We apply the asymptotic iteration method (AIM) (Ciftci, Hall and Saad 2003 J. Phys. A: Math. Gen. 36 11807) to solve new classes of second-order homogeneous linear differential equation. In particular, solutions are found for a general class of eigenvalue problems which includes Schrodinger problems with Coulomb, harmonic oscillator or Poschl-Teller potentials, as well as the special eigenproblems studied recently by Bender et al (2001 J. Phys. A: Math. Gen. 34 9835) and generalized in the present paper to arbitrary dimension.