The simplest differential equations of the first order are examples. First order differential equations

Educational Institution "Belarusian State

agricultural Academy"

Department of Higher Mathematics

FIRST ORDER DIFFERENTIAL EQUATIONS

Lecture summary for accounting students

correspondence form of education (NISPO)

Gorki, 2013

First order differential equations

The concept of a differential equation. General and particular solutions

When studying various phenomena, it is often not possible to find a law that directly connects the independent variable and the desired function, but it is possible to establish a connection between the desired function and its derivatives.

The relation connecting the independent variable, the desired function and its derivatives is called differential equation :

Here x is an independent variable, y is the desired function,
are the derivatives of the desired function. In this case, relation (1) requires the presence of at least one derivative.

The order of the differential equation is the order of the highest derivative in the equation.

Consider the differential equation

. (2)

Since this equation includes a derivative of only the first order, then it is called is a first-order differential equation.

If equation (2) can be solved with respect to the derivative and written as

, (3)

then such an equation is called a first-order differential equation in normal form.

In many cases it is expedient to consider an equation of the form

which is called a first-order differential equation written in differential form.

As
, then equation (3) can be written as
or
, where one can count
and
. This means that equation (3) has been converted to equation (4).

We write equation (4) in the form
. Then
,
,
, where one can count
, i.e. an equation of the form (3) is obtained. Thus, equations (3) and (4) are equivalent.

By solving the differential equation (2) or (3) any function is called
, which, when substituting it into equation (2) or (3), turns it into an identity:

or
.

The process of finding all solutions of a differential equation is called its integration , and the solution graph
differential equation is called integral curve this equation.

If the solution of the differential equation is obtained in implicit form
, then it is called integral given differential equation.

General solution differential equation of the first order is a family of functions of the form
, depending on an arbitrary constant With, each of which is a solution of the given differential equation for any admissible value of an arbitrary constant With. Thus, the differential equation has an infinite number of solutions.

Private decision differential equation is called the solution obtained from the general solution formula for a specific value of an arbitrary constant With, including
.

The Cauchy problem and its geometric interpretation

Equation (2) has an infinite number of solutions. In order to single out one solution from this set, which is called a particular solution, some additional conditions must be specified.

The problem of finding a particular solution to equation (2) under given conditions is called Cauchy problem . This problem is one of the most important in the theory of differential equations.

The Cauchy problem is formulated as follows: among all solutions of equation (2) find such a solution
, in which the function
takes a given numeric value if the independent variable x takes a given numeric value , i.e.

,
, (5)

where D is the domain of the function
.

Meaning called the initial value of the function , a – initial value of the independent variable . Condition (5) is called initial condition or Cauchy condition .

From a geometric point of view, the Cauchy problem for differential equation (2) can be formulated as follows: from the set of integral curves of equation (2) select the one that passes through a given point
.

Differential equations with separable variables

One of the simplest types of differential equations is a first-order differential equation that does not contain the desired function:

. (6)

Given that
, we write the equation in the form
or
. Integrating both sides of the last equation, we get:
or

. (7)

Thus, (7) is a general solution to equation (6).

Example 1 . Find the general solution of the differential equation
.

Decision . We write the equation in the form
or
. We integrate both parts of the resulting equation:
,
. Let's finally write down
.

Example 2 . Find a solution to the equation
given that
.

Decision . Let's find the general solution of the equation:
,
,
,
. By condition
,
. Substitute in the general solution:
or
. We substitute the found value of an arbitrary constant into the formula for the general solution:
. This is the particular solution of the differential equation that satisfies the given condition.

The equation

(8)

called a first-order differential equation that does not contain an independent variable . We write it in the form
or
. We integrate both parts of the last equation:
or
- general solution of equation (8).

Example . Find a general solution to the equation
.

Decision . We write this equation in the form:
or
. Then
,
,
,
. Thus,
is the general solution of this equation.

Type equation

(9)

integrated using separation of variables. To do this, we write the equation in the form
, and then, using the operations of multiplication and division, we bring it to such a form that one part includes only the function of X and differential dx, and in the second part - a function of at and differential dy. To do this, both sides of the equation must be multiplied by dx and divide by
. As a result, we obtain the equation

, (10)

in which the variables X and at separated. We integrate both parts of equation (10):
. The resulting relation is the general integral of equation (9).

Example 3 . Integrate Equation
.

Decision . Transform the equation and separate the variables:
,
. Let's integrate:
,
or is the general integral of this equation.
.

Let the equation be given in the form

Such an equation is called first-order differential equation with separable variables in symmetrical form.

To separate the variables, both sides of the equation must be divided by
:

. (12)

The resulting equation is called separated differential equation . We integrate equation (12):

.(13)

Relation (13) is a general integral of differential equation (11).

Example 4 . Integrate the differential equation.

Decision . We write the equation in the form

and divide both parts into
,
. The resulting equation:
is a separated variable equation. Let's integrate it:

,
,

,
. The last equality is the general integral of the given differential equation.

Example 5 . Find a particular solution of a differential equation
, satisfying the condition
.

Decision . Given that
, we write the equation in the form
or
. Let's separate the variables:
. Let's integrate this equation:
,
,
. The resulting relation is the general integral of this equation. By condition
. Substitute into the general integral and find With:
,With=1. Then the expression
is a particular solution of the given differential equation, written as a particular integral.

Linear differential equations of the first order

The equation

(14)

called linear differential equation of the first order . unknown function
and its derivative enter this equation linearly, and the functions
and
continuous.

If a
, then the equation

(15)

called linear homogeneous . If a
, then equation (14) is called linear inhomogeneous .

To find a solution to equation (14), one usually uses substitution method (Bernoulli) , the essence of which is as follows.

The solution of equation (14) will be sought in the form of a product of two functions

, (16)

where
and
- some continuous functions. Substitute
and derivative
into equation (14):

Function v will be chosen in such a way that the condition
. Then
. Thus, to find a solution to equation (14), it is necessary to solve the system of differential equations

The first equation of the system is a linear homogeneous equation and it can be solved by the method of separation of variables:
,
,
,
,
. As a function
one can take one of the particular solutions of the homogeneous equation, i.e. at With=1:
. Substitute into the second equation of the system:
or
.Then
. Thus, the general solution of a first-order linear differential equation has the form
.

Example 6 . solve the equation
.

Decision . We will seek the solution of the equation in the form
. Then
. Substitute into the equation:

or
. Function v choose in such a way that the equality
. Then
. We solve the first of these equations by the method of separation of variables:
,
,
,
,. Function v Substitute into the second equation:
,
,
,
. The general solution to this equation is
.

Questions for self-control of knowledge

What is a differential equation?

What is the order of a differential equation?

Which differential equation is called a first order differential equation?

How is a first-order differential equation written in differential form?

What is the solution of a differential equation?

What is an integral curve?

What is the general solution of a first order differential equation?

What is a particular solution of a differential equation?

How is the Cauchy problem formulated for a first-order differential equation?

What is the geometric interpretation of the Cauchy problem?

How is a differential equation written with separable variables in symmetric form?

Which equation is called a first-order linear differential equation?

What method can be used to solve a linear differential equation of the first order and what is the essence of this method?

Tasks for independent work

Solve differential equations with separable variables:

a)
; b)
;

in)
; G)
.

2. Solve first order linear differential equations:

a)
; b)
; in)
;

G)
; e)
.

Either already solved with respect to the derivative, or they can be solved with respect to the derivative .

General solution of differential equations of the type on the interval X, which is given, can be found by taking the integral of both sides of this equality.

Get .

If we look at the properties of the indefinite integral, we find the desired general solution:

y = F(x) + C,

where F(x)- one of the antiderivatives of the function f(x) in between X, a With is an arbitrary constant.

Please note that in most tasks the interval X do not indicate. This means that a solution must be found for everyone. x, for which and the desired function y, and the original equation make sense.

If you need to calculate a particular solution of a differential equation that satisfies the initial condition y(x0) = y0, then after calculating the general integral y = F(x) + C, it is still necessary to determine the value of the constant C=C0 using the initial condition. That is, a constant C=C0 determined from the equation F(x 0) + C = y 0, and the desired particular solution of the differential equation will take the form:

y = F(x) + C0.

Consider an example:

Find the general solution of the differential equation , check the correctness of the result. Let's find a particular solution of this equation that would satisfy the initial condition .

Decision:

After we have integrated the given differential equation, we get:

.

We take this integral by the method of integration by parts:

That., is a general solution of the differential equation.

Let's check to make sure the result is correct. To do this, we substitute the solution that we found into the given equation:

.

That is, at the original equation turns into an identity:

therefore, the general solution of the differential equation was determined correctly.

The solution we found is the general solution of the differential equation for each real value of the argument x.

It remains to calculate a particular solution of the ODE that would satisfy the initial condition . In other words, it is necessary to calculate the value of the constant With, at which the equality will be true:

.

.

Then, substituting C = 2 into the general solution of the ODE, we obtain a particular solution of the differential equation that satisfies the initial condition:

.

Ordinary differential equation can be solved with respect to the derivative by dividing the 2 parts of the equation by f(x). This transformation will be equivalent if f(x) does not go to zero for any x from the interval of integration of the differential equation X.

Situations are likely when, for some values ​​of the argument x ∈ X functions f(x) and g(x) turn to zero at the same time. For similar values x the general solution of the differential equation is any function y, which is defined in them, because .

If for some values ​​of the argument x ∈ X the condition is satisfied, which means that in this case the ODE has no solutions.

For all others x from interval X the general solution of the differential equation is determined from the transformed equation.

Let's look at examples:

Example 1

Let us find the general solution of the ODE: .

Decision.

From the properties of the basic elementary functions, it is clear that the natural logarithm function is defined for non-negative values ​​of the argument, therefore, the domain of the expression log(x+3) there is an interval x > -3 . Hence, the given differential equation makes sense for x > -3 . With these values ​​of the argument, the expression x + 3 does not vanish, so one can solve the ODE with respect to the derivative by dividing the 2 parts by x + 3.

We get .

Next, we integrate the resulting differential equation, solved with respect to the derivative: . To take this integral, we use the method of subsuming under the sign of the differential.

Ordinary differential equation called an equation that connects an independent variable, an unknown function of this variable and its derivatives (or differentials) of various orders.

The order of the differential equation is the order of the highest derivative contained in it.

In addition to ordinary ones, partial differential equations are also studied. These are equations relating independent variables, an unknown function of these variables and its partial derivatives with respect to the same variables. But we will only consider ordinary differential equations and therefore we will omit the word "ordinary" for brevity.

Examples of differential equations:

(1) ;

(3) ;

(4) ;

Equation (1) is of the fourth order, equation (2) is of the third order, equations (3) and (4) are of the second order, equation (5) is of the first order.

Differential equation n order does not have to explicitly contain a function, all its derivatives from first to n th order and an independent variable. It may not explicitly contain derivatives of some orders, a function, an independent variable.

For example, in equation (1) there are clearly no derivatives of the third and second orders, as well as functions; in equation (2) - second-order derivative and function; in equation (4) - independent variable; in equation (5) - functions. Only equation (3) explicitly contains all derivatives, the function, and the independent variable.

By solving the differential equation any function is called y = f(x), substituting which into the equation, it turns into an identity.

The process of finding a solution to a differential equation is called its integration.

Example 1 Find a solution to the differential equation.

Decision. We write this equation in the form . The solution is to find the function by its derivative. The original function, as is known from the integral calculus, is the antiderivative for, i.e.

That's what it is solution of the given differential equation . changing in it C, we will get different solutions. We found out that there are an infinite number of solutions to a first-order differential equation.

General solution of the differential equation n th order is its solution expressed explicitly with respect to the unknown function and containing n independent arbitrary constants, i.e.

The solution of the differential equation in example 1 is general.

Partial solution of the differential equation its solution is called, in which specific numerical values ​​are assigned to arbitrary constants.

Example 2 Find the general solution of the differential equation and a particular solution for .

Decision. We integrate both parts of the equation such a number of times that the order of the differential equation is equal.

,

.

As a result, we got the general solution -

given third-order differential equation.

Now let's find a particular solution under the specified conditions. To do this, we substitute their values ​​instead of arbitrary coefficients and obtain

.

If, in addition to the differential equation, the initial condition is given in the form , then such a problem is called Cauchy problem . The values ​​and are substituted into the general solution of the equation and the value of an arbitrary constant is found C, and then a particular solution of the equation for the found value C. This is the solution to the Cauchy problem.

Example 3 Solve the Cauchy problem for the differential equation from Example 1 under the condition .

Decision. We substitute into the general solution the values ​​from the initial condition y = 3, x= 1. We get

We write down the solution of the Cauchy problem for the given differential equation of the first order:

Solving differential equations, even the simplest ones, requires good skills in integrating and taking derivatives, including complex functions. This can be seen in the following example.

Example 4 Find the general solution of the differential equation.

Decision. The equation is written in such a form that both sides can be integrated immediately.

.

We apply the method of integration by changing the variable (substitution). Let , then .

Required to take dx and now - attention - we do it according to the rules of differentiation of a complex function, since x and there is a complex function ("apple" - extracting the square root or, which is the same - raising to the power "one second", and "minced meat" - the expression itself under the root):

We find the integral:

Returning to the variable x, we get:

.

This is the general solution of this differential equation of the first degree.

Not only skills from the previous sections of higher mathematics will be required in solving differential equations, but also skills from elementary, that is, school mathematics. As already mentioned, in a differential equation of any order there may not be an independent variable, that is, a variable x. The knowledge about proportions that has not been forgotten (however, anyone has it like) from the school bench will help to solve this problem. This is the next example.

The content of the article

DIFFERENTIAL EQUATIONS. Many physical laws that govern certain phenomena are written in the form of a mathematical equation expressing a certain relationship between some quantities. Often we are talking about the relationship between values ​​that change over time, for example, the efficiency of an 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; similarly, 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 soon 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 a 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 a 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 be 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 be 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 only be represented 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 section 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 laid off 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 allow solutions 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 verify 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. one.

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 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.

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, we consider the types of ordinary differential equations of the first order that can be solved with respect to the derivative, then we move on to second-order ODEs, then we 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.