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Cauchy–Euler Equation
In mathematics, an Euler–Cauchy equation, or Cauchy–Euler equation, or simply Euler's equation is a linear homogeneous ordinary differential equation with variable coefficients. It is sometimes referred to as an ''equidimensional'' equation. Because of its particularly simple equidimensional structure, the differential equation can be solved explicitly. The equation Let be the ''n''th derivative of the unknown function . Then a Cauchy–Euler equation of order ''n'' has the form a_ x^n y^(x) + a_ x^ y^(x) + \dots + a_0 y(x) = 0. The substitution x = e^u (that is, u = \ln(x); for x < 0, one might replace all instances of by , which extends the solution's domain to ) may be used to reduce this equation to a linear differential equation with constant coefficients. Alternatively, the trial solution may be used to directly solve for the basic solutions. Second o ...
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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 t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Linear Differential Equation
In mathematics, 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) where and are arbitrary differentiable functions that do not need to be linear, and are the successive derivatives of an unknown function of the variable . Such an equation is an ordinary differential equation (ODE). A ''linear differential equation'' may also be a linear partial differential equation (PDE), if the unknown function depends on several variables, and the derivatives that appear in the equation are partial derivatives. A linear differential equation or a system of linear equations such that the associated homogeneous equations have constant coefficients may be solved by quadrature, which means that the solutions may be expressed in terms of integrals. This is also true for a linear equation of order one, with non-co ... [...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 ... [...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 mathematic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Variable Coefficient
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 a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Equidimensionality
In mathematics, especially in topology, equidimensionality is a property of a space that the local dimension is the same everywhere. Definition (topology) A topological space ''X'' is said to be equidimensional if for all points ''p'' in ''X'', the dimension at ''p'', that is dim ''p''(''X''), is constant. The Euclidean space is an example of an equidimensional space. The disjoint union of two spaces ''X'' and ''Y'' (as topological spaces) of different dimension is an example of a non-equidimensional space. Definition (algebraic geometry) A scheme ''S'' is said to be equidimensional if every irreducible component has the same Krull dimension. For example, the affine scheme Spec k ,y,z(xy,xz), which intuitively looks like a line intersecting a plane, is not equidimensional. Cohen–Macaulay ring An affine algebraic variety whose coordinate ring is a Cohen–Macaulay ring In mathematics, a Cohen–Macaulay ring is a commutative ring with some of the algebraic geometry, alg ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Laplace's Equation
In mathematics and physics, Laplace's equation is a second-order partial differential equation named after Pierre-Simon Laplace, who first studied its properties. This is often written as \nabla^2\! f = 0 or \Delta f = 0, where \Delta = \nabla \cdot \nabla = \nabla^2 is the Laplace operator,The delta symbol, Δ, is also commonly used to represent a finite change in some quantity, for example, \Delta x = x_1 - x_2. Its use to represent the Laplacian should not be confused with this use. \nabla \cdot is the divergence operator (also symbolized "div"), \nabla is the gradient operator (also symbolized "grad"), and f (x, y, z) is a twice-differentiable real-valued function. The Laplace operator therefore maps a scalar function to another scalar function. If the right-hand side is specified as a given function, h(x, y, z), we have \Delta f = h. This is called Poisson's equation, a generalization of Laplace's equation. Laplace's equation and Poisson's equation are the simplest exa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Reduction Of Order
Reduction of order is a technique in mathematics for solving second-order linear ordinary differential equations. It is employed when one solution y_1(x) is known and a second linearly independent solution y_2(x) is desired. The method also applies to ''n''-th order equations. In this case the ansatz will yield an (''n''−1)-th order equation for v. Second-order linear ordinary differential equations An example Consider the general, homogeneous, second-order linear constant coefficient ordinary differential equation. (ODE) : a y''(x) + b y'(x) + c y(x) = 0, where a, b, c are real non-zero coefficients. Two linearly independent solutions for this ODE can be straightforwardly found using characteristic equations except for the case when the discriminant, b^2 - 4 a c, vanishes. In this case, : a y''(x) + b y'(x) + \frac y(x) = 0, from which only one solution, :y_1(x) = e^, can be found using its characteristic equation. The method of reduction of order is used to obtain ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Euler's Formula
Euler's formula, named after Leonhard Euler, is a mathematical formula in complex analysis that establishes the fundamental relationship between the trigonometric functions and the complex exponential function. Euler's formula states that for any real number : e^ = \cos x + i\sin x, where is the base of the natural logarithm, is the imaginary unit, and and are the trigonometric functions cosine and sine respectively. This complex exponential function is sometimes denoted ("cosine plus i sine"). The formula is still valid if is a complex number, and so some authors refer to the more general complex version as Euler's formula. Euler's formula is ubiquitous in mathematics, physics, and engineering. The physicist Richard Feynman called the equation "our jewel" and "the most remarkable formula in mathematics". When , Euler's formula may be rewritten as , which is known as Euler's identity. History In 1714, the English mathematician Roger Cotes presented a geometr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
