“Ordinary Differential Equations” come up whenever you have an exact relationship between variables and their rates. Therefore, you can find them in geometry, economics, technology, ecology, mechanics, physiology, and many other topics. For example, they describe geodesics in geometry and competing species in ecology.
(1) If a body heated to the temperature TA is placed in a medium whose temperature is equal to zero, then under certain conditions, we may presume that the increase I”T (negative when T > A 0) of its temperature over a short interval of time I”T can be expressed with sufficient accuracy by the expression:
Where kA is a constant. In the mathematical treatment of this physical problem, we assume that the corresponding relationship between the derivatives:
(1) vitamin D T=-k Td T holds.
In other words, we assume that the differential equation T=-k T holds, where T’A denotes the derivative with respect to time.
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=Ce -kt- (2)
where C is a constant. Formula (2) with an arbitrary constant CA is called the general solution of equation (1).
(2) Suppose a weight pA of mass mA attached to a spring is in a state of equilibrium. If we stretch the spring, 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). Therefore, 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. The theory of differential equations developed into an independent, fully elaborated scientific subject in the eighteenth century (the works of D. Bernoulli, 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, y’) = 0.
The relation between the independent variable t, the unknown function y, and its derivative y’ = dy/dt is called an ordinary differential equation of the first order in one unknown function (for the present we will analyze 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),
where the function f(x, y) is supposed to be single-valued. It is simpler to analyze many inquiries of the theory of differential equations for such equations.
Equation (B) can be written in the form of a relation between partial derivatives,
f(x, y) + y’ – x’ = 0.
Then it becomes a particular instance of equations of type
(C) P(x, y) + Q(x, y)y’ = 0.
In equations of type (B), it is natural to see the variables x and y as equivalent, that is, we are not interested in which of them is independent.
The general solution of this equation is y = 1/(C – t). The integral curves matching the values of the parameters C=0 and C=1 are drawn.
The graph of any single-valued function y = y(x) intersects every consecutive line parallel to the y-axis only one time. Such, accordingly, 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 assistance of the pair of continuous functions P(x, y) and Q(x, y), it is possible to specify any continuous path field. The problem of integrating equations of type (C) coincides with the strictly geometric (independent of the choice of coordinate axes) problem of finding the integral curves matching to a given path field in the plane.
It should be noted that no definite path corresponds to the points (x0, y0), at which both functions P(x, y) and Q(x, y) vanish. Such points are called singular points of the equation (C).
For example, consider the equation
y + xy’ = 0,
which can be written in the form
Strictly speaking, the right-hand side of the latter equation becomes meaningless for x = 0 and y = 0. The corresponding path field and the family of integral curves, which in this instance are the circles x^2 + y^2 = C. The origin (x = 0, y = 0) is a singular point of the differential equation. The integral curves of the equation
y’ = x = 0
They are the rays from the origin. The origin is a singular point of the equation.
The geometric interpretation of differential equations of the first order suggests that through each interior point MA of a sphere GA with a given uninterrupted field, a unique built-in curve passes. With regards to the existence of a built-in curve, the formulated hypothesis is valid, and this was confirmed by G. Peano.
However, the uniqueness part of this hypothesis, in general, is proven to be wrong. Even for such a simple equation, where the right-hand side is uninterrupted in the entire plane, the built-in curves have the form depicted. The singularity of the built-in curve passing through a given point is violated at all points of the O xA axis.
Uniqueness, that is, the claim that there is only one built-in curve passing through a given point, holds for equations of type (B) with an uninterrupted right-hand side under the extra premise that the function f(x, y) has a bounded derivative with respect to y in the sphere under consideration.
This requirement is a special case of the following, slightly broader Lipschitz condition: there exists a constant LA such that in the sphere under consideration, we have the inequality
|f(x, y1) – f(x, y2)| < L|y1 – y2|
This condition is most often cited in textbooks as a sufficient condition of singularity.
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, the value Y(x0) of the function Y(x) for an “initial value” x0 of the independent variable x singles out one definite solution from the family of all solutions y(x).
For example, if for equation (1) we require that at the initial time t0=0, the temperature of the body be equal to the initial value T0, then we will have singled out a definite solution fulfilling the given initial conditions from the infinite family of solutions of (2): T(T) = T0e^(-kappa).
This example is typical: in mechanics and physics, differential equations usually determine the general laws of the course of some phenomenon. However, to obtain definite quantitative results from these laws, it is necessary to specify data regarding the initial state of the physical system being studied at some definite “initial moment” t0.
If the conditions of singularity are fulfilled, then the solution Y(x) that satisfies the condition Y(x0) = y0 can be written in the form (5) Y(x) = I(x; x0, y0), in which x0 and y0 enter as parameters.
The function I(x; x0, y0) of the three variables x, x0, and y0 is determined unambiguously 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 I(x0, y0) changes randomly small over some finite interval as x 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 y is bounded (or satisfies a Lipschitz condition), then I(x; x0, y0) is again continuous with respect to x0 and y0.
If the conditions of singularity for equation (B) are satisfied in a vicinity of the point (x0, yJ), so all the built-in curves passing through a sufficiently small vicinity of the point (x0A, y0) intersect the perpendicular line x=x0A, and each of them is determined by the ordinate yA=CA of its point of intersection with this line.
Therefore, all these solutions belong to the household with the individual parametric quantity C:
YxA = F(x, C)
which is the general solution of the differential equation (B).
In the vicinity of points at which the conditions of singularity are violated. The inquiry of the behavior of the built-in curves “in the big” instead of in the vicinity of the point (x0A, y0) is also quite complex.
GENERAL INTEGRAL. SINGULAR SOLUTIONS
It is natural to present the converse job: given a household of curves depending on a parametric quantity C, find a differential equation for which the curves of the given household would function as built-in curves.
The general method of solving this job consists of the following. Consider the household of curves in the xOyA plane to be defined by the relation (6) F(x, y, C) = 0. We differentiate (6) keeping CA constant and obtain
Or in symmetric notation and eliminate the parametric quantity CA from the two equations (6) and (7) or (6) and (8). If a differential equation is obtained from the relation (6) in this mode, 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 is still necessary to check whether the differential equation has extra solutions not contained in the household of built-in curves (6).
For example, consider the household of curves
(9) A (x – C)3A – y = 0
If we keep CA constant and differentiate (9), then we obtain 3(x – C)2A – y’ = 0. After elimination of CA, we arrive at the differential equation,
(10) A 27y2A – (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 −a≤z≤C1≤C2≤a. This solution depends on the two parameters C1 and C2 but is formed from sections of curves of the one-parameter household (9) and a section of the remarkable solution (11). Solution (11) of equation (10) can serve as an illustration of a remarkable solution of a differential equation. As another example, we examine the household of lines
(12) A 4(y – Cx) + C2 = 0
These lines are built-in curves of the differential equation
4(y – xy’) + (y’)2A = 0
A remarkable built-in curve of this differential equation is the parabola x2=y, which is the envelope of the lines (12). This situation is typical: remarkable built-in curves are usually envelopes of the household of built-in curves of the general solution.
OTHER APPLICATIONS OF DIFFERENTIAL EQUATIONS:
In economic sciences, differential equations have applications in the field of econometrics.
Econometrics is concerned with developing and applying quantitative or statistical methods to survey and elucidate economic rules. Econometrics combines economic theory with statistics to analyze and prove economic relationships. Theoretical econometrics considers inquiries about the statistical properties of calculators and trials, while applied econometrics is concerned with applying econometric methods to measure economic theories.
While many econometric methods represent applications of standard statistical models, there are some particular characteristics of economic data that distinguish econometrics from other subdivisions of statistics. Economic data is generally experimental, instead of being derived from controlled experiments.
Because the individual units in an economic system interact with each other, the observed data tends to reflect complex economic equilibrium conditions instead of simple behavioral relationships based on preferences or engineering. Consequently, the field of econometrics has developed methods for identifying and estimating coincidence. These methods allow researchers to make causal inferences in the absence of controlled experiments.
Geometric modeling (including computer graphics) and computer-aided geometric design draw on ideas from differential geometry. Geometric modeling is a subdivision of applied mathematics and computational geometry that studies methods and algorithms for the mathematical description of forms.
The forms studied in geometric modeling are largely two- or three-dimensional, although many of its tools and rules can be applied to sets of any finite dimension. Today, most geometric modeling is done with computers and for computer-based applications. Planar models are important in computer typography and technical drawing.
In technology, differential geometry can be applied to solve problems in digital signal processing. Digital signal processing (DSP) is concerned with the representation of signals by a sequence of numbers or symbols and the processing of these signals. Digital signal processing and analog signal processing are subfields of signal processing.
DSP includes subfields like audio and speech signal processing, echo sounder and radar signal processing, sensor array processing, spectral appraisal, statistical signal processing, 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 parallel signals. The first step is usually to convert the signal from an analog to a digital form, by sampling it using an analog-to-digital converter (ADC), which turns the analog signal into a stream of numbers.
However, often, the required output signal is another analog output signal, which requires a digital-to-analog converter (DAC). Even if this process is more complex than analog processing and has a distinct value range, the application of computational power to digital signal processing allows for many advantages over analog processing in many applications, such as error detection and correction in transmission as well as data compression.
DSP algorithms have long been run on standard computers, on specialized processors called digital signal processors (DSPs), or on purpose-made hardware such as application-specific integrated circuits (ASICs).
Today there are additional technologies used for digital signal processing including more powerful general-purpose microprocessors, field-programmable gate arrays (FPGAs), digital signal controllers (mostly for industrial apps such as motor control), and stream processors, among others.
In probability, statistics, and information theory, one can interpret various structures as Riemannian manifolds, which yields the field of information geometry, particularly via the Fisher information metric.
In structural geology, differential geometry is used to analyze and describe geologic structures. Structural geology is the study of the 3-dimensional distribution of rock units with regard to their deformational histories. The primary goal of structural geology is to use measurements of contemporary rock geometries to reveal information about the history of deformation (strain) in the rocks, and ultimately, to understand the stress field 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 development of a particular area with regard to regionally widespread patterns of rock deformation (e.g., mountain building, rifting) due to plate tectonics.
In computer vision, differential geometry is used to analyze forms. Computer vision is the science and technology of machines that see, where “see” 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 multidimensional data from a medical scanner. ( Relation between computer vision and various fields )