Non-meromorphic solutions to differential equations

The growth order of solutions to differential equations has many known results for entire and meromorphic solutions by using Wiman-Valiron theory or Nevanlinna theory. In this dissertation we consider the case when a solution has essential singularities in the complex plane. In this case, the growth of the solution may not only occur as |z| approaches infinity but also may occur as |z| approaches singularities in the plane. So we investigate the growth order at essential singularities of non-meromorphic solutions.;In this pursuit of determining the growth order we prove a Wiman-Valiron type theorem for non-meromorphic functions with only one essential singularity in the complex plane. We then use this to determine the growth order of solutions to homogeneous differential equations of nth order.