  Approximate analytical solutions of the modified Langevin equation are
obtained. These solutions are relatively simple and enough accurate. They are
illustrated by considering a mean-field model of a system with interacting
superparamagnetic particles. Within the framework of this model system we
derived analytical approximate formulas for the temperature dependencies of the
saturation and remnant magnetization, coercive force, initial magnetic
susceptibility as well as for the law of approach to saturation. We obtained
also some exact analytical relationships for the coercive force. We found
remarkable similarity between the approximate cubic equation, which is resulted
from the modified Langevin equation, and the exact equation resulting from the
divergence condition of a solution derivative. The analytical formulas obtained
in this work can be used in various models (not only magnetic ones), where the
modified Langevin equation is applied.
