The goal of the present paper is to obtain new, more general sufficient conditions for univalency and quasiconformal extendibility of analytic functions in the domainby means of the general nonlinear differential operators of the form, whereFis an analytic function of its arguments (n ≥ 2). It is connected with the solution of the question about the existence of univalent solutions of nonlinear differential equations with analytic coefficients. The most natural conditions are obtained in the case of the operatorsFis nonlinear. In the paper the operators of such kind in simply connected domains with quasiconformal boundary are considered, in the case of the disk the operator with the linear part a1f(n)/f′ + a2f(n)(n= 2, 3) is also regarded.