In this paper, we determine the forbidden set and give an explicit formula for the solutions of the difference equation xn+1 = axnxn−1 ⁄−bxn + cxn−2 , n ∈ N0 where a, b, c are positive real numbers and the initial conditions x−2, x−1, x0 are real numbers. We show that every admissible solution of that equation converges to zero if either a < c or a > c with (a − c)/b < 1. When a > c with (a − c)/b > 1, we prove that every admissible solution is unbounded. Finally, when a = c, we prove that every admissible solution converges to zero.
In this paper we consider positive unbounded solutions of second order quasilinear ordinary differential equations. Our objective is to determine the asymptotic forms of unbounded solutions. An application to exterior Dirichlet problems is also given.