In our preceding articles, by applying the results of Bieberbach, Fatou and Picard, we have constructed an example such that the domain of holomorphy of the solution admits an exterior point. Next we have studied the domain of holomorphy of the solution for differential operators with polynomial coefficients that concern the differential equations of Darboux-Halphen, of Chazy and of the modular function. In this article, we make some studies on the analytic continuation of the solution of the Cauchy problem for certain differential operators of principal part with polynomial coefficients.