## First integrals for spherically symmetric shear-free perfect fluid distributions.

##### Abstract

In this dissertation we study spherically symmetric shear-free spacetimes. In particular
we analyse the integrability of and find exact solutions to the Emden-
Fowler equation yxx = f(x)y2; which is the master equation governing the behaviour
of shear-free neutral perfect fluid distributions. We first review the study
of Maharaj et al (1996) by finding a first integral to this master equation. This first
integral is subject to the integrability condition which we use to find restrictions
on the function f(x): We show that this first integral is a generalisation of particular
solutions obtained by Stephani (1983) and Srivastava (1987). Furthermore,
we use a similar method to obtain a new first integral of the master equation. This
is achieved by multiplying the Emden-Fowler equation by an integrating factor.
We then study the integrability condition, which is an integral equation, related to
the new first integral. We find that the integrability condition can be written as a
third order differential equation whose solution can be expressed in terms of elementary
functions and elliptic integrals. In general the solution of the integrability
condition is given parametrically. We believe that this is a new result. A particular
form of f(x) is identified which corresponds to repeated roots of a cubic equation
giving an explicit solution.