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List Of Linear Ordinary Differential Equations
This is a list of named linear ordinary differential equations. A–Z :{, class="wikitable sortable" style="background: white; color: black; text-align: left" , -style="background: #eee" !Name !Order !Equation !Applications , - , Airy function, Airy , 2 , \frac{d^2y}{dx^2} - xy = 0 , Optics , - , Bessel function, Bessel , 2 , x^2 \frac{d^2 y}{dx^2} + x \frac{dy}{dx} + \left(x^2 - \alpha^2 \right)y = 0 , Wave, Wave propagation , - , Cauchy-Euler equation, Cauchy-Euler , n , a_{n} x^n y^{(n)}(x) + a_{n-1} x^{n-1} y^{(n-1)}(x) + \dots + a_0 y(x) = 0 , , - , Chebyshev polynomials, Chebyshev , 2 , (1 - x^2)y'' - xy' + n^2 y = 0,\quad(1 - x^2)y'' - 3xy' + n(n + 2) y = 0 , Orthogonal polynomial, Orthogonal polynomials , - , Harmonic oscillator, Damped harmonic oscillator , 2 , m \frac{\mathrm{d}^2x}{\mathrm{d}t^2}+ c\frac{\mathrm{d}x}{\mathrm{d}t} +kx =0 , Damping , - , Frenet-Serret formulas, Frenet-Serret , 1 , \dfrac{ \mathrm{d} \mathbf{T} }{ \mathrm{d} s } =\kappa \mathbf{N}, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Linear Ordinary Differential Equations
Linearity is the property of a mathematical relationship (''function (mathematics), function'') that can be graph of a function, graphically represented as a straight Line (geometry), line. Linearity is closely related to ''Proportionality (mathematics), proportionality''. Examples in physics include rectilinear motion, the linear relationship of voltage and Electric current, current in an electrical conductor (Ohm's law), and the relationship of mass and weight. By contrast, more complicated relationships are ''nonlinear''. Generalized for functions in more than one dimension (mathematics), dimension, linearity means the property of a function of being compatible with addition and scale analysis (mathematics), scaling, also known as the superposition principle. The word linear comes from Latin ''linearis'', "pertaining to or resembling a line". In mathematics In mathematics, a linear map or linear function ''f''(''x'') is a function that satisfies the two properties: * Addi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hill Differential Equation
In mathematics, the Hill equation or Hill differential equation is the second-order linear ordinary differential equation : \frac + f(t) y = 0, where f(t) is a periodic function by minimal period \pi . By these we mean that for all t :f(t+\pi)=f(t), and :\int_0^\pi f(t) \,dt=0, and if p is a number with 0 < p < \pi , the equation must fail for some . It is named after , who introduced it in 1886. Because has period , the Hill equation can be rewritten using the of : : |
List Of Nonlinear Ordinary Differential Equations
See also List of nonlinear partial differential equations and List of linear ordinary differential equations This is a list of named linear ordinary differential equations. A–Z :{, class="wikitable sortable" style="background: white; color: black; text-align: left" , -style="background: #eee" !Name !Order !Equation !Applications , - , Airy function, .... A–F : G–K : L–Q : R–Z : References {{Reflist differential, ordinary, nonlinear ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Applied Mathematics
Applied mathematics is the application of mathematical methods by different fields such as physics, engineering, medicine, biology, finance, business, computer science, and industry. Thus, applied mathematics is a combination of mathematical science and specialized knowledge. The term "applied mathematics" also describes the professional specialty in which mathematicians work on practical problems by formulating and studying mathematical models. In the past, practical applications have motivated the development of mathematical theories, which then became the subject of study in pure mathematics where abstract concepts are studied for their own sake. The activity of applied mathematics is thus intimately connected with research in pure mathematics. History Historically, applied mathematics consisted principally of applied analysis, most notably differential equations; approximation theory (broadly construed, to include representations, asymptotic methods, variation ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quantum Mechanics
Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, quantum field theory, quantum technology, and quantum information science. Classical physics, the collection of theories that existed before the advent of quantum mechanics, describes many aspects of nature at an ordinary ( macroscopic) scale, but is not sufficient for describing them at small (atomic and subatomic) scales. Most theories in classical physics can be derived from quantum mechanics as an approximation valid at large (macroscopic) scale. Quantum mechanics differs from classical physics in that energy, momentum, angular momentum, and other quantities of a bound system are restricted to discrete values ( quantization); objects have characteristics of both particles and waves (wave–particle duality); and there are limit ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quantum Harmonic Oscillator
量子調和振動子 は、調和振動子, 古典調和振動子 の 量子力学, 量子力学 類似物です。任意の滑らかな ポテンシャル エネルギー, ポテンシャル は通常、安定した 平衡点 の近くで 調和振動子#単純調和振動子, 調和ポテンシャル として近似できるため、最も量子力学における重要なモデル系。さらに、これは正確な量子力学システムのリスト, 解析解法が知られている数少ない量子力学系の1つである。 author=Griffiths, David J. , title=量子力学入門 , エディション=2nd , 出版社=プレンティス・ホール , 年=2004 , isbn=978-0-13-805326-0 , author-link=David Griffiths (物理学者) , URL アクセス = 登録 , url=https://archive.org/details/introductiontoel00grif_0 One-dimensional harmonic oscillator Hamiltonian and energy eigenstates 粒子の ハミルトニアン (量子力学), ハミルトニアン は次の� ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Riemann's Differential Equation
In mathematics, Riemann's differential equation, named after Bernhard Riemann, is a generalization of the hypergeometric differential equation, allowing the regular singular points to occur anywhere on the Riemann sphere, rather than merely at 0, 1, and \infty. The equation is also known as the Papperitz equation. The hypergeometric differential equation is a second-order linear differential equation which has three regular singular points, 0, 1 and \infty. That equation admits two linearly independent solutions; near a singularity z_s, the solutions take the form x^s f(x), where x = z-z_s is a local variable, and f is locally holomorphic with f(0)\neq0. The real number s is called the exponent of the solution at z_s. Let ''α'', ''β'' and ''γ'' be the exponents of one solution at 0, 1 and \infty respectively; and let ''α''', ''β''' and ''γ''' be those of the other. Then :\alpha + \alpha' + \beta + \beta' + \gamma + \gamma' = 1. B ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Elliptic Curves
In mathematics, an elliptic curve is a smooth, projective, algebraic curve of genus one, on which there is a specified point . An elliptic curve is defined over a field and describes points in , the Cartesian product of with itself. If the field's characteristic is different from 2 and 3, then the curve can be described as a plane algebraic curve which consists of solutions for: :y^2 = x^3 + ax + b for some coefficients and in . The curve is required to be non-singular, which means that the curve has no cusps or self-intersections. (This is equivalent to the condition , that is, being square-free in .) It is always understood that the curve is really sitting in the projective plane, with the point being the unique point at infinity. Many sources define an elliptic curve to be simply a curve given by an equation of this form. (When the coefficient field has characteristic 2 or 3, the above equation is not quite general enough to include all non-singular cubi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Matrix Differential Equation
A differential equation is a mathematical equation for an unknown function of one or several variables that relates the values of the function itself and its derivatives of various orders. A matrix differential equation contains more than one function stacked into vector form with a matrix relating the functions to their derivatives. For example, a first-order matrix ordinary differential equation is : \mathbf(t) = \mathbf(t)\mathbf(t) where \mathbf(t) is an n \times 1 vector of functions of an underlying variable t, \mathbf(t) is the vector of first derivatives of these functions, and \mathbf(t) is an n \times n matrix of coefficients. In the case where \mathbf is constant and has ''n'' linearly independent eigenvectors, this differential equation has the following general solution, : \mathbf(t) = c_1 e^ \mathbf_1 + c_2 e^ \mathbf_2 + \cdots + c_n e^ \mathbf_n ~, where are the eigenvalues of A; are the respective eigenvectors of A; and are constants. More generally, if \math ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Legendre Polynomials
In physical science and mathematics, Legendre polynomials (named after Adrien-Marie Legendre, who discovered them in 1782) are a system of complete and orthogonal polynomials, with a vast number of mathematical properties, and numerous applications. They can be defined in many ways, and the various definitions highlight different aspects as well as suggest generalizations and connections to different mathematical structures and physical and numerical applications. Closely related to the Legendre polynomials are associated Legendre polynomials, Legendre functions, Legendre functions of the second kind, and associated Legendre functions. Definition by construction as an orthogonal system In this approach, the polynomials are defined as an orthogonal system with respect to the weight function w(x) = 1 over the interval 1,1/math>. That is, P_n(x) is a polynomial of degree n, such that \int_^1 P_m(x) P_n(x) \,dx = 0 \quad \text n \ne m. With the additional standardization condi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
