Several Types Of The Differential Equation Of Vibration Study

The oscillation of ordinary differential equation is one of important branches of differential equations. With the development of science and technology, there are many problems relating to differential equation derived from lots of practical applications, such as whether differential equation has a oscillating solution or not, and whether all of its solutions are oscillatories or not for all the structures of its emergence have deep physical background and realistic mathematical models. Along with the development of modern physics and applied mathematics, the importance of the theory of nonlinear differential equations become increasingly apparent, not only in the field of engineering, space technology and automatic con-trol, but also in the field of computer science, population dynamics and financial. Therefore, the theory of differential equations attracts the interests of domestic and foreign scholars. And its basic theory and application significance are more and more attended by the scholars. Especially in recent years, the study of oscill-tion of the ordinary differential equations is developing quite rapidly. From linear to half-linear and nonlinear, first-order to higher order, there are a lot of results. Comparatively speaking, the researches of oscillation of third-order and high or-der differential equations are less than the second order differential equations. In this paper, we study the oscillation of third-order and high order differential equations.This paper is divided into two chapters according to the contents.In Chapter1, we study the oscillation of third order differential equation (c(t)([x(t)+p(t)x(Ï„(t))](n-1)α))’+q(t)xα(σ(t))=0, under the condition∫t0∞a-1/γ(s)ds=∞.And the equation satisfies:(H1) a,p, q are positive,a,p, q∈C(t0,∞)),and0≤p(t)≤p≤∞;(H2) σ(t)≤t,and σ(t) is nondecreasing.limtâ†'∞τ(t)=∞,limtâ†'∞σ(t)=∞,τ’(t)≥τ0>0,and το σ=σ ο Ï„.τ∈C1([t0,∞))(H3)∫t0∞a-1/γ(s)ds=∞. In this paper,we obtain the oscillation of the third-order differential equation through the oscillation of the special first order differential equation. In Chapter2,we study the oscillation of the odd order differential equation (c(t)[x(t)+p(t)x((?)(t))](n-1)’+q(t)xα(σ(t))=0,(H1H2H3) and (c(t)([x(t)+p(t)x((?)(t))](n-1)α))’+q(t)xα(σ(t))=0,(H1H2H4) where (K1)q(t)>0,0≤p(t)≤p0<∞(K2)(?)(t)=a+bt,b>0,σ(t)∈C([t0,∞)),(?)(t)≤t,(?)οσ=σο(?),limt∞σ(t)=0ï¼›(k3)c’(t)>0,c(t)>0,limtâ†'∞∫t1t c(s)/1ds=∞;(K4)c’(t)>0,c(t)>0,limtâ†'∞∫t1t c-1/γ(s)ds=∞. In this paper,we obtain the oscillation of odd order differential equation through H(t,s)and h(t,s).