The classical inexact Newton method was presented as a tool for solving nonlinear differential algebraic equations (DAEs) in a fully implicit form F(y, y, t) = 0. This is especially in chemical engineering where describing the DAE system in a different form can be difficult or even impossible to realize. The appropriate rewriting of the DAEs using the backward Euler method makes it possible to present the differentialalgebraic system as a large-scale system of nonlinear equations. To solve the obtained system of nonlinear equations, the inexact Newton backtracking method was proposed. Because the convergence of the inexact Newton algorithm is strongly affected by the choice of the forcing terms, new variants of the inexact Newton method were presented and tested on the catalyst mixing problem.