Types Of Differential Equations,

The oscillation of differential equation is one of important branche of differential equations. With the increasing development of science and tech-nology, there are many problems relating to differential equation derived from lots of practical applications, such as whether differential equation has a oscil-lating solution or not, and whether all of its solutions are oscillatory or not. Because all the structures of its emergence have deep physical background and realistic mathematical models. Many scholars take on the research of this field, they have achieved many good results. The study of dynamic equations on time scales is an area of mathematics that has recently received a lot of attention. In very resent years, great changes of this field have taken place. The differential equation has been paid more attentions and investigated in various classes by using different methods(see [1]-[32]).The present paper employs average function, generalized Riccati trans-formation to investigate the oscillation criteria for some class of differential equations, the results of which generalized and improved some known oscilla-tion criteria. The thesis is divided into two sections according to contents.In Chapter 1, we study the oscillation of the forced second-order nonlineardifferential equations of the form: we mainly employed average function shallfurther the investigation and im-prove the main results of Sun [6], we obtained several new oscillation criteria at the end of this section.In Chapter 2, we study the oscillation of the second-order nonlinear damped dynamic equation: (r(t)xΔ(t))Δ+p(t)xΔ(t)+q(t)f(x(τ-(t)))=0, (2.1.1)we mainly employed a generalized Riccati transformation technique shallfur-ther the investigation and improve the main results of Erbe and Peterson[31,32],we obtained several new oscillation criteria at the end of this paper.