A functional analytic method was developed by E.K.Ifantis in 1987 to prove that certain non-linear ordinary differential equations (ODEs) have a unique power series solution. We first introduced the resistant virus during single-drug therapy model of HIV infection. Then, we extended the functional analytic method to certain systems of two non-linear ordinary differential equations and applied the result to the HIV infection model governed by a system of two non-linear ordinary differential equations and obtained the power series solutions.