Solution sets determine differential equations

It is proved that if the differential equations have the same particular solutions in suitable regions where f and g are continuous real-valued functions of two real variables, then the functions f and g are equal. Special attention is paid to the case in which f and g satisfy the classical Lipschitz condition. Analogous results for systems of ordinary differential equations and nth-order differential equations are also established. This note could find classroom use in a course on differential equations as enrichment material relative to the standard existence and uniqueness theorems for solutions of ordinary differential equations.