The study of random vibration problems, using the concepts of stochastic process theory, is a relatively new engineering discipline. Interest in the field has grown rapidly in the last few decades, due to the need to design structures and machinery which can operate reliably when subjected to random environmental load.In many practical applications, the system of concern has non-linearity, which must be taken into account if one is to predict its performance in a realistic way. For such non-linear problems the classical linear theory, which can be found in a number of textbooks, is not directly applicable and new techniques are required. Much of the recent research in the field of random vibration has been directed towards developing suitable methods for analyzing non-linear systems.The present thesis deals with the solution of nonlinear mathematical and dynamic stochastic problems. Two approaches are considered, one of them is a sample of the approximate methods which called statitical linearization. And the other approach is an exact method which called Fokker-Planck. The two methods are applied to many problems to emphasize the difference between the two categories.4.2 ConclusionsIt is concluded that the Fokker-Planck method get an exact solution in simple problems when the equations parameters and excitation takes any complex form. While in the case of large problems, the exact solution can’t be obtained easily analytically.Comparing the solution of nonlinear stochastic problems that obtained using the statistical linearization method as an approximate method with the exact solution that obtained using the Fokker-Planck method, it is found that the approximate solution converges to the exact solutionwhen the nonlinearity is low, while the approximate solution diverges from the exact solution for high nonlinearity.4.3 Future worksIt is recommended for the future works to include the following points:1- Study more applications using the Fokker-Planck method and the approximate methods to clarify the error in the approximate methods.2- Studying another exact method for the stochastic problems as Markov vector approach.3- Studying the non-stationary cases for the random variables.4- Studying the shot noise of the excitation.