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Method Of Undetermined Coefficients
In mathematics, the method of undetermined coefficients is an approach to finding a particular solution to certain nonhomogeneous ordinary differential equations and recurrence relations. It is closely related to the annihilator method, but instead of using a particular kind of differential operator (the annihilator) in order to find the best possible form of the particular solution, an ansatz or 'guess' is made as to the appropriate form, which is then tested by differentiating the resulting equation. For complex equations, the annihilator method or variation of parameters is less time-consuming to perform. Undetermined coefficients is not as general a method as variation of parameters, since it only works for differential equations that follow certain forms.Ralph P. Grimaldi (2000). "Nonhomogeneous Recurrence Relations". Section 3.3.3 of ''Handbook of Discrete and Combinatorial Mathematics''. Kenneth H. Rosen, ed. CRC Press. . Description of the method Consider a linear non-ho ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ordinary Differential Equation
In mathematics, an ordinary differential equation (ODE) is a differential equation whose unknown(s) consists of one (or more) function(s) of one variable and involves the derivatives of those functions. The term ''ordinary'' is used in contrast with the term partial differential equation which may be with respect to ''more than'' one independent variable. Differential equations A linear differential equation is a differential equation that is defined by a linear polynomial in the unknown function and its derivatives, that is an equation of the form :a_0(x)y +a_1(x)y' + a_2(x)y'' +\cdots +a_n(x)y^+b(x)=0, where , ..., and are arbitrary differentiable functions that do not need to be linear, and are the successive derivatives of the unknown function of the variable . Among ordinary differential equations, linear differential equations play a prominent role for several reasons. Most elementary and special functions that are encountered in physics and applied mathematics are ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Recurrence Relation
In mathematics, a recurrence relation is an equation according to which the nth term of a sequence of numbers is equal to some combination of the previous terms. Often, only k previous terms of the sequence appear in the equation, for a parameter k that is independent of n; this number k is called the ''order'' of the relation. If the values of the first k numbers in the sequence have been given, the rest of the sequence can be calculated by repeatedly applying the equation. In ''linear recurrences'', the th term is equated to a linear function of the k previous terms. A famous example is the recurrence for the Fibonacci numbers, F_n=F_+F_ where the order k is two and the linear function merely adds the two previous terms. This example is a linear recurrence with constant coefficients, because the coefficients of the linear function (1 and 1) are constants that do not depend on n. For these recurrences, one can express the general term of the sequence as a closed-form expression o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Annihilator Method
In mathematics, the annihilator method is a procedure used to find a particular solution to certain types of non-homogeneous ordinary differential equations (ODE's). It is similar to the method of undetermined coefficients, but instead of guessing the particular solution in the method of undetermined coefficients, the particular solution is determined systematically in this technique. The phrase ''undetermined coefficients'' can also be used to refer to the step in the annihilator method in which the coefficients are calculated. The annihilator method is used as follows. Given the ODE P(D)y=f(x), find another differential operator A(D) such that A(D)f(x) = 0. This operator is called the annihilator, hence the name of the method. Applying A(D) to both sides of the ODE gives a homogeneous ODE \big(A(D)P(D)\big)y = 0 for which we find a solution basis \ as before. Then the original inhomogeneous ODE is used to construct a system of equations restricting the coefficients of the line ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Differential Operator
In mathematics, a differential operator is an operator defined as a function of the differentiation operator. It is helpful, as a matter of notation first, to consider differentiation as an abstract operation that accepts a function and returns another function (in the style of a higher-order function in computer science). This article considers mainly linear differential operators, which are the most common type. However, non-linear differential operators also exist, such as the Schwarzian derivative. Definition An order-m linear differential operator is a map A from a function space \mathcal_1 to another function space \mathcal_2 that can be written as: A = \sum_a_\alpha(x) D^\alpha\ , where \alpha = (\alpha_1,\alpha_2,\cdots,\alpha_n) is a multi-index of non-negative integers, , \alpha, = \alpha_1 + \alpha_2 + \cdots + \alpha_n, and for each \alpha, a_\alpha(x) is a function on some open domain in ''n''-dimensional space. The operator D^\alpha is interpreted as D^\alp ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ansatz
In physics and mathematics, an ansatz (; , meaning: "initial placement of a tool at a work piece", plural Ansätze ; ) is an educated guess or an additional assumption made to help solve a problem, and which may later be verified to be part of the solution by its results. Use An ansatz is the establishment of the starting equation(s), the theorem(s), or the value(s) describing a mathematical or physical problem or solution. It typically provides an initial estimate or framework to the solution of a mathematical problem, and can also take into consideration the boundary conditions (in fact, an ansatz is sometimes thought of as a "trial answer" and an important technique in solving differential equations). After an ansatz, which constitutes nothing more than an assumption, has been established, the equations are solved more precisely for the general function of interest, which then constitutes a confirmation of the assumption. In essence, an ansatz makes assumptions about the form o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Variation Of Parameters
In mathematics, variation of parameters, also known as variation of constants, is a general method to solve inhomogeneous linear ordinary differential equations. For first-order inhomogeneous linear differential equations it is usually possible to find solutions via integrating factors or undetermined coefficients with considerably less effort, although those methods leverage heuristics that involve guessing and do not work for all inhomogeneous linear differential equations. Variation of parameters extends to linear partial differential equations as well, specifically to inhomogeneous problems for linear evolution equations like the heat equation, wave equation, and vibrating plate equation. In this setting, the method is more often known as Duhamel's principle, named after Jean-Marie Duhamel (1797–1872) who first applied the method to solve the inhomogeneous heat equation. Sometimes variation of parameters itself is called Duhamel's principle and vice versa. History Th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Homogeneous Differential Equation
A differential equation can be homogeneous in either of two respects. A first order differential equation is said to be homogeneous if it may be written :f(x,y) \, dy = g(x,y) \, dx, where and are homogeneous functions of the same degree of and . In this case, the change of variable leads to an equation of the form :\frac = h(u) \, du, which is easy to solve by integration of the two members. Otherwise, a differential equation is homogeneous if it is a homogeneous function of the unknown function and its derivatives. In the case of linear differential equations, this means that there are no constant terms. The solutions of any linear ordinary differential equation of any order may be deduced by integration from the solution of the homogeneous equation obtained by removing the constant term. History The term ''homogeneous'' was first applied to differential equations by Johann Bernoulli in section 9 of his 1726 article ''De integraionibus aequationum differentialium'' (On t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Homogeneous Differential Equation
A differential equation can be homogeneous in either of two respects. A first order differential equation is said to be homogeneous if it may be written :f(x,y) \, dy = g(x,y) \, dx, where and are homogeneous functions of the same degree of and . In this case, the change of variable leads to an equation of the form :\frac = h(u) \, du, which is easy to solve by integration of the two members. Otherwise, a differential equation is homogeneous if it is a homogeneous function of the unknown function and its derivatives. In the case of linear differential equations, this means that there are no constant terms. The solutions of any linear ordinary differential equation of any order may be deduced by integration from the solution of the homogeneous equation obtained by removing the constant term. History The term ''homogeneous'' was first applied to differential equations by Johann Bernoulli in section 9 of his 1726 article ''De integraionibus aequationum differentialium'' (On t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Superposition Principle
The superposition principle, also known as superposition property, states that, for all linear systems, the net response caused by two or more stimuli is the sum of the responses that would have been caused by each stimulus individually. So that if input ''A'' produces response ''X'' and input ''B'' produces response ''Y'' then input (''A'' + ''B'') produces response (''X'' + ''Y''). A function F(x) that satisfies the superposition principle is called a linear function. Superposition can be defined by two simpler properties: additivity F(x_1+x_2)=F(x_1)+F(x_2) \, and homogeneity F(a x)=a F(x) \, for scalar . This principle has many applications in physics and engineering because many physical systems can be modeled as linear systems. For example, a beam can be modeled as a linear system where the input stimulus is the load on the beam and the output response is the deflection of the beam. The importance of linear systems is that they are easier to analyze mathematically; the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
