Ordinary Differential Equations (ODEs for short) come up whenever you have an exact relationship between variables and their rates. Therefore you can find them in geometry, economics, engineering, ecology, mechanics, phys- iology, and many other subjects. For instance, they describe geodesics in geometry, and competing species in ecology
(1) If a body heated to the temperature TÂ is placed in a medium whose temperature is equal to zero, then under certain conditions we may assume that the increment Î”T (negative when T>Â 0) of its temperature over a short interval of time Î” t can be expressed with sufficient accuracy by the formula
T=-k T Î” t
Where kÂ is a constant. In the mathematical treatment of this physical problem we assume that the exactly corresponding limit relation between the differentials
(1)d T=-k Td t holds.
In other words, we assume that the differential equation T=-k T holds, where T'Â denotes the derivative with respect to t.
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To solve this differential equation, or, as we say, to integrate it, is to find the functions that satisfy it. For equation (1) all such functions (that is, all its particular solutions) have the form
T=C e -k t-(2)
where C is a constant. Formula (2) with an arbitrary constant CÂ is called the general
solution of equation-(1)
(2) Suppose a weight pÂ of mass mÂ attached to a spring is in a state of equilibrium (Figure 1, a). If we stretch the spring (Figure 1, b), then the equilibrium is disturbed and the weight is set in motion. If x(t) denotes the magnitude of the body's deviation from the state of equilibrium at time t, then the acceleration of the body is expressed by the second derivative x" (t). If the spring is stretched by a small amount, then, according to the theory of elasticity, the force m x"(t) is proportional to the deviation x(t). Thus, one obtains the differential equation m x"(t) = -k x(t) Its solution has the form and shows that the body will undergo harmonic oscillations (see Figure 1, c). The theory of differential equations developed into an independent, fully elaborated scientific discipline in the 18th century (the works of D. Bernouilli, J. d'Alembert, and L.
Euler). Differential equations are divided into ordinary differential equations, which involve the derivatives of one or several functions of a single independent variable, and partial differential equations, which involve partial derivatives of functions of several independent variables. The order of the differential equation is the highest order of the derivative appearing in it.
Ordinary differential equation of first order:::
A) F(x ,y ,z")=0
The relation between the independent variable x, the unknown function y, and its derivativeÂ y' = d y/ d x is called an ordinary differential equation of the first order in one unknown function (for the present we will examine only equations of this type). If equation (A) can be solved for the derivative, then we obtain an equation of the form
(B) y' = f(x ,y)
The function f(x, y) is supposed single-valued. It is simpler to examine many questions of the theory of differential equations for such equations.
Equation (B) can be written in the form of a relation between differentials
f(x ,y)d x - d y = 0
Then it becomes a particular case of equations of type
(C) P(x, y)d x + Q(x ,y)d y = 0
In equations of type (B), it is natural to consider the variables xÂ and yÂ as equivalent, that is, we are not interested in which of them is independent.
the general solution of this equation is yÂ = 1/(C - x). In Figure 2 the integral curves corresponding to the values of the parameters C=Â 0 and C=Â 1 are drawn
The graph of any single-valued functionÂ y = y(x) intersects every straight line parallel to The O yÂ axis only once. Such, consequently, are the integral curves of any equation (B) with a single-valued continuous function on the right-hand side. New possibilities for the form of integral curves arise in connection with equations of type (C). With the aid of the pair of continuous functionsÂ P(x, y) andÂ Q(x, y), it is possible to define any continuous direction field. The problem of integrating equations of type (C) coincides with the purely geometric (independent of the choice of coordinate axes) problem of finding the integral curves corresponding to a given direction field in the plane. It should be noted that no definite direction corresponds to the points (x0, y0), at which both functionsÂ P(x, y) and Q(x, y) vanish. Such points are called singular points of the equation (C).
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For example, consider the equation
Y d x + x d y = 0
that can be written in the form
Strictly speaking, the right-hand side of the latter equation becomes meaningless for xÂ = 0 and y = 0. The corresponding direction field and the family of integral curves, which in this case are the circles x2Â + y2Â = C, are shown in Figure 3. The origin (xÂ = 0,y-Â 0) is a
Singular point of the differential equation . The integral curves of the equation
y d x = x d y =0
are depicted in Figure 4. They are the rays from the origin. The origin is a singular point of the equation
are depicted in Figure 4. They are the rays from the origin. The origin is a singular point of the equation.
INITIAL CONDITIONS :
The geometric interpretation of differential equations of the first order suggests that through each interior point MÂ of a domain GÂ with a given continuous direction field there passes a unique integral curve. As regards the existence of an integral curve, the formulated hypothesis is valid. The existence proof was supplied by G. Peano. On the other hand, the uniqueness part of this hypothesis proves, generally speaking, to be incorrect. Even for such a simple equation as whose right-hand side is continuous in the entire plane, the integral curves have the form depicted in Figure 5. Uniqueness of the integral curve passing through a given point is violated at all points of the O xÂ axis. Uniqueness, that is, the assertion that there is just one integral curve passing through a given point, holds for equations of type (B) with a continuous right-hand side under the
additional assumption that the function/(x ,y) has a bounded derivative with respect toy in the domain under consideration
This requirement is a special case of the following, somewhat broader Lipschitz condition: there exists a constant LÂ such that in the domain under consideration we have the inequality
|f(x ,y 1) - f(x,y2)|<L|y1Â - y2|
This condition is most frequently cited in textbooks as a sufficient condition of uniqueness.
From the analytic point of view, the existence and uniqueness theorems for equations of type (B) signify the following: if the appropriate conditions are fulfilled(fig-5)
y(x0) of the functionÂ y(x) for an "initial value" x0Â of the independent variableÂ xÂ singles out one definite solution from the family of all solutionsy(x). For example, if for equation (1) we require that at the initial time t0Â =0Â the temperature of the body be equal to the initial value T0, then we will have singled out a definite solution satisfying the given initial
conditions from the infinite family of solutions of (2): T(t) = T0e- kt
This example is typical: in mechanics and physics differential equations usually determine the general laws of the course of some phenomenon. However, in order to obtain definite quantitative results from these laws, it is necessary to specify data
pertaining to the initial state of the physical system being studied at some definite "initial moment" t0.
If the conditions of uniqueness are fulfilled, then the solution y(x) that satisfies the condition y(x0) =y0Â can be written in the form (5)Â y(x) = Î¦(x;x0+y0)
in which x0Â andy0Â enter as parameters. The function Î¦(x; x0,y0) of the three variables x, x0, andÂ y0Â is determined uniquely by equation (B). It is important to note that given a sufficiently small change in the field (the right-hand side of the differential equation), the function Î¦ x0,y0) changes arbitrarily little over some finite interval as xÂ varies-in other words, there is a continuous dependence of the solution on the right-hand side of the differential equation. If the right-hand
side f(x, y) of the differential equation is continuous and its derivative with respect to yÂ is bounded (or satisfies a Lipschitz condition), thenÂ Î¦(x; x0, y0) is again continuous with respect to x0Â andy0.
If the conditions of uniqueness for equation (B) are satisfied in a neighborhood of the point (x0, y j, then all the integral curves passing through a sufficiently small
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neighborhood of the point (x0Â ,y0) intersect the vertical line x=x0Â and each of themÂ is determined by the ordinate yÂ =CÂ of its point of intersection with this line (see Figure6).
Thus, all these solutions belong to the family with the single parameter C:
Y xÂ = F(x ,C)
which is the general solution of the differential equation (B).
In the neighbourhood of points at which the conditions of uniqueness are violated,
(Fig-6) picture can be more complex. The question of the behaviour of the integral curves "in the large" rather than in the neighbourhood of the point (x0Â ,y 0) is also quite complex....
GENERAL INTEGRAL. SINGULAR SOLUTIONS. It is natural to pose the converse problem: given a family of curves depending on a parameter C, find a differential equation for which the curves of the given family would serve as integral curves. The general method of solving this problem consists in the following. Considering the family of curves in the
X O yÂ plane to be defined by the relation (6)Â F(x,y,C) = 0 we differentiate (6) keepingCÂ constant and obtain
or in symmetric notation and eliminate the parameter CÂ from the two equations (6) and (7) or (6) and (8). If a
differential equation is obtained from the relation (6) in this manner, then this relation is called the general integral of the differential equation. The same differential equation can have many different general integrals. After finding the general integral for a given differential equation, it still proves necessary, generally speaking, to check whether the
differential equation has additional solutions not contained in the family of integral curves (6).
Let, for example, the family of curves
(9)Â (x - C)3Â - y = 0 be given. If we keep CÂ constant and differentiate (9), then we obtain 3(x - c)2Â - y' = 0
After elimination of CÂ we arrive at the differential equation
(10)Â 27y2Â - (y')3 = 0
which is equivalent to equation (4). It is easy to see that, in addition to the solutions (9),
equation (10) has the solution
The most general solution of equation (10) is where -âˆž â‰¤C1Â â‰¤C2Â â‰¤ +âˆž (Figure 7). This solution depends on the two parametersC1Â and C2Â but is formed from segments of curves of the one-parameter family (9) and a segment
of the singular solution (11). Solution (11) of equation (10) can serve as an example of a singular solution of a
differential equation. As another example we examine the family of lines
(12)Â 4(y - Cx) + C2Â = 0
These lines are integral curves of the differential equation
4(y - xy') + (y')2Â = 0
A singular integral curve of this differential equation is the parabola x2=y which is the envelope of the lines (12) (Figure 8). This situation is typical: singular integral curves are usually envelopes of the family of integral curves of the general solution
OTHER APPLICATIONS OF DIFFERENTIAL EQUATIONS:
InÂ economics, differential geometry has applications to the field ofÂ econometric
EconometricsÂ is concerned with the tasks of developing and applyingÂ quantitativeÂ orÂ statisticalÂ methods to the study and elucidation of economic principles. Econometrics combinesÂ economic theoryÂ withÂ statisticsÂ to analyze and test economic relationships. Theoretical econometrics considers questions about the statistical properties of estimators and tests, while applied econometrics is concerned with the application of econometric methods to assess economic theories.
While many econometric methods represent applications of standardÂ statistical models, there are some special features ofÂ economic dataÂ that distinguish econometrics from other branches of statistics. Economic data are generallyÂ observational, rather than being derived fromÂ controlled experiments. Because the individual units in an economy interact with each other, the observed data tend to reflect complexÂ economic equilibriumÂ conditions rather than simple behavioural relationships based onÂ preferencesÂ orÂ technology. Consequently, the field of econometrics has developed methods forÂ identificationÂ andÂ estimationÂ of simultaneous. These methods allow researchers to make causal inferences in the absence of controlled experiments.
2..Geometric modellingÂ (includingÂ computer graphics) andÂ computer-aided geometric designÂ draw on ideas from differential geometry.
Geometric modelling Â is a branch ofÂ applied mathematicsÂ andÂ computational geometryÂ that studies methods andÂ algorithmsÂ for the mathematical description of shapes.
The shapes studied in geometric modelling are mostly two- or three-dimensional, although many of its tools and principles can be applied to sets of any finite dimension. Today most geometric modelling is done with computers and for computer-based applications.Â Two-dimensional modelsÂ are important in computerÂ typographyÂ andÂ technical drawing.
3..InÂ engineering, differential geometry can be applied to solve problems inÂ digital signal processing
Digital signal processingÂ (DSP) is concerned with the representation ofÂ signalsÂ by a sequence of numbers or symbols and the processing of these signals. Digital signal processing andÂ analog signal processingÂ are subfields ofÂ signal processing. DSP includes subfields like:Â audioÂ andÂ speech signal processing, sonar and radar signal processing, sensor array processing, spectral estimation, statistical signal processing,Â digital image processing, signal processing for communications, control of systems, biomedical signal processing, seismic data processing, etc.
The goal of DSP is usually to measure, filter and/or compress continuous real-world analog signals. The first step is usually to convert the signal from an analog to a digital form, byÂ samplingÂ it using an analog-to-digital converterÂ (ADC), which turns the analog signal into a stream of numbers. However, often, the required output signal is another analog output signal, which requires aÂ digital-to-analog converterÂ (DAC). Even if this process is more complex than analog processing and has aÂ discrete value range, the application of computational power to digital signal processing allows for many advantages over analog processing in many applications, such asÂ error detection and correctionÂ in transmission as well asÂ data compression.
DSPÂ algorithmsÂ have long been run on standard computers, on specialized processors calledÂ digital signal processorsÂ (DSPs), or on purpose-built hardware such asÂ application-specific integrated circuitÂ (ASICs). Today there are additional technologies used for digital signal processing including more powerful general purposeÂ microprocessors,Â field-programmable gate arraysÂ (FPGAs),Â digital signal controllersÂ (mostly for industrial apps such as motor control), andÂ stream processors, among others
4...InÂ probability,Â statistics, andÂ information theory, one can interpret various structures as Riemannian manifolds, which yields the field ofÂ information geometry, particularly via theÂ Fisher information metric.
InÂ structural geology, differential geometry is used to analyze and describe geologic structures.
Structural geologyÂ is the study of the three-dimensional distribution ofÂ rockÂ units with respect to their deformational histories. The primary goal of structural geology is to use measurements of present-day rock geometries to uncover information about the history of deformation (strain) in the rocks, and ultimately, to understand theÂ stress fieldÂ that resulted in the observed strain and geometries. This understanding of the dynamics of the stress field can be linked to important events in the regional geologic past; a common goal is to understand the structural evolution of a particular area with respect to regionally widespread patterns of rock deformation (e.g.,Â mountain building,Â rifting) due toÂ plate tectonics. Structural geologyÂ is the study of the three-dimensional distribution ofÂ rockÂ units with respect to their deformational histories. The primary goal of structural geology is to use measurements of present-day rock geometries to uncover information about the history of deformation (strain) in the rocks, and ultimately, to understand theÂ stress fieldÂ that resulted in the observed strain and geometries. This understanding of the dynamics of the stress field can be linked to important events in the regional geologic past; a common goal is to understand the structural evolution of a particular area with respect to regionally widespread patterns of rock deformation (e.g.,Â mountain building,Â rifting) due toÂ plate tectonics.
InÂ computer vision, differential geometry is used to analyze shapes.
computer visionÂ is the science and technology of machines that see, whereÂ seeÂ in this case means that the machine is able to extract information from an image that is necessary to solve some task. As a scientific discipline, computer vision is concerned with the theory behind artificial systems that extract information from images. The image data can take many forms, such as video sequences, views from multiple cameras, or multi-dimensional data from a medical scanner.
(RELATION BETWEEN COMPUTER VISION AND VARIOUS FIELD)