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Singular Solution
A singular solution ''ys''(''x'') of an ordinary differential equation is a solution that is singular or one for which the initial value problem (also called the Cauchy problem by some authors) fails to have a unique solution at some point on the solution. The set on which a solution is singular may be as small as a single point or as large as the full real line. Solutions which are singular in the sense that the initial value problem fails to have a unique solution need not be singular functions. In some cases, the term ''singular solution'' is used to mean a solution at which there is a failure of uniqueness to the initial value problem at every point on the curve. A singular solution in this stronger sense is often given as tangent to every solution from a family of solutions. By ''tangent'' we mean that there is a point ''x'' where ''ys''(''x'') = ''yc''(''x'') and ''y's''(''x'') = ''y'c''(''x'') where ''yc'' is a solution in a family of solutions parameterized by ''c''. This m ... [...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] |
Mathematical Singularity
In mathematics, a singularity is a point at which a given mathematical object is not defined, or a point where the mathematical object ceases to be well-behaved in some particular way, such as by lacking differentiability or analyticity. For example, the real function : f(x) = \frac has a singularity at x = 0, where the numerical value of the function approaches \pm\infty so the function is not defined. The absolute value function g(x) = , x, also has a singularity at x = 0, since it is not differentiable there. The algebraic curve defined by \left\ in the (x, y) coordinate system has a singularity (called a cusp) at (0, 0). For singularities in algebraic geometry, see singular point of an algebraic variety. For singularities in differential geometry, see singularity theory. Real analysis In real analysis, singularities are either discontinuities, or discontinuities of the derivative (sometimes also discontinuities of higher order derivatives). There are four kinds of discon ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Initial Value Problem
In multivariable calculus, an initial value problem (IVP) is an ordinary differential equation together with an initial condition which specifies the value of the unknown function at a given point in the domain. Modeling a system in physics or other sciences frequently amounts to solving an initial value problem. In that context, the differential initial value is an equation which specifies how the system evolves with time given the initial conditions of the problem. Definition An initial value problem is a differential equation :y'(t) = f(t, y(t)) with f\colon \Omega \subset \mathbb \times \mathbb^n \to \mathbb^n where \Omega is an open set of \mathbb \times \mathbb^n, together with a point in the domain of f :(t_0, y_0) \in \Omega, called the initial condition. A solution to an initial value problem is a function y that is a solution to the differential equation and satisfies :y(t_0) = y_0. In higher dimensions, the differential equation is replaced with a family of equati ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tangent
In geometry, the tangent line (or simply tangent) to a plane curve at a given point is the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points on the curve. More precisely, a straight line is said to be a tangent of a curve at a point if the line passes through the point on the curve and has slope , where ''f'' is the derivative of ''f''. A similar definition applies to space curves and curves in ''n''-dimensional Euclidean space. As it passes through the point where the tangent line and the curve meet, called the point of tangency, the tangent line is "going in the same direction" as the curve, and is thus the best straight-line approximation to the curve at that point. The tangent line to a point on a differentiable curve can also be thought of as a '' tangent line approximation'', the graph of the affine function that best approximates the original function at the given point. Similarly, t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Envelope (mathematics)
In geometry, an envelope of a planar family of curves is a curve that is tangent to each member of the family at some point, and these points of tangency together form the whole envelope. Classically, a point on the envelope can be thought of as the intersection of two "infinitesimally adjacent" curves, meaning the limit of intersections of nearby curves. This idea can be generalized to an envelope of surfaces in space, and so on to higher dimensions. To have an envelope, it is necessary that the individual members of the family of curves are differentiable curves as the concept of tangency does not apply otherwise, and there has to be a smooth transition proceeding through the members. But these conditions are not sufficient – a given family may fail to have an envelope. A simple example of this is given by a family of concentric circles of expanding radius. Envelope of a family of curves Let each curve ''C''''t'' in the family be given as the solution of an equation ''f'''' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
0 (number)
0 (zero) is a number representing an empty quantity. In place-value notation such as the Hindu–Arabic numeral system, 0 also serves as a placeholder numerical digit, which works by multiplying digits to the left of 0 by the radix, usually by 10. As a number, 0 fulfills a central role in mathematics as the additive identity of the integers, real numbers, and other algebraic structures. Common names for the number 0 in English are ''zero'', ''nought'', ''naught'' (), ''nil''. In contexts where at least one adjacent digit distinguishes it from the letter O, the number is sometimes pronounced as ''oh'' or ''o'' (). Informal or slang terms for 0 include ''zilch'' and ''zip''. Historically, ''ought'', ''aught'' (), and ''cipher'', have also been used. Etymology The word ''zero'' came into the English language via French from the Italian , a contraction of the Venetian form of Italian via ''ṣafira'' or ''ṣifr''. In pre-Islamic time the word (Arabic ) had the meanin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Picard–Lindelöf Theorem
In mathematics – specifically, in differential equations – the Picard–Lindelöf theorem gives a set of conditions under which an initial value problem has a unique solution. It is also known as Picard's existence theorem, the Cauchy–Lipschitz theorem, or the existence and uniqueness theorem. The theorem is named after Émile Picard, Ernst Lindelöf, Rudolf Lipschitz and Augustin-Louis Cauchy. Theorem Let D \subseteq \R \times \R^nbe a closed rectangle with (t_0, y_0) \in D. Let f: D \to \R^n be a function that is continuous in t and Lipschitz continuous in y. Then, there exists some such that the initial value problem y'(t)=f(t,y(t)),\qquad y(t_0)=y_0. has a unique solution y(t) on the interval _0-\varepsilon, t_0+\varepsilon/math>. Note that D is often instead required to be open but even under such an assumption, the proof only uses a closed rectangle within D. Proof sketch The proof relies on transforming the differential equation, and applying ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Peano Existence Theorem
In mathematics, specifically in the study of ordinary differential equations, the Peano existence theorem, Peano theorem or Cauchy–Peano theorem, named after Giuseppe Peano and Augustin-Louis Cauchy, is a fundamental theorem which guarantees the existence of solutions to certain initial value problems. History Peano first published the theorem in 1886 with an incorrect proof. In 1890 he published a new correct proof using successive approximations. Theorem Let D be an open subset of \mathbb\times\mathbb with f\colon D \to \mathbb a continuous function and y'(x) = f\left(x,y(x)\right) a continuous, explicit first-order differential equation defined on ''D'', then every initial value problem y\left(x_0\right) = y_0 for ''f'' with (x_0, y_0) \in D has a local solution z\colon I \to \mathbb where I is a neighbourhood of x_0 in \mathbb, such that z'(x) = f\left(x,z(x)\right) for all x \in I . The solution need not be unique: one and the same initial value (x_0,y_0) may give ris ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Clairaut's Equation
In mathematical analysis, Clairaut's equation (or the Clairaut equation) is a differential equation of the form :y(x)=x\frac+f\left(\frac\right) where ''f'' is continuously differentiable. It is a particular case of the Lagrange differential equation. It is named after the French mathematician Alexis Clairaut, who introduced it in 1734. Definition To solve Clairaut's equation, one differentiates with respect to ''x'', yielding :\frac=\frac+x\frac+f'\left(\frac\right)\frac, so :\left +f'\left(\frac\right)\rightfrac = 0. Hence, either :\frac = 0 or :x+f'\left(\frac\right) = 0. In the former case, ''C'' = ''dy''/''dx'' for some constant ''C''. Substituting this into the Clairaut's equation, one obtains the family of straight line functions given by :y(x)=Cx+f(C),\, the so-called ''general solution'' of Clairaut's equation. The latter case, :x+f'\left(\frac\right) = 0, defines only one solution ''y''(''x''), the so-called ''singular solution'', whose graph is the en ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebra
Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics. Elementary algebra deals with the manipulation of variables (commonly represented by Roman letters) as if they were numbers and is therefore essential in all applications of mathematics. Abstract algebra is the name given, mostly in education, to the study of algebraic structures such as groups, rings, and fields (the term is no more in common use outside educational context). Linear algebra, which deals with linear equations and linear mappings, is used for modern presentations of geometry, and has many practical applications (in weather forecasting, for example). There are many areas of mathematics that belong to algebra, some having "algebra" in their name, such as commutative algebra, and some not, such as Galois theory. The word ''algebra'' is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Derivative
In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. For example, the derivative of the position of a moving object with respect to time is the object's velocity: this measures how quickly the position of the object changes when time advances. The derivative of a function of a single variable at a chosen input value, when it exists, is the slope of the tangent line to the graph of the function at that point. The tangent line is the best linear approximation of the function near that input value. For this reason, the derivative is often described as the "instantaneous rate of change", the ratio of the instantaneous change in the dependent variable to that of the independent variable. Derivatives can be generalized to functions of several real variables. In this generalization, the derivativ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Chandrasekhar Equation
Chandrasekhar, Chandrashekhar or Chandra Shekhar is an Indian name and may refer to a number of individuals. The name comes from the name of an incarnation of the Hindu god Shiva. In this form he married the goddess Parvati. Etymologically, the name comes from the Sanskrit words "चन्द्र (''candra'')", meaning "moon", and "शेखर (''śekhara'')", meaning "crest" or "crown", which is an epithet of the Shiva. Notable people with this name In politics and activism: * Chandra Shekhar Azad (1906–1931), known as Azad ("The Free"), Indian revolutionary who organised the Hindustan Socialist Republican Army (HSRA) * K. M. Chandrasekhar (born 1948), Indian Cabinet Secretary * Sripati Chandrasekhar (1918–2001), an Indian demographer and politician * Chandra Shekhar Dubey (born 1946), member of the 14th Lok Sabha of India * Chandrashekhar Prasad (died 1997), assassinated Indian student activist * Kalvakuntla Chandrashekar Rao (born 1954), known as K.C.R., Indian politic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
