We study the generalized anharmonic oscillator in three dimensions described by the potentials of the form ∑2m+1k=1bkr 2k. An asymptotic analysis of the Schrödinger equation yields the leading asymptotic behavior of the energy eigenfunctions in terms of the dominant (m+1) coupling constants bk, m+1≤k≤2m+1. Using an ansatz which incorporates this asymptotic behavior, we reduce the eigenvalue equation to an (m+2)-term difference equation. The corresponding Hill determinant may be made to factorize with a finite determinant as a factor if a set of constraints on the couplings is satisfied; an infinite sequence of such sets exists. The exact energy eigenvalues appear as the real roots of the finite factor of the Hill determinant; the corresponding wavefunctions are Gaussian weighted polynomials. We consider the potentials ∑31bkr 2k and ∑51bkr 2k explicitly; potentials of the form ∑2m1bjr j and ∑2m1bjr j+δ/r containing both even and odd terms are also considered. Finally, we show that this method of constructing exact solutions fails for anharmonic potentials of the form ∑2m1bkr 2k, of which the quartic anharmonic oscillator is the simplest example.