Necessary and sufficient conditions have been found to force all solutions of the equation \[ (r(t)y^{\prime }(t))^{(n-1)} + a(t)h(y(g(t))) = f(t), \] to behave in peculiar ways. These results are then extended to the elliptic equation \[ |x|^{p-1} \Delta y(|x|) + a(|x|)h(y(g(|x|))) = f(|x|) \] where $ \Delta $ is the Laplace operator and $p \ge 3$ is an integer.