高阶变系数线性微分方程的一些新的可积类型

借助双变换—未知函数的变换和自变量的变换,将几类高阶变系数线性微分方程化为相应的常系数线性微分方程,从而顺利求得它们的通解,得到了变系数线性微分方程新的可积类型,所得结果极大地推广了著名的Euler方程及前人的一些的工作,并给出了相应的实例加以佐证.     

The Connection between the Characteristic Roots and the Corresponding Solutions of a Single Linear Differential Equation with Comparable Coefficients

Given a linear differential equation of the form x (ⁿ) + a₁ (t) x (n-1) + …+ a_n (t) x = 0 with variable coefficients defined on the positive semi -axis for t ? 1. We denote its fundamental set of solutions (FSS) by {exp [∫ ri (t) dt] } (i = 1, 2,…,n). In this paper we look for the asymptotic connection (as t → ∞) between the logarithmic derivatives ri (t) of an FSS and of the roots of the characteristic equation yⁿ + a₁ (t) y^n-1 +… + a_n (t) = 0. We mainly consider the case when the coefficients of the equation and the characteristic roots are comparable and have the power order of growth for t → ∞. We discuss the conditions when the functions λ_ii(t) are equivalent to the corresponding roots λ_ii(t) of the characteristic equation as t → ∞.     

常系数线性方程组的一种新解法

对于常系数线性微分方程组:dx/dt=Ax(A是n阶实常数矩阵)通过特征根λ和对应的特征行向量K:K~T(A-λE)=0将微分方程组化为线性方程组:1°当有n个互异的特征根λ_1,λ_2,…,λ_n,对应的线性无关的特征行向量为K_1,K_2,…,K_n,若记K_i=(k_1,k_2,…,k_n)(i=1,2,…,n),则有方程组:(n∑i=1 k_ix_i)′=λ_j(n∑i=1 k_ix_I)(j=1,2,…,n);2°当有不同的特征根λ_1,λ_2,…,λ_m其重数分别为n_1,n_2,…,n_m,n_1+n_2+…+n_m=n,对应的线性无关的特征行向量为K_i=(k_1,K_2,…,k_n)(i=1,2,…,m),则有方程组:(n∑i=1 k_rx_r)′=λ_k(n∑i=1 k_rx_r)((A-λ_jE)x_(n_i)=0;i=1),(n∑i=1 k_rx_r)′=λ_j(n∑i=1k_rx_r)+c_(n_i)e~(λ_jt)((A-λ_kE)x_(i-1)=Ex_i,i=2,…,n_i).     

一类变系数线性微分方程及其解

根据常系数线性微分方程的求解原理,通过一个适当变换,研究了一类变系数线性微分方程及其解的问题,从而可以得到这类方程在特征根都是互异单根时的解法和通解,并对三阶方程的各种情况进行了较为详尽的讨论.     

一类系数为间隙级数的高阶线性微分方程解的增长性

In this paper, we investigate the growth of solutions of a class of higher order linear differential equations with coefficients being gap series. In this case, we remove the condition that the order of coefficients in equations is less than 1/2, and obtain some results which improve the previous results.     

一类二阶变系数线性微分方程及其解的构造方法

利用构造法构造二阶变系数线性齐次微分方程及其解,根据这种方法也能求得某些二阶变系数线性齐次微分方程的非零解,并给出了二阶变系数线性齐次微分方程存在非零解的充要条件.     

一类二阶整函数非齐次线性微分方程的复振荡

研究具有整函数函数系数的二阶非齐次线性微分方程:f″+A(z)e~(az)f′+B(z)e~(P(z))f=F(z)解的复振荡,其中P(z)为非常数多项式且deg(P)=n,A(z),B(x),F(z)均为整函数且max{ρ(A),ρ(B)}n.我们将看到方程的任一非零解具有无穷增长级.     

一阶常系数线性微分方程组的另一种向量解法

讨论一阶常系数线性微分方程组通解问题,给出一种新的向量解法.     

多时滞微分方程数值稳定性

考虑了时滞微分方程的初值问题,分析了用线性多步法求解一类滞后型微分系统数值解的稳定性,在一定的Lagrange插值条件下,给出并证明了求解滞后型微分系统的线性多步法数值稳定的充分必要条件.     

一类高阶线性变系数常微分方程的通解

利用升阶法研究了一类高阶线性变系数常微分方程,给出了齐次方程的通解公式,并讨论了非齐次方程待定的特解.     

一类亚纯系数高阶线性微分方程解的增长性

In this paper, we investigate the growth of solutions of higher order linear differ-ential equations with meromorphic coefficients. Under certain conditions, we obtain precise estimation of growth order and hyper-order of solutions of the equation.     

On the Relation between Linear Difference and Differential Equations with Polynomial Coefficients

This paper represents the third part of a contribution to the “dictionary” of homogeneous linear differential equations with polynomial coefficients on one hand and corresponding difference equations on the other. In the first part (cf. [4]) we studied the case that the differential equation (D) has at most regular singularities at O and at ∞, and arbitrary singularities in the rest of the complex plane. We constructed fundamental systems of solutions of a corresponding difference equation (A), using integral transforms of microsolutions of (D) at its singular points in ?. In the second part ([5]) we considered differential equations having at most a regular singularity at O and an irregular one at O. We used integral transforms of asymptotically flat solutions of (D) to define it fundamental system of solutions of (Δ), holomorphic in a right half plane, and integral transforms of sections of the sheaf of solutions of (D) modulo solutions with moderate growth as t → 0 in some sector, to define a fundamental system of (Δ), holomorphic in a left half plane. In this final part we combine the techniques and results of the preceding papers to deal with the general case.     

具变系数时滞微分方程解的振动性

研究了时滞微分方程解的振动性，其中P（t）、τ（t）非负连续．我们证明了：如果对充分大的，且，则方程（*）每一解振动．该结论改进和推广了许多已知结果．     

亚纯系数的非齐次线性微分方程的振荡解

本文研究关于亚纯系数的非齐次线性微分方程的复振荡,得到方程f(k)+ak-1fk-1+…+a0f=F(a0,a1,…,ak-1和F是亚纯函数)具有一个振荡解空间,其空间中所有解的零点收敛指数为∞,至多除去一个例外值.     

一类高阶线性微分方程解的复振荡

In this paper,we shall use Nevanlinna theory of meromorphic functions to investigate the complex oscillation theory of solutions of some higher order linear differential equation.Suppose that A is a transcendental entire function with ρ(A)<1/2.Suppose that k≥2 and f(k)+A(z)f=0 has a solution f with λ(f)<ρ(A),and suppose that A1=A+h,where h≡0 is an entire function with ρ(h)<ρ(A).Then g(k)+A1(z)g=0 does not have a solution g with λ(g)<∞.     

Homoclinic Solutions and Chaos in Ordinary Differential Equations with Singular Perturbations

Ordinary differential equations are considered which contain a singular perturbation. It is assumed that when the perturbation parameter is zero, the equation has a hyperbolic equilibrium and homoclinic solution. No restriction is placed on the dimension of the phase space or on the dimension of intersection of the stable and unstable manifolds. A bifurcation function is established which determines nonzero values of the perturbation parameter for which the homoclinic solution persists. It is further shown that when the vector field is periodic and a transversality condition is satisfied, the homoclinic solution to the perturbed equation produces a transverse homoclinic orbit in the period map. The techniques used are those of exponential dichotomies, Lyapunov-Schmidt reduction and scales of Banach spaces. A much simplified version of this latter theory is developed suitable for the present case. This work generalizes some recent results of Battelli and Palmer.

     

二阶非线性变系数奇异微分方程的周期解

利用拓扑度理论和不动点指数理论,研究了二阶非线性变系数奇异微分方程u″(t)+a(t)u(t)=r(t)f(u(t))的周期解的存在性.特别地,本文没有假设a(t)和f(u)的非负性.     

二阶周期线性微分方程解的几个性质

In this paper,the zeros of solutions of periodic second order linear differential equation y ＋ Ay = 0,where A（z） = B（e z ）,B（ζ） = g（ζ） ＋ p j=1 b ？j ζ ？j ,g（ζ） is a transcendental entire function of lower order no more than 1/2,and p is an odd positive integer,are studied.It is shown that every non-trivial solution of above equation satisfies the exponent of convergence of zeros equals to infinity.     

变系数二阶线性微分方程一个新的可解类型再讨论

利用二阶线性微分方程的不变量,给出二阶线性微分方程常系数与变系数、齐次与非齐次的统一解法,而且扩大了自由项函数的形式.