用观察法求二元一次不定方程的特解初探

任何事物的矛盾都是普遍性和特殊性的辩证统一 ,普遍性存在于特殊性之中 ,特殊性里包含着普遍性 ,因此认识“特殊”是认识“一般”的入门向导 .求二元一次不定方程的一个特解 ,是求该方程通解的关键 ,也是求二元一次不定方程的正整数解或二元一次不定方程应用题的重要基础 .用观察法求特解 ,简便易行 ,快速敏捷 .下面 ,举例说明用观察法求特解的一点技巧 .1　对于二元一次不定方程 ax by=c( a,b,c为整数 ,( a,b) =1) ) ,当 a|c或 b|c的情况 .例 1　求 3 x 5 y=9的一个特解 .解　∵　 ( 3 ,5 ) |9,此方程有整数解 .求特解∵　 a=3 ,3 |9　可…     

大系数二元一次不定方程解法探讨

大系数二元一次不定方程解法探讨朱志嘉，周定远（四川乐山市教科所６１４０００）我们知道，求一元一次不定方程ａｘ±ｂｙ＝ｃ（其中ａ，ｂ，ｃ为非零整数）①的整数解（通解）的关键步骤是先求出其对应方程ａｘ±ｂｙ＝１②的一组特解，求②的特解一般常用辗转相除法或...     

一类常系数线性微分方程特解的求法

针对一类常系数线性微分方程,运用观察法或特征函数导数法,得到了方程特解的计算公式.当方程非齐次项Aeαx中α=0时,可采用观察法;当α≠0时,可采用特征函数导数法得到方程的一个公式化特解.该方法简便易行,特解形式直观,避免了以前方法计算量过大的不足.     

用常数变易法求微分方程特解的简单处理

简化了用"常数变易"法求常系数非齐次线性微分方程特解的过程,给出了求二阶常系数非齐次线性微分方程特解的一般公式.并将该方法推广到对n阶方程的降阶,从而求其特解.此方法简单实用,且运算量小.     

常系数非齐次线性微分方程的变量替换法

常系数非齐次线性微分方程求解是微积分教学的一个难点,主要困难是特解形式复杂和计算量大.本文用变量替换的思想研究这类微分方程的特解,从最简单的情形出发,通过类比得出复杂情形下方程的解法.使用变量替换把复杂问题简单化,求解不需要特解形式,有效降低了计算量.     

常系数非齐次线性差分方程的特解

本文讨论了二阶常系数非齐次线性差分方程特解的求法,给出了用升阶法和常数变易法求特解的两种方法.     

关于常微分方程特解的一个注记

本文结合具体例子,对利用微分方程的通解求特解时需要注意的问题进行了分析讨论,补充完善了特解的求法.     

求一类非齐次微分方程特解的待定算子法

给出了求一类非齐次微分方程L(D)y=f(x)特解的待定微分算子解法.即通过求与方程相关的待定微分算子R(D),从而得出非齐次微分方程的特解y=R(D)f(x).     

微分方程组(dX)/(dt)=AX+e~(αt)P_m(t)具有重特征根α时的特解结构定理及应用

对于常系数非齐线性微分方程组(dX)/(dt)=AX+eαt sum (Bktk) from k=0 to m,当n级方阵A具有s≥1重特征根α时,本文给出了其特解X(t)的结构定理和计算方法,将求特解X(t)的积分运算转化为简单的代数运算,解决了利用计算机求特解X(t)的计算问题.     

对“关于y″+py′+qy=Ae~(ax)的特解”一文的意见

文[1]利用待定系数法导出二阶常系数微分方程y″ py′ qy=Aeax(1)特解的一般公式,有实用价值,但有几个问题提出来与作者商榷.1.在“主要结果”中,当令f(r)=r2 pr q后所列原方程的三个特解式中,后两个解式都是错的,第2和第3个解式中分别少写了x和x2.2.在“结果的证明”中只证明α是特征方程r2 pr q=0的单根时的情形,即令(1)的特解为y*=Bxeax,代入方程(1)合并同类项后得(2Bα pB)eax B(α2 pα q)xeax=Aeax文中说“比较系数得B=A2α p”.实际上应说明由于α是特征方程的单根,即有f(α)=α2 pα q=0,f′(α)=2α p≠0,从而约去非零因子eax…     

On the relation between the continuous and discrete Painlevé equations

A method for deriving difference equations (the discrete Painlevé equations in particular) from the Bäcklund transformations of the continuous Painlevé equations is discussed. This technique can be used to derive several of the known discrete painlevé equations (in particular, the first and second discrete Painlevé equations and some of their alternative versions). The Painlevé equations possess hierarchies of rational solutions and one-parameter families of solutions expressible in terms of the classical special functions for special values of the parameters. Hence, the aforementioned relations can be used to generate hierarchies of exact solutions for the associated discrete Painlevé equations. Exact solutions of the Painlevé equations simultaneously satisfy both a differential equation and a difference equation, analogously to the special functions.     

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.     

On reduction of some classes of partial differential equations to equations with fewer variables and exact solutions

We establish a connection between the fundamental solutions to some classes of linear nonstationary partial differential equations and the fundamental solutions to other nonstationary equations with fewer variables. In particular, reduction enables us to obtain exact formulas for the fundamental solutions of some spatial nonstationary equations of mathematical physics (for example, the Kadomtsev-Petviashvili equation, the Kelvin-Voigt equation, etc.) from the available fundamental solutions to one-dimensional stationary equations.     

Properties of set-valued stochastic differential equations

The paper is devoted to properties of set-valued stochastic differential equations. The main result of the paper deals with existence and uniqueness of solutions. Furthermore, a connection between solutions of stochastic differential inclusions and solutions of set-valued stochastic differential equations are given. The result of the paper extends a lot of particular results dealing with such type equations.     

On the existence of multiple steady-state solutions in the theory of electrodiffusion. Part I: the nonelectroneutral case. Part II: a constructive method for the electroneutral case

We give a constructive method for giving examples of doping functions and geometry of the device for which the nonelectroneutral voltage driven equations have multiple solutions. We show in particular, by performing a singular perturbation analysis of the current driven equations that if the electroneutral voltage driven equations have multiple solutions then the nonelectroneutral voltage driven equations have multiple solutions for sufficiently small normed Debye length. We then give a constructive method for giving examples of data for which the electroneutral voltage driven equations have multiple solutions.

     

From gas dynamics to pressureless gas dynamics

This paper is devoted to the convergence of solutions of the compressible Euler equations towards solutions of the pressureless gas dynamics system, when the pressure tends to 0. The goal is to prove accurate uniform bounds for particular solutions of the Euler equations.

     

Construction of solutions to quasilinear partial differential equations

We propose a method for constructing solutions to a class of quasilinear parabolic partial differential equations (PDEs) basing on a new property of these equations. The method applies to quasilinear hyperbolic and elliptic equations as well. The results of this article broaden the class of exact solutions to the quasilinear equations, in particular, to the nonlinear heat equations, the equations of chemical kinetics and mathematical biology.     

Regularity for solutions of nonlinear second order evolution equations

This paper deals with the existence of solutions for the class of nonlinear second order evolution equations. The regularity and a variation of solutions of the given equations are also given. As particular cases of our general formulation, some results for Volterra integrodifferential equations of the hyperbolic type are given.     

Radial Equivalence of Nonhomogeneous Nonlinear Diffusion Equations

We establish one-to-one transformations and self-maps between nonlinear diffusion equations in nonhomogeneous media, where the density function is given by a power. We use these transformations to deduce new interesting self-similar, radially symmetric solutions of the equations. In particular, Barenblatt, dipole and focusing Aronson-Graveleau type solutions are deduced, and some equations with singular potentials are studied. The new solutions are example of interesting or unexpected mathematical features of these equations, providing also natural candidates for the asymptotic behavior.