Generalization of the simplest equation method for nonlinear non-autonomous differential equations

• Generalization of the simplest equation method for non-autonomous differential equations is suggested.
• Using the method special solutions of three Painlevé equations are found.
• The obtained special solutions are expressed in terms of the special functions defined by linear second order ODEs.

It is known that the simplest equation method is applied for finding exact solutions of autonomous nonlinear differential equations. In this paper we extend this method for finding exact solutions of non-autonomous nonlinear differential equations (DEs). We applied the generalized approach to look for exact special solutions of three Painlevé equations. As ODE of lower order than Painlevé equations the Riccati equation is taken. The obtained exact special solutions are expressed in terms of the special functions defined by linear ODEs of the second order.