UBC Mathematics Department
As is well known from elementary differential equations courses, for a first order ODE, finding an integrating factor is equivalent to solving it in closed form. But how does one find integrating factors systematically? In the late 19th century Sophus Lie showed that each point symmetry admitted by a first order ODE yields an explicit formula for an integrating factor and, conversely, each integrating factor yields an infinite point symmetry group admitted by the ODE.
Until recently, it has been an open question how to algorithmically find integrating factors for higher order ODEs or systems of ODEs. It turns out that the situation for a first order ODE is "flukey". In particular, one observes that integrating factors are related to symmetries if and only if the linearized system of a given system of ODEs is self-adjoint. In this case an integrating factor must be a symmetry but the converse is not true. In general it will be shown that integrating factors must be solutions of the adjoint system of the linearized system of a given system of ODEs and, in addition, must satisfy an "adjoint invariance condition".
An algorithm will be presented to find all integrating factors of a given system of ODEs. Moreover an explicit construction formula will be presented to find resulting first integrals.
Techniques will be presented to implement the algorithm. In particular it will be shown how to utilize known first integrals and symmetries to find new integrating factors. Illustrative examples will be exhibited.