Differential Equations- a branch of mathematics that studies the theory and methods for solving equations containing the desired function and its derivatives of various orders of one argument (ordinary differential) or several arguments (partial differential equations). Differential equations are widely used in practice, in particular, to describe transient processes.

Theory of differential equations- a branch of mathematics dealing with the study of differential equations and related problems. Their results are used in many natural sciences, especially widely in physics.

Simply put, differential equation- this is an equation in which some function is an unknown quantity. At the same time, not only an unknown function, but also its various derivatives participate in the equation itself. A differential equation describes the relationship between an unknown function and its derivatives. Such connections are found in various fields of knowledge: in mechanics, physics, chemistry, biology, economics, etc.

There are ordinary differential equations and partial differential equations. More complex are integro-differential equations.

First, differential equations arose from the problems of mechanics, in which the coordinates of bodies, their velocities and accelerations, considered as functions of time, participated.

The differential equation is called integrable in quadratures, if the problem of finding all tie-breaks can be reduced to calculating a finite number of integrals of known functions and simple algebraic operations.

Story

Leonard Euler

Joseph Louis Lagrange

Pierre-Simon Laplace

Joseph Liouville

Henri Poincare

Differential equations invented by Newton (1642-1727). Newton considered this invention of his so important that he coded it as an anagram, the meaning of which in modern terms can be freely expressed as: "the laws of nature are expressed by differential equations."

Newton's main analytical achievement was the expansion of all kinds of functions into power series (the meaning of Newton's second, long anagram is that to solve any equation, you need to substitute a series into the equation and equate terms of the same degree). Of particular importance here was the Newton binomial formula discovered by him (of course, not only with integer exponents, for which, for example, Viet (1540-1603) knew the formula, but also, which is especially important, with fractional and negative exponents). Newton decomposed all the basic elementary functions into "Taylor series". This, together with the primitive table he compiled (which passed almost unchanged into modern textbooks of analysis), allowed him, according to him, to compare the areas of any figures "in half a quarter of an hour."

Newton pointed out that the coefficients of his series are proportional to the successive derivatives of the function, but did not dwell on this in detail, since he rightly believed that it was more convenient to carry out all calculations in analysis not with the help of multiple differentiations, but by calculating the first terms of the series. For Newton, the relationship between the coefficients of a series and derivatives was more of a means of calculating derivatives than a means of compiling a series. One of Newton's most important achievements is his theory of the solar system, set forth in the "Mathematical Principles of Natural Philosophy" ("Principia") without the aid of mathematical analysis. It is generally believed that Newton discovered the law of universal gravitation with the help of his analysis. In fact, Newton (1680) owns only the proof of the ellipticity of orbits in the field of attraction according to the inverse square law: this law itself was indicated to Newton by Hooke (1635-1703) and, perhaps, was guessed by several other scientists.

Of the huge number of works of the 18th century on differential equations, the works of Euler (1707-1783) and Lagrange (1736-1813) stand out. In these works, the theory of small oscillations was first developed, and consequently, the theory of linear systems of differential equations; along the way, the basic concepts of linear algebra arose (eigenvalues ​​and vectors in the n-dimensional case). The characteristic equation of a linear operator has long been called secular, since it is from such an equation that secular (age-related, i.e., slow compared to annual motion) perturbations of planetary orbits are determined according to the theory of small oscillations of Lagrange. Following Newton, Laplace and Lagrange, and later Gauss (1777-1855), also developed the methods of perturbation theory.

When the unsolvability of algebraic equations in radicals was proved, Joseph Liouville (1809-1882) built a similar theory for differential equations, establishing the impossibility of solving a number of equations (in particular, such classical ones as second-order linear equations) in elementary functions and quadratures. Later, Sophus Lie (1842-1899), analyzing the question of integrating equations in quadratures, came to the need to study in detail the groups of dipheomorphisms (which later received the name of Lie groups) - this is how one of the most fruitful areas of modern mathematics arose in the theory of differential equations, the further development of which was is closely related to completely different issues (Lie algebras were considered even earlier by Simeon-Denis Poisson (1781-1840) and, especially, Carl Gustav Jacob Jacobi (1804-1851)).

A new stage in the development of the theory of differential equations begins with the work of Henri Poincare (1854-1912), the "qualitative theory of differential equations" he created, together with the theory of functions of complex variables, led to the foundation of modern topology. The qualitative theory of differential equations, or, as it is now more commonly called, the theory of dynamical systems, is currently developing most actively and has the most important applications of the theory of differential equations in natural science.

Ordinary differential equations

Ordinary differential equations are equations of the form F (t, x, x ", x "",..., x(n)) = 0 , Where x = x (t) is an unknown function (possibly a vector function; in this case one often speaks of a system of differential equations) depending on the time variable t, prime means differentiation with respect to t. Number n is called the order of the differential equation.

A solution (or solution) of a differential equation is a function that is differentiated n times, and satisfies the equation at all points of its domain of definition. Usually there are a whole set of such functions, and to choose one of the outcomes, one must impose additional conditions on it: for example, require that the decisions take a certain value at a certain point.

The main problems and results of the theory of differential equations: the existence and uniqueness of the solution of various problems for ODEs, methods for expanding simple ODEs, a qualitative study of solutions to ODEs without finding their explicit form.

Partial Differential Equations

Partial Differential Equations are equations containing unknown functions of several variables and their partial derivatives.

The general form of such equations can be represented as:

,

where are independent variables and is a function of these variables.

Nonlinear differential equations

Nonlinear differential equations - a branch of mathematics that studies the theory and methods for solving nonlinear equations containing the desired function and its derivatives of various orders of one argument (ordinary nonlinear differential equations) or several arguments (nonlinear partial differential equations). Differential equations are widely used in practice, in particular, to describe transient processes.

The theory of nonlinear differential equations is a branch of mathematics that deals with the study of differential equations and related problems. Their results are used in many natural sciences: mechanics, physics, thermoelasticity, optics.

A non-linear differential equation is an equation in which some function is an unknown quantity. The differential equation itself involves not only an unknown function, but also its various derivatives in a non-linear form. A nonlinear differential equation describes the relationship between an unknown function and its derivatives. Such connections are found in various fields of knowledge: in mechanics, physics, chemistry, biology, economics, etc.

A distinction is made between ordinary non-linear differential equations and non-linear partial differential equations.

Nonlinear differential equations arose from the problems of nonlinear mechanics, in which the coordinates of bodies, their velocities and accelerations, considered as functions of time, participated.

Examples

  • Newton's second law can be written in the form of a differential equation
,

Where m- body mass, x- its coordinate, F (x, t) - force acting on a body with coordinate x at the time t. Its solution is the trajectory of the body under the action of the specified force.

  • The vibration of a string is given by the equation
,

Where u = u (x, t) - string deflection at a point with coordinate x at the time t, parameter a sets the properties of the string.


In some problems of physics, a direct connection between the quantities describing the process cannot be established. But there is a possibility to obtain an equality containing the derivatives of the functions under study. This is how differential equations arise and the need to solve them in order to find an unknown function.

This article is intended for those who are faced with the problem of solving a differential equation in which the unknown function is a function of one variable. The theory is built in such a way that with a zero understanding of differential equations, you can do your job.

Each type of differential equations is associated with a solution method with detailed explanations and solutions of typical examples and problems. You just have to determine the type of differential equation of your problem, find a similar analyzed example and carry out similar actions.

To successfully solve differential equations, you will also need the ability to find sets of antiderivatives (indefinite integrals) of various functions. If necessary, we recommend that you refer to the section.

First, consider the types of first-order ordinary differential equations that can be solved with respect to the derivative, then we will move on to second-order ODEs, then we will dwell on higher-order equations and finish with systems of differential equations.

Recall that if y is a function of the argument x .

First order differential equations.

    The simplest differential equations of the first order of the form .

    Let us write down several examples of such DE .

    Differential Equations can be resolved with respect to the derivative by dividing both sides of the equality by f(x) . In this case, we arrive at the equation , which will be equivalent to the original one for f(x) ≠ 0 . Examples of such ODEs are .

    If there are values ​​of the argument x for which the functions f(x) and g(x) simultaneously vanish, then additional solutions appear. Additional solutions to the equation given x are any functions defined for those argument values. Examples of such differential equations are .

Second order differential equations.

    Second Order Linear Homogeneous Differential Equations with Constant Coefficients.

    LODE with constant coefficients is a very common type of differential equations. Their solution is not particularly difficult. First, the roots of the characteristic equation are found . For different p and q, three cases are possible: the roots of the characteristic equation can be real and different, real and coinciding or complex conjugate. Depending on the values ​​of the roots of the characteristic equation, the general solution of the differential equation is written as , or , or respectively.

    For example, consider a second-order linear homogeneous differential equation with constant coefficients. The roots of his characteristic equation are k 1 = -3 and k 2 = 0. The roots are real and different, therefore, the general solution to the LDE with constant coefficients is

    Linear Nonhomogeneous Second Order Differential Equations with Constant Coefficients.

    The general solution of the second-order LIDE with constant coefficients y is sought as the sum of the general solution of the corresponding LODE and a particular solution of the original inhomogeneous equation, that is, . The previous paragraph is devoted to finding a general solution to a homogeneous differential equation with constant coefficients. And a particular solution is determined either by the method of indefinite coefficients for a certain form of the function f (x) , standing on the right side of the original equation, or by the method of variation of arbitrary constants.

    As examples of second-order LIDEs with constant coefficients, we present

    To understand the theory and get acquainted with the detailed solutions of examples, we offer you on the page linear inhomogeneous differential equations of the second order with constant coefficients.

    Linear Homogeneous Differential Equations (LODEs) and second-order linear inhomogeneous differential equations (LNDEs).

    A special case of differential equations of this type are LODE and LODE with constant coefficients.

    The general solution of the LODE on a certain interval is represented by a linear combination of two linearly independent particular solutions y 1 and y 2 of this equation, that is, .

    The main difficulty lies precisely in finding linearly independent partial solutions of this type of differential equation. Usually, particular solutions are chosen from the following systems of linearly independent functions:

    However, particular solutions are not always presented in this form.

    An example of a LODU is .

    The general solution of the LIDE is sought in the form , where is the general solution of the corresponding LODE, and is a particular solution of the original differential equation. We just talked about finding, but it can be determined using the method of variation of arbitrary constants.

    An example of an LNDE is .

Higher order differential equations.

    Differential equations admitting order reduction.

    Order of differential equation , which does not contain the desired function and its derivatives up to k-1 order, can be reduced to n-k by replacing .

    In this case , and the original differential equation reduces to . After finding its solution p(x), it remains to return to the replacement and determine the unknown function y .

    For example, the differential equation after the replacement becomes a separable equation , and its order is reduced from the third to the first.

Often, the mere mention of differential equations makes students feel uncomfortable. Why is this happening? Most often, because when studying the basics of the material, a gap in knowledge arises, due to which the further study of difurs becomes simply torture. Nothing is clear what to do, how to decide where to start?

However, we will try to show you that difurs is not as difficult as it seems.

Basic concepts of the theory of differential equations

From school, we know the simplest equations in which we need to find the unknown x. In fact differential equations only slightly different from them - instead of a variable X they need to find a function y(x) , which will turn the equation into an identity.

Differential equations are of great practical importance. This is not abstract mathematics that has nothing to do with the world around us. With the help of differential equations, many real natural processes are described. For example, string vibrations, the movement of a harmonic oscillator, by means of differential equations in the problems of mechanics, find the speed and acceleration of a body. Also DU are widely used in biology, chemistry, economics and many other sciences.

Differential equation (DU) is an equation containing the derivatives of the function y(x), the function itself, independent variables and other parameters in various combinations.

There are many types of differential equations: ordinary differential equations, linear and non-linear, homogeneous and non-homogeneous, differential equations of the first and higher orders, partial differential equations, and so on.

The solution to a differential equation is a function that turns it into an identity. There are general and particular solutions of remote control.

The general solution of the differential equation is the general set of solutions that turn the equation into an identity. A particular solution of a differential equation is a solution that satisfies additional conditions specified initially.

The order of a differential equation is determined by the highest order of the derivatives included in it.


Ordinary differential equations

Ordinary differential equations are equations containing one independent variable.

Consider the simplest ordinary differential equation of the first order. It looks like:

This equation can be solved by simply integrating its right side.

Examples of such equations:

Separable Variable Equations

In general, this type of equation looks like this:

Here's an example:

Solving such an equation, you need to separate the variables, bringing it to the form:

After that, it remains to integrate both parts and get a solution.


Linear differential equations of the first order

Such equations take the form:

Here p(x) and q(x) are some functions of the independent variable, and y=y(x) is the desired function. Here is an example of such an equation:

When solving such an equation, most often they use the method of variation of an arbitrary constant or represent the desired function as a product of two other functions y(x)=u(x)v(x).

To solve such equations, a certain preparation is required, and it will be quite difficult to take them “on a whim”.

An example of solving a DE with separable variables

So we have considered the simplest types of remote control. Now let's take a look at one of them. Let it be an equation with separable variables.

First, we rewrite the derivative in a more familiar form:

Then we will separate the variables, that is, in one part of the equation we will collect all the “games”, and in the other - the “xes”:

Now it remains to integrate both parts:

We integrate and obtain the general solution of this equation:

Of course, solving differential equations is a kind of art. You need to be able to understand what type an equation belongs to, and also learn to see what transformations you need to make with it in order to bring it to one form or another, not to mention just the ability to differentiate and integrate. And it takes practice (as with everything) to succeed in solving DE. And if at the moment you don’t have time to figure out how differential equations are solved or the Cauchy problem has risen like a bone in your throat or you don’t know how to properly format a presentation, contact our authors. In a short time, we will provide you with a ready-made and detailed solution, the details of which you can understand at any time convenient for you. In the meantime, we suggest watching a video on the topic "How to solve differential equations":

The content of the article

DIFFERENTIAL EQUATIONS. Many physical laws, which are subject to certain phenomena, are written in the form of a mathematical equation that expresses a certain relationship between some quantities. Often we are talking about the relationship between values ​​that change over time, for example, the efficiency of the engine, measured by the distance that a car can travel on one liter of fuel, depends on the speed of the car. The corresponding equation contains one or more functions and their derivatives and is called a differential equation. (The rate of change of distance with time is determined by speed; therefore, speed is a derivative of distance; likewise, acceleration is a derivative of speed, since acceleration sets the rate of change of speed with time.) The importance that differential equations have for mathematics and especially for its applications , are explained by the fact that the study of many physical and technical problems is reduced to solving such equations. Differential equations play an essential role in other sciences, such as biology, economics, and electrical engineering; in fact, they arise wherever there is a need for a quantitative (numerical) description of phenomena (as long as the surrounding world changes in time, and conditions change from one place to another).

Examples.

The following examples provide a better understanding of how various problems are formulated in terms of differential equations.

1) The law of decay of some radioactive substances is that the rate of decay is proportional to the available amount of this substance. If x is the amount of matter at a given point in time t, then this law can be written as follows:

Where dx/dt is the decay rate, and k is some positive constant characterizing the given substance. (The minus sign on the right side indicates that x decreases with time; the plus sign, always implied when the sign is not explicitly stated, would mean that x increases over time.)

2) The container initially contains 10 kg of salt dissolved in 100 m 3 of water. If pure water is poured into a container at a rate of 1 m 3 per minute and is evenly mixed with a solution, and the resulting solution flows out of the container at the same speed, then how much salt will be in the container at any subsequent point in time? If x- the amount of salt (in kg) in the container at the time t, then at any time t 1 m 3 of the solution in the container contains x/100 kg of salt; so the amount of salt decreases at a rate x/100 kg/min, or

3) Let the mass on the body m suspended from the end of a spring, a restoring force acts proportional to the amount of tension in the spring. Let x- the amount of deviation of the body from the equilibrium position. Then, according to Newton's second law, which states that acceleration (the second derivative of x in time, denoted d 2 x/dt 2) in proportion to strength:

The right side is with a minus sign because the restoring force reduces the extension of the spring.

4) The law of body cooling states that the amount of heat in the body decreases in proportion to the temperature difference between the body and the environment. If a cup of coffee heated to a temperature of 90 ° C is in a room whose temperature is 20 ° C, then

Where T– coffee temperature at the time t.

5) The Minister of Foreign Affairs of the State of Blefuscu claims that the armaments program adopted by Lilliput is forcing his country to increase military spending as much as possible. Similar statements are made by the Minister of Foreign Affairs of Lilliput. The resulting situation (in its simplest interpretation) can be accurately described by two differential equations. Let x And y- the cost of arming Lilliput and Blefuscu. Assuming that Lilliputia increases its armament spending at a rate proportional to the rate of increase in Blefuscu's armament spending, and vice versa, we get:

where the members are ax And - by describe the military spending of each country, k And l are positive constants. (This problem was first formulated in this way in 1939 by L. Richardson.)

After the problem is written in the language of differential equations, one should try to solve them, i.e. find quantities whose rates of change are included in the equations. Sometimes the solutions are found in the form of explicit formulas, but more often they can be represented only in an approximate form or obtain qualitative information about them. It is often difficult to establish whether a solution exists at all, let alone find one. An important part of the theory of differential equations is the so-called "existence theorems", which prove the existence of a solution for one or another type of differential equations.

The original mathematical formulation of a physical problem usually contains simplifying assumptions; the criterion of their reasonableness can be the degree of consistency of the mathematical solution with the available observations.

Solutions of differential equations.

Differential equation, for example dy/dx = x/y, satisfies not a number, but a function, in this particular case such that its graph at any point, for example, at a point with coordinates (2,3), has a tangent with a slope equal to the ratio of the coordinates (in our example 2/3). This is easy to verify if a large number of points are constructed and a short segment with a corresponding slope is drawn from each. The solution will be a function whose graph touches each of its points on the corresponding segment. If there are enough points and segments, then we can approximately outline the course of decision curves (three such curves are shown in Fig. 1). There is exactly one solution curve passing through every point with y No. 0. Each individual solution is called a particular solution of the differential equation; if it is possible to find a formula containing all particular solutions (with the possible exception of a few special ones), then we say that a general solution has been obtained. A particular solution is a single function, while a general solution is a whole family of them. To solve a differential equation means to find either its particular or general solution. In our example, the general solution has the form y 2 – x 2 = c, Where c- any number; the particular solution passing through the point (1,1) has the form y = x and is obtained when c= 0; the particular solution passing through the point (2.1) has the form y 2 – x 2 = 3. The condition requiring that the solution curve pass, for example, through the point (2,1), is called the initial condition (because it specifies the starting point on the solution curve).

It can be shown that in example (1) the general solution has the form x = ce –kt, Where c- a constant that can be determined, for example, by indicating the amount of substance at t= 0. The equation from example (2) is a special case of the equation from example (1), corresponding to k= 1/100. Initial condition x= 10 at t= 0 gives a particular solution x = 10e –t/100 . The equation from example (4) has a general solution T = 70 + ce –kt and a particular solution 70 + 130 – kt; to determine the value k, additional data is needed.

Differential equation dy/dx = x/y is called a first-order equation, since it contains the first derivative (it is customary to consider the order of the highest derivative included in it as the order of a differential equation). For most (though not all) differential equations of the first kind that arise in practice, only one solution curve passes through each point.

There are several important types of first-order differential equations that can be solved in the form of formulas containing only elementary functions - powers, exponents, logarithms, sines and cosines, etc. These equations include the following.

Equations with separable variables.

Equations of the form dy/dx = f(x)/g(y) can be solved by writing it in differentials g(y)dy = f(x)dx and integrating both parts. In the worst case, the solution can be represented as integrals of known functions. For example, in the case of the equation dy/dx = x/y we have f(x) = x, g(y) = y. By writing it in the form ydy = xdx and integrating, we get y 2 = x 2 + c. The equations with separable variables include the equations from examples (1), (2), (4) (they can be solved by the method described above).

Equations in total differentials.

If the differential equation has the form dy/dx = M(x,y)/N(x,y), Where M And N are two given functions, it can be represented as M(x,y)dx – N(x,y)dy= 0. If the left side is the differential of some function F(x,y), then the differential equation can be written as dF(x,y) = 0, which is equivalent to the equation F(x,y) = const. Thus, equation-solution curves are "lines of constant levels" of a function, or locus of points that satisfy the equations F(x,y) = c. The equation ydy = xdx(Fig. 1) - with separable variables, and it is the same - in total differentials: to make sure of the latter, we write it in the form ydy – xdx= 0, i.e. d(y 2 – x 2) = 0. Function F(x,y) in this case is equal to (1/2)( y 2 – x 2); some of its constant level lines are shown in Fig. 1.

Linear equations.

Linear equations are "first degree" equations - the unknown function and its derivatives are included in such equations only in the first degree. Thus, the first-order linear differential equation has the form dy/dx + p(x) = q(x), Where p(x) And q(x) are functions depending only on x. Its solution can always be written using integrals of known functions. Many other types of first-order differential equations are solved using special techniques.

Equations of higher orders.

Many of the differential equations that physicists deal with are second-order equations (i.e., equations containing second derivatives). Such, for example, is the simple harmonic motion equation from example (3), md 2 x/dt 2 = –kx. Generally speaking, one would expect a second-order equation to have particular solutions satisfying two conditions; for example, one may require that the solution curve pass through a given point in a given direction. In cases where the differential equation contains some parameter (a number whose value depends on circumstances), solutions of the required type exist only for certain values ​​of this parameter. For example, consider the equation md 2 x/dt 2 = –kx and we require that y(0) = y(1) = 0. Function yє 0 is certainly a solution, but if is an integer multiple p, i.e. k = m 2 n 2 p 2, where n is an integer, and in fact only in this case, there are other solutions, namely: y= sin npx. The parameter values ​​for which the equation has special solutions are called characteristic or eigenvalues; they play an important role in many tasks.

The equation of simple harmonic motion exemplifies an important class of equations, namely linear differential equations with constant coefficients. A more general example (also second order) is the equation

Where a And b are given constants, f(x) is a given function. Such equations can be solved in various ways, for example, using the Laplace integral transform. The same can be said about linear equations of higher orders with constant coefficients. Linear equations with variable coefficients also play a significant role.

Nonlinear differential equations.

Equations containing unknown functions and their derivatives higher than the first or in some more complex way are called non-linear. In recent years, they have attracted more and more attention. The point is that physical equations are usually linear only in the first approximation; further and more accurate investigation, as a rule, requires the use of non-linear equations. In addition, many problems are inherently non-linear. Since the solutions of nonlinear equations are often very complex and difficult to represent with simple formulas, a significant part of modern theory is devoted to a qualitative analysis of their behavior, i.e. the development of methods that make it possible, without solving the equations, to say something significant about the nature of the solutions as a whole: for example, that they are all limited, or have a periodic character, or depend in a certain way on the coefficients.

Approximate solutions of differential equations can be found numerically, but this takes a lot of time. With the advent of high-speed computers, this time has been greatly reduced, which has opened up new possibilities for the numerical solution of many problems that were previously not amenable to such a solution.

Existence theorems.

An existence theorem is a theorem stating that under certain conditions a given differential equation has a solution. There are differential equations that do not have solutions or have more solutions than expected. The purpose of the existence theorem is to convince us that a given equation does have a solution, and most often to assure that it has exactly one solution of the required type. For example, the equation we have already encountered dy/dx = –2y has exactly one solution passing through every point of the plane ( x,y), and since we have already found one such solution, we have completely solved this equation. On the other hand, the equation ( dy/dx) 2 = 1 – y 2 has many solutions. Among them are direct y = 1, y= –1 and curves y= sin( x + c). The solution may consist of several segments of these straight lines and curves, passing into each other at the points of contact (Fig. 2).

Partial Differential Equations.

An ordinary differential equation is a statement about the derivative of an unknown function of one variable. A partial differential equation contains a function of two or more variables and the derivatives of that function in at least two different variables.

In physics, examples of such equations are the Laplace equation

X , y) inside the circle if the values u are given at each point of the bounding circle. Since problems with more than one variable in physics are the rule rather than the exception, it is easy to imagine how broad the subject of the theory of partial differential equations is.

A differential equation is an equation that includes a function and one or more of its derivatives. In most practical problems, functions are physical quantities, derivatives correspond to the rates of change of these quantities, and the equation determines the relationship between them.


This article discusses methods for solving some types of ordinary differential equations, the solutions of which can be written in the form elementary functions, that is, polynomial, exponential, logarithmic and trigonometric functions, as well as their inverse functions. Many of these equations occur in real life, although most other differential equations cannot be solved by these methods, and for them the answer is written as special functions or power series, or found by numerical methods.


To understand this article, you need to know differential and integral calculus, as well as have some understanding of partial derivatives. It is also recommended to know the basics of linear algebra as applied to differential equations, especially second-order differential equations, although knowledge of differential and integral calculus is sufficient to solve them.

Preliminary information

  • Differential equations have an extensive classification. This article talks about ordinary differential equations, that is, about equations that include a function of one variable and its derivatives. Ordinary differential equations are much easier to understand and solve than partial differential equations, which include functions of several variables. This article does not consider partial differential equations, since the methods for solving these equations are usually determined by their specific form.
    • Below are some examples of ordinary differential equations.
      • d y d x = k y (\displaystyle (\frac ((\mathrm (d) )y)((\mathrm (d) )x))=ky)
      • d 2 x d t 2 + k x = 0 (\displaystyle (\frac ((\mathrm (d) )^(2)x)((\mathrm (d) )t^(2)))+kx=0)
    • Below are some examples of partial differential equations.
      • ∂ 2 f ∂ x 2 + ∂ 2 f ∂ y 2 = 0 (\displaystyle (\frac (\partial ^(2)f)(\partial x^(2)))+(\frac (\partial ^(2 )f)(\partial y^(2)))=0)
      • ∂ u ∂ t − α ∂ 2 u ∂ x 2 = 0 (\displaystyle (\frac (\partial u)(\partial t))-\alpha (\frac (\partial ^(2)u)(\partial x ^(2)))=0)
  • Order differential equation is determined by the order of the highest derivative included in this equation. The first of the above ordinary differential equations is of the first order, while the second is of the second order. Degree of a differential equation is called the highest power to which one of the terms of this equation is raised.
    • For example, the equation below is third order and second power.
      • (d 3 y d x 3) 2 + d y d x = 0 (\displaystyle \left((\frac ((\mathrm (d) )^(3)y)((\mathrm (d) )x^(3)))\ right)^(2)+(\frac ((\mathrm (d) )y)((\mathrm (d) )x))=0)
  • The differential equation is linear differential equation if the function and all its derivatives are in the first power. Otherwise, the equation is nonlinear differential equation. Linear differential equations are remarkable in that linear combinations can be made from their solutions, which will also be solutions to this equation.
    • Below are some examples of linear differential equations.
      • d y d x + p (x) y = q (x) (\displaystyle (\frac ((\mathrm (d) )y)((\mathrm (d) )x))+p(x)y=q(x) )
      • x 2 d 2 y d x 2 + a x d y d x + b y = 0 (\displaystyle x^(2)(\frac ((\mathrm (d) )^(2)y)((\mathrm (d) )x^(2) ))+ax(\frac ((\mathrm (d) )y)((\mathrm (d) )x))+by=0)
    • Below are some examples of non-linear differential equations. The first equation is non-linear due to the sine term.
      • d 2 θ d t 2 + g l sin ⁡ θ = 0 (\displaystyle (\frac ((\mathrm (d) )^(2)\theta )((\mathrm (d) )t^(2)))+( \frac (g)(l))\sin \theta =0)
      • d 2 x d t 2 + (d x d t) 2 + t x 2 = 0 (\displaystyle (\frac ((\mathrm (d) )^(2)x)((\mathrm (d) )t^(2)))+ \left((\frac ((\mathrm (d) )x)((\mathrm (d) )t))\right)^(2)+tx^(2)=0)
  • Common decision ordinary differential equation is not unique, it includes arbitrary constants of integration. In most cases, the number of arbitrary constants is equal to the order of the equation. In practice, the values ​​of these constants are determined by given initial conditions, that is, by the values ​​of the function and its derivatives at x = 0. (\displaystyle x=0.) The number of initial conditions that are needed to find private decision differential equation, in most cases is also equal to the order of this equation.
    • For example, this article will look at solving the equation below. This is a second order linear differential equation. Its general solution contains two arbitrary constants. To find these constants, it is necessary to know the initial conditions at x (0) (\displaystyle x(0)) And x′ (0) . (\displaystyle x"(0).) Usually the initial conditions are given at the point x = 0 , (\displaystyle x=0,), although this is not required. This article will also consider how to find particular solutions for given initial conditions.
      • d 2 x d t 2 + k 2 x = 0 (\displaystyle (\frac ((\mathrm (d) )^(2)x)((\mathrm (d) )t^(2)))+k^(2 )x=0)
      • x (t) = c 1 cos ⁡ k x + c 2 sin ⁡ k x (\displaystyle x(t)=c_(1)\cos kx+c_(2)\sin kx)

Steps

Part 1

First order equations

When using this service, some information may be transferred to YouTube.

This page has been viewed 69,354 times.

Was this article helpful?