This paper investigates the approximate solution of Riccati differential equations, and DNA repair model which is formed by Riccati equation. The existence result of the solution is obtained by Adomian decomposition method (ADM). Some initial value problems are solved by the method to demonstrate the main results. The obtained results are compared with the Runge-Kutta method to verify the validity of the ADM. It shows that the ADM has better results than the Runge-Kutta method. This results confirm that the ADM is a suitable method for approaching the exact solution of Riccati differential equations and the proposed model. © Published under licence by IOP Publishing Ltd.