\documentstyle{article}
\begin{document}

\centerline{\bf An Algorithm for Constructing Integrating Factors}
\centerline{\bf and Corresponding First Integrals for Systems of ODE's}

\vskip 0.3cm

\centerline{Stephen Anco and George Bluman}

\centerline{Department of Mathematics}
\centerline{University of British Columbia}
\centerline{Vancouver, BC, CANADA, V6T 1Z2}
\centerline{Email: bluman@math.ubc.ca}

\vskip 0.3cm

For first order scalar ODEs, Sophus Lie showed how to construct an
integrating factor for each admitted point symmetry; conversely, Lie
showed that each integrating factor yields admitted point symmetries.

In general, for systems of ODEs, integrating factors are factors
yielding conservation laws. If a system of ODEs is self-adjoint (i.e.,
its linearized system is self-adjoint), then  integrating factors must
be symmetries of the system.  If a given system of ODEs is not
self-adjoint, then integrating factors must be solutions of the adjoint
system of its linearized system.

In this talk we present a formulation which yields all integrating
factors and corresponding first integrals for any given system of ODEs.
The first integrals are obtained algorithmically through an adjoint
invariance condition together with a formula which directly yields a
conservation law for any solution of the condition.  When the given
system is self-adjoint, this algorithm is computationally simpler than
the classical Noether's Theorem for obtaining conservation laws.

Through an example we will show that our formulation has advantages over
Lie's method for reducing the order of ODEs.

\end{document}
