In this paper a simplified version is given of the proof of Schiffer's differential equation for univalent functions solving an extremal problem. First an elementary variation of the boundary is made to show without any passage to the limit that it consists of curves satisfying a Lipschitz condition locally. Then the differential equation is obtained by an examination of the limit superior of the difference quotient that avoids the usual appeal to a topological lemma of Haslam Jones.