governing the orbits of gravitational objects. Only numerical
results are possible in this case. We also perform a perturbative analysis of the relativistic
equation and determine the amount of perihelion shift. The second equation
considered is the Emden-Fowler equation of order two which arises in many physical
problems, including certain inhomogeneous cosmological applications. The analytical
features of this equation are investigated using Mathematica and the Lie analysis
of differential equations. Different cases of the related autonomous form of the
Emden-Fowler equation are investigated and graphically represented. Thereafter, we
generate a number of profiles of the energy density and the pressure for a particular
solution which demonstrates that a numerical approach for studying inhomogeneity,
in cosmological models in general relativity, is feasible.