In this paper, firstly a rational one-parameter family of Falkner-type explicit methods is presented for directly solving numerically special second order initial value problems in ordinary differential equations. The proposed family of methods has second algebraic order of convergence. Imposing that the principal term of the local truncation error of the proposed family vanishes, we get an expression for the free parameter at the grid point (xn, yn). By substituting this value of the free parameter in the family, a new rational third order method is obtained. Further, by combining the third order method with any member of the second order family, their variable step-size formulation as an embedded pair is considered. Some numerical experiments are given to illustrate the performance and efficiency of the proposed methods.