It is proved that if the differential equations "y[(n)] = f(x,y,y[prime],...,y[(n-1)])" and "y[(m)] = g(x,y,y[prime],...,y[(m-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.