**The connection of ordinary differential equations and partial differential
equations through a common realisation of the algebra of their complete symmetry groups**

**Abstract:**

The complete symmetry group of a second-order 1+1 steady-state partial differential equation has been
demonstrated to be represented by the eight-dimensional Lie algebra of point symmetries for which the eight-dimensional
algebra is sl(3,R). We construct a steady-state partial differential equation using the idea of complete symmetry groups
from a second-order free particle ordinary differential equation y" = 0. The equation immediately realized turn out to be
equivalent (in symmetry structure) to a system of
second-order ordinary differential equation and a first-order ordinary differential equation. Also we construct further
second-order equations from higher order equations of the form y^{n} = 0.