The rigid rod model has been proven successful in predicting the rheological response to rodlike polymers in the nematic phase. The model can be reduced to constitutive equations written in terms of a set of first-order ordinary differential equations, provided that a mathematical approximation required to close the problem is introduced. These constitutive equations are attractive as they are suitable for fast numerical processing. The required closure approximation, however, may have profound effects on the capabilities of the constitutive equation to retain the features of the original model. A method based on the bifurcation analysis is proposed as an effective test of constitutive equations. As an example, the bifurcation analysis of a particular constitutive equation is presented, and the results are compared with the bifurcation pattern of the exact model.