On the well-definedness of the order of an ordinary differential equation

It is proved that if the differential equations y ^{( n )}=f(x, y, y′, …, y ^{( n ?1 )}) and y ^{( m )}=g(x, y, y′, …, y ^{( n ?1 )}) have the same particular solutions in a suitable region where f and g are continuous real-valued functions with continuous partial derivatives (alternatively, continuous functions satisfying the classical Lipschitz condition), then n?=?m and the functions f and g are equal. This note could find classroom use in a course on differential equations as enrichment material for the unit on the existence and uniqueness theorems for solutions of initial value problems.