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From Wikipedia

Differential equation

A differential equation is a mathematicalequation for an unknown function of one or several variables that relates the values of the function itself and its derivatives of various orders. Differential equations play a prominent role in engineering, physics, economics, and other disciplines.

Differential equations arise in many areas of science and technology, specifically whenever a deterministic relation involving some continuously varying quantities (modeled by functions) and their rates of change in space and/or time (expressed as derivatives) is known or postulated. This is illustrated in classical mechanics, where the motion of a body is described by its position and velocity as the time varies. Newton's laws allow one to relate the position, velocity, acceleration and various forces acting on the body and state this relation as a differential equation for the unknown position of the body as a function of time. In some cases, this differential equation (called an equation of motion) may be solved explicitly.

An example of modelling a real world problem using differential equations is determination of the velocity of a ball falling through the air, considering only gravity and air resistance. The ball's acceleration towards the ground is the acceleration due to gravity minus the deceleration due to air resistance. Gravity is constant but air resistance may be modelled as proportional to the ball's velocity. This means the ball's acceleration, which is the derivative of its velocity, depends on the velocity. Finding the velocity as a function of time involves solving a differential equation.

Differential equations are mathematically studied from several different perspectives, mostly concerned with their solutions—the set of functions that satisfy the equation. Only the simplest differential equations admit solutions given by explicit formulas; however, some properties of solutions of a given differential equation may be determined without finding their exact form. If a self-contained formula for the solution is not available, the solution may be numerically approximated using computers. The theory of dynamical systems puts emphasis on qualitative analysis of systems described by differential equations, while many numerical methods have been developed to determine solutions with a given degree of accuracy.

Directions of study

The study of differential equations is a wide field in pure and applied mathematics, physics, meteorology, and engineering. All of these disciplines are concerned with the properties of differential equations of various types. Pure mathematics focuses on the existence and uniqueness of solutions, while applied mathematics emphasizes the rigorous justification of the methods for approximating solutions. Differential equations play an important role in modelling virtually every physical, technical, or biological process, from celestial motion, to bridge design, to interactions between neurons. Differential equations such as those used to solve real-life problems may not necessarily be directly solvable, i.e. do not have closed form solutions. Instead, solutions can be approximated using numerical methods.

Mathematicians also study weak solutions (relying on weak derivatives), which are types of solutions that do not have to be differentiable everywhere. This extension is often necessary for solutions to exist, and it also results in more physically reasonable properties of solutions, such as possible presence of shocks for equations of hyperbolic type.

The study of the stability of solutions of differential equations is known as stability theory.

Nomenclature

The theory of differential equations is quite developed and the methods used to study them vary significantly with the type of the equation.

  • An ordinary differential equation (ODE) is a differential equation in which the unknown function (also known as the dependent variable) is a function of a single independent variable. In the simplest form, the unknown function is a real or complex valued function, but more generally, it may be vector-valued or matrix-valued: this corresponds to considering a system of ordinary differential equations for a single function. Ordinary differential equations are further classified according to the order of the highest derivative of the dependent variable with respect to the independent variable appearing in the equation. The most important cases for applications are first-order and second-order differential equations. In the classical literature also distinction is made between differential equations explicitly solved with respect to the highest derivative and differential equations in an implicit form.
  • A partial differential equation (PDE) is a differential equation in which the unknown function is a function of multiple independent variables and the equation involves its partial derivatives. The order is defined similarly to the case of ordinary differential equations, but further classification into elliptic, hyperbolic, and parabolic equations, especially for second-order linear equations, is of utmost importance. Some partial differential equations do not fall into any of these categories over the whole domain of the independent variables and they are said to be of mixed type.

Both ordinary and partial differential equations are broadly classified as linear and nonlinear. A differential equation is linear if the unknown function and its derivatives appear to the power

Differential calculus

In mathematics, differential calculus is a subfield of calculus concerned with the study of the rates at which quantities change. It is one of the two traditional divisions of calculus, the other being integral calculus.

The primary objects of study in differential calculus are the derivative of a function, related notions such as the differential, and their applications. The derivative of a function at a chosen input value describes the rate of change of the function near that input value. The process of finding a derivative is called differentiation. Geometrically, the derivative at a point equals the slope of the tangent line to the graph of the function at that point. For a real-valued function of a single real variable, the derivative of a function at a point generally determines the best linear approximation to the function at that point.

Differential calculus and integral calculus are connected by the fundamental theorem of calculus, which states that differentiation is the reverse process to integration.

Differentiation has applications to all quantitative disciplines. In physics, the derivative of the displacement of a moving body with respect to time is the velocity of the body, and the derivative of velocity with respect to time is acceleration. Newton's second law of motion states that the derivative of the momentum of a body equals the force applied to the body. The reaction rate of a chemical reaction is a derivative. In operations research, derivatives determine the most efficient ways to transport materials and design factories. By applying game theory, differentiation can provide best strategies for competing corporations.

Derivatives are frequently used to find the maxima and minima of a function. Equations involving derivatives are called differential equations and are fundamental in describing natural phenomena. Derivatives and their generalizations appear in many fields of mathematics, such as complex analysis, functional analysis, differential geometry, measure theory and abstract algebra.

The derivative

Suppose that x and y are real numbers and that y is a function of x, that is, for every value of x, we can determine the value of y. This relationship is written as: y = f(x). Where f(x) is the equation for a straight line, y = mx + b, where m and b are real numbers that determine the locus of the line in Cartesian coordinates. m is called the slope and is given by:

m={\mbox{change in } y \over \mbox{change in } x} = {\Delta y \over{\Delta x}},

where the symbol Δ (the uppercase form of the Greek letter Delta) is an abbreviation for "change in". It follows that Δy = mΔx.

In linear functions the derivative of f at the point x is the best possible approximation to the idea of the slope of f at the point x. It is usually denoted f'(x) or dy/dx. Together with the value of f at x, the derivative of f determines the best linear approximation, orlinearization, of f near the point x. This latter property is usually taken as the definition of the derivative. Derivatives cannot be calculated in nonlinear functions because they do not have a well-defined slope.

A closely related notion is the differential of a function.

When x and y are real variables, the derivative of f at x is the slope of the tangent line to the graph of f' at x. Because the source and target of f are one-dimensional, the derivative of f is a real number. If x and y are vectors, then the best linear approximation to the graph of f depends on how f changes in several directions at once. Taking the best linear approximation in a single direction determines a partial derivative, which is usually denoted ∂y/∂x. The linearization of f in all directions at once is called thetotal derivative. It is alinear transformation, and it determines the hyperplane that most closely approximates the graph of f. This hyperplane is called the osculating hyperplane; it is conceptually the same idea as taking tangent lines in all directions at once.



From Yahoo Answers

Question:Hi. Please, could you give me specific examples of the application of differential equations to 1) Physics 2) thermodynamics thanks in advance!

Answers:The distribution of load over the length of a beam, second moment of area of an object, the rate of decay of a radioactive sample, the rate of heat transfer between two objects... minds gone blank :/

Question:If a derivative means to find the "rate of change" then: (1) What is a differential equation mean, or used for? (2) What is linear algebra used for?

Answers:Yes... the derivative of a function F(x) is another function f(x) that outputs the instantaneous slope of F(x) at x. Does that make sense? if y = x then y' = 2x... The parabola y=x , at every value x, has a slope of 2x I dont know how much you know about derivative calculus... but you seem to understand what it does and whats its for already. === 1) What are differential equations? DEs are equations, functions, that relate a function f(x) and its varying degrees of differentiations to each other and to other functions of x. Often times, the objective of a DE is to find out what f(x) is, explicitly, in terms of 'x'; without relating f(x) to any of its derivatives Like, for example, if the DE was: y + 2y' - x = 4 And I asked you to figure out what the function y(x) was in terms of x... The answer would not be y(x) = 4 + x - 2y'(x)... because that is just the same DE rearranged. In many instances, from physics to statistics to economics... differential equations show up. DE's are equations that relate a functions output, quantity and its rate of change over the course of time 't' in a single equation... these relationships might be easy to determine with the data they have... but the mathematician might want to figure out what the quantity is at time t, or what its rate of change is... without having to know the other first

Question:Differentiation of PV=RT In my physics textbook, PV=RT is differentiated as PdV+VdP=RdT, from which the value of dT is taken and used to derive the equation PV(power gamma) is constant for adiabatic conditions. Now, can you explain the differentiation process in the equation (only one mole of gas is considered), and with respect to what quantity differentiation is being carried out?

Answers:Really speaking this is a form called differential,where we can separate differentials dP,dV and dT.We can still use the rules of differentiation.Hence d(PV)=RdT or dP V+P dV=RdT Note:In calculus we know that dy/dx is a limit and it is not dy(divided)dx.Thermodynamics can be formulated in differential form. IVAN

Question:I will be taking a physical chemistry that is geared toward biochemistry majors and is a two quarter sequence rather than the normal three quarter sequence for chemistry/engineer majors. It is more of a biological approach. I have up to calculus III (third quarter calculus covering vectors), integral and differential calculus. The description for the course says that matrix algebra and differential equations is recommended (but not necessarily needed or required as a pre-req). What should I do?

Answers:You have a good background for the courses. Although those math courses are not required, a course in differential equations beforehand (or even at the same time) will give you a huge benefit in your studies. You will certainly study the Schrodinger equation, for example, which is a differential equation. You will also study kinetics (more diffy-Qs). You won't use nearly as much matrix algebra and so I would recommend the ODE course over the matrix algebra if you must choose. Good luck!

From Youtube

Application of a second order differential equation :Solution of Question 1 in the Exercises 5.3.11 from the notes for MS1S03. Note that the condition for k being larger than delta has been corrected in this clip.

Lecture 31 - Solving Ordinary Differential Equations :Numerical Methods and Programing by PBSunil Kumar, Dept, of physics, IIT Madras