Summary: We apply the asymptotic iteration method (AIM) [Ciftci, Hall and Saad, J. Phys. A, Math. Gen. 36, No. 47, 11807–11816 (2003;

Zbl 1070.34113)] 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 Schrödinger problems with Coulomb, harmonic oscillator or Pöschl-Teller potentials, as well as the special eigenproblems studied recently by

*C. M. Bender* and

*Q. Wang* [J. Phys. A, Math. Gen. 34, 9835–9847 (2001;

Zbl 1006.34076)] and generalized in the present paper to arbitrary dimension.