### Abstract:

In Chapter One the theoretical basis for infinitesimal transformations is
presented with particular emphasis on the central theme of this thesis which
is the invariance of ordinary differential equations, and their first integrals,
under infinitesimal transformations. The differential operators associated with
these infinitesimal transformations constitute an algebra under the operation
of taking the Lie Bracket. Some of the major results of Lie's work are recalled.
The way to use the generators of symmetries to reduce the order of a differential
equation and/or to find its first integrals is explained. The chapter concludes
with a summary of the state of the art in the mid-seventies just before the
work described here was initiated.
Chapter Two describes the growing awareness of the algebraic properties of
the paradigms of differential equations. This essentially ad hoc period demonstrated
that there was value in studying the Lie method of extended groups
for finding first integrals and so solutions of equations and systems of equations.
This value was emphasised by the application of the method to a class of
nonautonomous anharmonic equations which did not belong to the then pantheon
of paradigms. The generalised Emden-Fowler equation provided a route
to major development in the area of the theory of the conditions for the linearisation
of second order equations. This was in addition to its own interest.
The stage was now set to establish broad theoretical results and retreat from
the particularism of the seventies.
Chapters Three and Four deal with the linearisation theorems for second
order equations and the classification of intrinsically nonlinear equations according
to their algebras. The rather meagre results for systems of second
order equations are recorded.
In the fifth chapter the investigation is extended to higher order equations
for which there are some major departures away from the pattern established
at the second order level and reinforced by the central role played by these
equations in a world still dominated by Newton. The classification of third
order equations by their algebras is presented, but it must be admitted that
the story of higher order equations is still very much incomplete.
In the sixth chapter the relationships between first integrals and their algebras
is explored for both first order integrals and those of higher orders. Again
the peculiar position of second order equations is revealed.
In the seventh chapter the generalised Emden-Fowler equation is given a
more modern and complete treatment.
The final chapter looks at one of the fundamental algebras associated with
ordinary differential equations, the three element 8£(2, R), which is found in all
higher order equations of maximal symmetry, is a fundamental feature of the
Pinney equation which has played so prominent a role in the study of nonautonomous
Hamiltonian systems in Physics and is the signature of Ermakov
systems and their generalisations.