A fresh look at linear ordinary differential equations with constant coefficients. Revisiting the impulsive response method using factorization

We present an approach to the impulsive response method for solving linear constant-coefficient ordinary differential equations of any order based on the factorization of the differential operator. The approach is elementary, we only assume a basic knowledge of calculus and linear algebra. In particular, we avoid the use of distribution theory, as well as of the other more advanced approaches: Laplace transform, linear systems, the general theory of linear equations with variable coefficients and variation of parameters. The approach presented here can be used in a first course on differential equations for science and engineering majors.     

Frobenius integrable decompositions for ninth-order partial differential equations of specific polynomial type

Frobenius integrable decompositions are presented for a kind of ninth-order partial differential equations of specific polynomial type. Two classes of such partial differential equations possessing Frobenius integrable decompositions are connected with two rational Bäcklund transformations of dependent variables. The presented partial differential equations are of constant coefficients, and the corresponding Frobenius integrable ordinary differential equations possess higher-order nonlinearity. The proposed method can be also easily extended to the study of partial differential equations with variable coefficients.     

变系数三阶线性微分方程的一种解法

通过变量替换,将三阶变系数微分方程转化为微分方程组形式,再根据刘维尔公式及必要的积分运算最终得出方程的通解.这提供了一种解变系数高阶微分方程的方法,用一个例子说明了此方法的有效性.     

关于常系数齐次线性微分方程组的解法研究

利用凯莱-哈密顿定理给出矩阵指数函数eAt的简洁计算方法;同时利用约当标准形推导出求常系数齐次线性微分方程组通解的循环公式.     

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.     

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.     

Multipoint boundary value problems of Neumann type for functional differential equations

We obtain the expression of the explicit solution to a class of multipoint boundary value problems of Neumann type for linear ordinary differential equations and apply these results to study sufficient conditions for the existence of solution to linear functional differential equations with multipoint boundary conditions, considering the particular cases of equations with delay and integro-differential equations.     

n阶常系数非齐次线性微分方程的降阶解法

给出一阶线性非齐次微分方程的积分因子解法,避免了常数变易法带来的不便和不自然;给出,n阶常系数非齐次线性微分方程的降阶解法,可以看出,高阶常系数线性非齐次微分方程最终都可以归结为求解一阶线性微分方程,从而避免了待定系数法求非齐次方程特解的繁琐,并最终说明了一般微积分教材中只给出两种类型常系数非齐次线性微分方程的待定系数解法的原因．     

一类二阶变系数线性微分方程的解法

介绍如何通过变换把二阶变系数线性微分方程转化为一阶非线性微分方程,进而利用待定系数法对其求解,并对二阶变系数线性微分方程与一阶常系数非线性微分方程的内在的关系进行讨论.     

Tropical differential equations

Tropical differential equations are introduced and an algorithm is designed which tests solvability of a system of tropical linear differential equations within the complexity polynomial in the size of the system and in the absolute values of its coefficients. Moreover, we show that there exists a minimal solution, and the algorithm constructs it (in case of solvability). This extends a similar complexity bound established for tropical linear systems. In case of tropical linear differential systems in one variable a polynomial complexity algorithm for testing its solvability is designed.We prove also that the problem of solvability of a system of tropical non-linear differential equations in one variable is NP-hard, and this problem for arbitrary number of variables belongs to NP. Similar to tropical algebraic equations, a tropical differential equation expresses the (necessary) condition on the dominant term in the issue of solvability of a differential equation in power series.     

On admissible perturbations for exponential dichotomy

In this work we obtain an optimal upper bound for exponential dichotomy roughness in infinite-dimensional Banach spaces. Unlike some previous works, we do not assume bounded growth. We consider linear, non-autonomous ordinary differential equations with bounded and unbounded coefficients.     

零化多项式的一个应用

利用矩阵的零化多项式 ,给出计算标准基解矩阵 e At的一个公式 .利用向量关于矩阵的零化多项式 ,给出常系数齐次线性微分方程组初值问题的一个求解公式 .相应地 ,可以推出常系数齐次线性差分方程组在给定的初始条件下的一个求解公式 .     

常系数线性微分方程组的基解矩阵的一种新求法

在求解常系数线性微分方程组时,关键是基解矩阵的计算.给出了利用哈密顿—凯莱定理计算基解矩阵的一种方法,并通过实例说明了这种方法的特点和在简化计算方面的有效性.     

On differential equations for orthogonal polynomials on the unit circle

In this paper we characterize sequences of orthogonal polynomials on the unit circle whose corresponding Carathéodory function satisfies a Riccati differential equation with polynomial coefficients, in terms of second order matrix differential equations. In the semi-classical case, a characterization in terms of second order linear differential equations with polynomial coefficients is deduced.     

A pair of difference differential equations of Euler-Cauchy type

We study two classes of linear difference differential equations analogous to Euler-Cauchy ordinary differential equations, but in which multiple arguments are shifted forward or backward by fixed amounts. Special cases of these equations have arisen in diverse branches of number theory and combinatorics. They are also of use in linear control theory. Here, we study these equations in a general setting. Building on previous work going back to de Bruijn, we show how adjoint equations arise naturally in the problem of uniqueness of solutions. Exploiting the adjoint relationship in a new way leads to a significant strengthening of previous uniqueness results. Specifically, we prove here that the general Euler-Cauchy difference differential equation with advanced arguments has a unique solution (up to a multiplicative constant) in the class of functions bounded by an exponential function on the positive real line. For the closely related class of equations with retarded arguments, we focus on a corresponding class of solutions, locating and classifying the points of discontinuity. We also provide an explicit asymptotic expansion at infinity.

     

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.     

Systems of linear ordinary differential equations with bounded coefficients may have very oscillating solutions

An elementary example shows that the number of zeroes of a component of a solution of a system of linear ordinary differential equations cannot be estimated through the norm of coefficients of the system.

     

关于多级一阶线性常系数微分方程组的一点注记

有O初始条件的多级一阶线性常系数微分方程组,可以用拉氏变换求解,并且解还可以用Green矩阵来表达.早已为人所知.但关于Green矩阵的一个在实际中有用的表达形式,因为推导上有毛病,未被普遍接受.本文对此毛病作了点注记.予以克服.     

A theorem on energy integrals for linear second-order ordinary differential equations with variable coefficients

We aim at demonstrating a novel theorem on the derivation of energy integrals for linear second-order ordinary differential equations with variable coefficients. Namely, in this context, we will present a possible and consistent method to overcome the traditional difficulty of deriving energy integrals for Lagrangian functions that explicitly exhibit the independent variable. Our theorem is such that it appropriately governs the arbitrariness of the variable coefficients in order to have energy integrals ensured. In view of the theoretical framework in which the theorem will be embedded, we will also demonstrate that it can be applied as a mathematical method to solve linear second-order ordinary differential equations with variable coefficients. These results are expected to have a generalized fundamental character.     

Conditional linearizability criteria for a system of third-order ordinary differential equations

We provide linearizability criteria for a class of systems of two third-order ordinary differential equations that is cubically nonlinear in the first derivative, by differentiating a system of second-order quadratically nonlinear ordinary differential equations and using the original system to replace the second derivatives. The procedure developed splits into two cases: those for which the coefficients are constant and those for which they are variables. Both cases are discussed and examples given.