We consider a class of systems of nonlinear ordinary differential equations with parameters. In particular, systems of such type arise when modeling the multistage synthesis of a substance. We study properties of solutions to the systems and propose a method for approximate solving the systems in the case of very large coefficients. We establish approximation estimates and show that the convergence rate depends on the parameters characterizing the nonlinearity of the systems. Moreover, the larger the coefficients of the systems, the more exact the approximate solutions. Thereby this method allows us to avoid difficulties arising inevitably when solving systems of nonlinear differential equations with very large coefficients.