  We propose a physical analogy between finding the solution of an ordinary
differential equation (ODE) and a $N$ particle problem in statistical
mechanics. It uses the fact that the solution of an ODE is equivalent to obtain
the minimum of a functional. Then, we link these two notions, proposing this
functional to be the interaction potential energy or thermodynamic potential of
an equivalent particle problem. Therefore, solving this statistical mechanics
problem amounts to solve the ODE. If only one solution exists, our method
provides the unique solution of the ODE. In case we treat an eigenvalue
equation, where infinite solutions exist, we obtain the absolute minimum of the
corresponding functional or fundamental mode. As a result, it is possible to
establish a general relationship between statistical mechanics and ODEs which
allows not only to solve them from a physical perspective but also to obtain
all relevant thermodynamical equilibrium variables of that particle system
related to the differential equation.
