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.

We get .

Looking at the properties indefinite integral, then 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.

Note that for most tasks, the interval X do not indicate. This means that a solution must be found for everyone. x for which the required function y, and the original equation makes sense.

If you need to calculate a particular solution of a differential equation that satisfies the initial condition y (x 0) = y 0, then after calculating the general integral y = F (x) + C, it is also necessary to determine the value of the constant C = C 0 using the initial condition. That is, the constant C = C 0 determined from the equation F (x 0) + C = y 0, and the sought-for particular solution of the differential equation takes the form:

y = F (x) + C 0.

Let's consider an example:

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

Solution:

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

.

Let us take this integral by the method of integration by parts:

That., is an common decision differential equation.

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

.

That is, for the original equation becomes an identity:

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

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

It remains to calculate a particular solution to 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 for the derivative by dividing the 2 parts of the equality by f (x)... This transformation will be equivalent if f (x) does not vanish 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) simultaneously vanish. For similar values x the general solution of the differential equation will be any function y, which is defined in them, since ...

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 take a look at the examples:

Example 1.

Let's find the general solution to the ODE: .

Solution.

Of the properties of the main elementary functions it is clear that the function natural logarithm is defined for non-negative argument values, so the scope of the expression ln (x + 3) there is an interval x > -3 ... Hence, the given differential equation makes sense for x > -3 ... For 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 bringing the differential under the sign.

Educational institution "Belarusian State

agricultural Academy"

Department higher mathematics

DIFFERENTIAL EQUATIONS OF THE FIRST ORDER

Lecture notes for accounting students

extramural education (NISPO)

Gorki, 2013

First order differential equations

Differential equation concept. General and specific 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- independent variable, y- the required function,
- derivatives of the required function. In this case, the presence of at least one derivative is required in relation (1).

The order of the differential equation is called the order of the highest derivative entering the equation.

Consider the differential equation

. (2)

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

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

, (3)

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

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

which is called differential equation of the first order, written in differential form.

Because
, then equation (3) can be written in the form
or
where can be considered
and
... This means that equation (3) is transformed into equation (4).

We write equation (4) in the form
... Then
,
,
where can be considered
, 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 substituted into equation (2) or (3), turns it into an identity:

or
.

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

If the solution to the differential equation is obtained implicitly
, then it is called integral this differential equation.

By general decision of a first-order differential equation is a family of functions of the form
depending on an arbitrary constant WITH, each of which is a solution to this differential equation for any admissible value of an arbitrary constant WITH... Thus, the differential equation has countless solutions.

By private decision differential equation is called the solution obtained from the general solution formula for a specific value of an arbitrary constant WITH including
.

Cauchy problem and its geometric interpretation

Equation (2) has countless solutions. In order to single out one solution from this set, which is called a particular solution, it is necessary to set some additional conditions.

The problem of finding a particular solution of equation (2) under given conditions is called the 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 the given numerical value if the independent variable x takes the given numerical value , i.e.

,
, (5)

where D- function definition area
.

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

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

Separable Differential Equations

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

. (6)

Considering 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 to the differential equation
.

Solution ... We write the equation in the form
or
... We integrate both sides of the resulting equation:
,
... We will finally write down
.

Example 2 ... Find a solution to the equation
provided
.

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

The equation

(8)

Called differential equation of the first order that does not contain an independent variable ... Let's write it in the form
or
... We integrate both sides of the last equation:
or
- general solution of equation (8).

Example ... Find the general solution to the equation
.

Solution ... We write this equation in the form:
or
... Then
,
,
,
... In this way,
- common decision this equation.

Equation of the form

(9)

integrates using variable separation. For this, we write the equation in the form
, and then, using the operations of multiplication and division, we reduce it to such a form that 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 split into
... As a result, we obtain the equation

, (10)

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

Example 3 ... Integrate Equation
.

Solution ... Let's transform the equation and separate the variables:
,
... Let's integrate:
,
or - the general integral of the given 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, you need to divide both sides of the equation by
:

. (12)

The resulting equation is called differential equation with separated variables ... Let us integrate equation (12):

.(13)

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

Example 4 ... Integrate the differential equation.

Solution ... We write the equation in the form

and divide both parts into
,
... The resulting equation:
is an equation with separated variables. Let's integrate it:

,
,

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

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

Solution ... Considering that
, we write the equation in the form
or
... Let's split the variables:
... Let's integrate this equation:
,
,
... The resulting relationship is the general integral of this equation. By condition
... Substitute in the general integral and find WITH:
,WITH= 1. Then the expression
is a particular solution of this differential equation, written in the form of a particular integral.

Linear differential equations of the first order

The equation

(14)

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

If
, then the equation

(15)

called linear homogeneous ... If
, then equation (14) is called linear non-uniform .

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

The solution to 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 selected 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 can be solved by the method of separation of variables:
,
,
,
,
... As a function
one of the particular solutions of the homogeneous equation can be taken, i.e. at WITH=1:
... Let's substitute in 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
.

Solution ... We will seek a solution to the equation in the form
... Then
... Substitute into the equation:

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

Questions for self-control of knowledge

What is called a differential equation?

What is called the order of a differential equation?

What differential equation is called a first order differential equation?

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

What is called a solution to a differential equation?

What is called an integral curve?

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

What is called a particular solution of a differential equation?

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

What is the geometric interpretation of the Cauchy problem?

How is a separable differential equation written in symmetric form?

Which equation is called a first order linear differential equation?

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

Self-study assignments

Solve differential equations with separable variables:

a)
; b)
;

v)
; G)
.

2. Solve linear differential equations of the first order:

a)
; b)
; v)
;

G)
; e)
.

Solving differential equations. Thanks to our online service, you can solve differential equations of any kind and complexity: inhomogeneous, homogeneous, nonlinear, linear, first, second order, with separable or non-separable variables, etc. You get the solution of differential equations in analytical form with detailed description... Many people wonder: why do you need to solve differential equations online? This kind equations are very common in mathematics and physics, where it will be impossible to solve many problems without calculating the differential equation. Differential equations are also common in economics, medicine, biology, chemistry and other sciences. Solving such an equation online greatly facilitates your assigned tasks, makes it possible to better assimilate the material and test yourself. Benefits of solving differential equations online. A modern mathematical service site allows you to solve differential equations online of any complexity. As you know, there are a large number of types of differential equations and each of them has its own solutions. On our service you can find solutions to differential equations of any order and type online. To obtain a solution, we suggest you fill in the initial data and click the "Solution" button. Errors in the service are excluded, so you can be 100% sure that you received the correct answer. Solve differential equations with our service. Solve differential equations online. By default, in such an equation, the function y is a function of the x variable. But you can also specify your own variable designation. For example, if you specify y (t) in the differential equation, then our service will automatically determine that y is a function of the t variable. The order of the entire differential equation will depend on the maximum order of the derivative of the function present in the equation. To solve such an equation means to find the required function. Our service will help you solve differential equations online. It doesn't take much effort on your part to solve the equation. You just need to enter the left and right sides of your equation in the required fields and click the "Solution" button. When entering, the derivative of a function must be denoted with an apostrophe. In a matter of seconds, you will receive a ready detailed solution differential equation. Our service is absolutely free. Differential Equations with separable variables. If in a differential equation on the left side there is an expression that depends on y, and on the right side there is an expression that depends on x, then such a differential equation is called with separable variables. On the left side there can be a derivative of y, the solution of differential equations of this kind will be in the form of a function y, expressed through the integral of the right side of the equation. If the differential of the function of y is on the left-hand side, then both sides of the equation are integrated. When the variables in a differential equation are not separated, they will need to be split to obtain a split differential equation. Linear differential equation. A linear differential equation is a differential equation in which the function and all its derivatives are in the first degree. General form equations: y '+ a1 (x) y = f (x). f (x) and a1 (x) are continuous functions of x. The solution of differential equations of this type is reduced to the integration of two differential equations with separated variables. The order of the differential equation. The differential equation can be of the first, second, n-th order. The order of a differential equation determines the order of the highest derivative it contains. In our service you can solve differential equations online first, second, third, etc. order. The solution to the equation will be any function y = f (x), substituting it into the equation, you get the identity. The process of finding a solution to a differential equation is called integration. Cauchy problem. If, in addition to the differential equation itself, the initial condition y (x0) = y0 is specified, then this is called the Cauchy problem. The indices y0 and x0 are added to the solution of the equation and determine the value of an arbitrary constant C, and then a particular solution of the equation at this value of C. This is the solution to the Cauchy problem. The Cauchy problem is also called a problem with boundary conditions, which is very common in physics and mechanics. You also have the opportunity to set the Cauchy problem, that is, of all possible solutions the equation to choose a quotient that meets the given initial conditions.

Today, one of the most important skills for any specialist is the ability to solve differential equations. Solving differential equations - no applied problem can do without it, be it the calculation of a physical parameter or the modeling of changes as a result of the adopted macroeconomic policy. These equations are also important for a number of other sciences such as chemistry, biology, medicine, etc. Below we will give an example of the use of differential equations in economics, but before that, we will briefly talk about the main types of equations.

Differential equations - the simplest types

The sages said that the laws of our universe are written in mathematical language. Of course there are many examples in algebra different equations, but these are, for the most part, educational examples that are not applicable in practice. The really interesting mathematics begins when we want to describe the processes taking place in real life... But how to reflect the time factor to which real processes are subject - inflation, production or demographic indicators?

Let us recall one important definition from the course of mathematics concerning the derivative of a function. The derivative is the rate of change of a function, therefore, it can help us reflect the time factor in the equation.

That is, we compose an equation with a function that describes the indicator of interest to us and add the derivative of this function to the equation. This is the differential equation. Now let's move on to the simplest types of differential equations for dummies.

The simplest differential equation has the form $ y ’(x) = f (x) $, where $ f (x) $ is some function, and $ y’ (x) $ is the derivative or rate of change of the required function. It is solved by the usual integration: $$ y (x) = \ int f (x) dx. $$

Second simplest type is called a differential equation with separable variables. Such an equation looks like this $ y ’(x) = f (x) \ cdot g (y) $. You can see that the dependent variable $ y $ is also part of the constructed function. The equation is solved very simply - you need to "separate the variables", that is, bring it to the form $ y ’(x) / g (y) = f (x) $ or $ dy / g (y) = f (x) dx $. It remains to integrate both sides $$ \ int \ frac (dy) (g (y)) = \ int f (x) dx $$ - this is the solution of a separable differential equation.

The last simple type is a first order linear differential equation. It has the form $ y ’+ p (x) y = q (x) $. Here $ p (x) $ and $ q (x) $ are some functions, and $ y = y (x) $ is the required function. To solve such an equation, they already use special methods(Lagrange method of variation of an arbitrary constant, Bernoulli substitution method).

There are more complex types of equations - equations of the second, third, and generally arbitrary order, homogeneous and inhomogeneous equations, as well as a system of differential equations. To solve them you need preliminary preparation and experience in solving simpler problems.

Of great importance for physics and, unexpectedly, finance are the so-called partial differential equations. This means that the required function depends on several variables at the same time. For example, the Black-Scholes equation from the field of financial engineering describes the value of an option (type of security) depending on its yield, the amount of payments, and the timing of the beginning and end of payments. The solution of a partial differential equation is quite difficult, usually you need to use special programs such as Matlab or Maple.

An example of the application of a differential equation in economics

Let us give, as promised, a simple example of solving a differential equation. Let's set the task first.

For a certain firm, the function of marginal revenue from the sale of its products has the form $ MR = 10-0.2q $. Here, $ MR $ is the company's marginal revenue, and $ q $ is the volume of production. You need to find the total revenue.

As you can see from the problem, this is an applied example from microeconomics. Many firms and enterprises are constantly faced with such calculations in the course of their activities.

Let's get down to the solution. As is known from microeconomics, marginal revenue is a derivative of total revenue, and revenue is zero when zero level sales.

From a mathematical point of view, the problem has been reduced to solving the differential equation $ R '= 10-0.2q $ under the condition $ R (0) = 0 $.

Let us integrate the equation, taking the antiderivative function of both sides, and obtain the general solution: $$ R (q) = \ int (10-0.2q) dq = 10 q-0.1q ^ 2 + C. $$

To find the constant $ C $, recall the condition $ R (0) = 0 $. Substitute: $$ R (0) = 0-0 + C = 0. $$ So C = 0 and our total revenue function takes the form $ R (q) = 10q-0.1q ^ 2 $. The problem has been solved.

Other examples on different types DUs are collected on the page:

Instructions

If the equation is presented in the form: dy / dx = q (x) / n (y), refer them to the category of differential equations with separable variables. They can be solved by writing the condition in the differentials as follows: n (y) dy = q (x) dx. Then integrate both parts. In some cases, the solution is written in the form of integrals taken from known functions. For example, in the case dy / dx = x / y, you get q (x) = x, n (y) = y. Write it down as ydy = xdx and integrate. You should get y ^ 2 = x ^ 2 + c.

Linear equations relate the equations "first". An unknown function with its derivatives is included in such an equation only to the first degree. The linear one has the form dy / dx + f (x) = j (x), where f (x) and g (x) are functions depending on x. The solution is written using integrals taken from known functions.

Please note that many differential equations are second-order equations (containing second derivatives). For example, there is an equation of simple harmonic motion written as a general one: md 2x / dt 2 = –kx. Such equations have, in, particular solutions. The equation of simple harmonic motion is an example of something quite important: linear differential equations that have constant coefficient.

If in the conditions of the problem there is only one linear equation, so you are given additional conditions, thanks to which a solution can be found. Read the problem carefully to find these conditions. If variables x and y indicate distance, speed, weight - feel free to set the limit x≥0 and y≥0. It is quite possible that the number of apples, etc. is hidden under x or y. - then only values ​​can be. If x is the age of the son, it is clear that he cannot be older than the father, so indicate this in the conditions of the problem.

Sources:

• how to solve an equation in one variable

Differential and integral calculus problems are important elements consolidation of the theory of mathematical analysis, a section of higher mathematics studied in universities. Differential the equation is solved by the integration method.

Instructions

Differential calculus explores properties. Conversely, integration of a function allows for given properties, i.e. derivatives or differentials of a function find it itself. This is the solution to the differential equation.

Any is the relationship between the unknown and the known data. In the case of a differential equation, the role of the unknown is played by a function, and the role of known quantities is played by its derivatives. In addition, the relation can contain an independent variable: F (x, y (x), y '(x), y' '(x), ..., y ^ n (x)) = 0, where x is an unknown variable, y (x) is the function to be determined, the order of the equation is the maximum order of the derivative (n).

Such an equation is called an ordinary differential equation. If there are several independent variables in the relation and the partial derivatives (differentials) of the function with respect to these variables, then the equation is called a partial differential equation and has the form: x∂z / ∂y - ∂z / ∂x = 0, where z (x, y) is the required function.

So, in order to learn how to solve differential equations, you need to be able to find antiderivatives, i.e. solve the problem inverse to differentiation. For example: Solve the first order equation y '= -y / x.

Solution Replace y 'with dy / dx: dy / dx = -y / x.

Reduce the equation to a form convenient for integration. To do this, multiply both sides by dx and divide by y: dy / y = -dx / x.

Integrate: ∫dy / y = - ∫dx / x + Сln | y | = - ln | x | + C.

This solution is called a general differential equation. C is a constant, the set of values ​​of which determines the set of solutions to the equation. For any specific value of C, the solution will be unique. This solution is a particular solution to the differential equation.

Solving most equations of higher degrees does not have a clear formula like finding the roots of a square equations... However, there are several casting methods that allow you to transform the equation the highest degree to a more visual form.

Instructions

The most common method for solving higher-order equations is expansion. This approach is a combination of the selection of integer roots, divisors of the intercept, and the subsequent division of the general polynomial by the form (x - x0).

For example, solve the equation x ^ 4 + x³ + 2 · x² - x - 3 = 0. Solution: The free term of this polynomial is -3, therefore, its integer divisors can be ± 1 and ± 3. Substitute them one by one into the equation and find out if you get the identity: 1: 1 + 1 + 2 - 1 - 3 = 0.

The second root is x = -1. Divide by expression (x + 1). Write down the resulting equation (x - 1) · (x + 1) · (x² + x + 3) = 0. The degree has decreased to the second, therefore, the equation can have two more roots. To find them, solve the quadratic equation: x² + x + 3 = 0D = 1 - 12 = -11

The discriminant is negative, which means that the equation no longer has real roots. Find the complex roots of the equation: x = (-2 + i √11) / 2 and x = (-2 - i √11) / 2.

Another method for solving an equation of the highest degree is by changing variables to bring it to the square. This approach is used when all powers of the equation are even, for example: x ^ 4 - 13 x² + 36 = 0

Now find the roots of the original equation: x1 = √9 = ± 3; x2 = √4 = ± 2.

Tip 10: How to Determine Redox Equations

A chemical reaction is a process of transformation of substances that occurs with a change in their composition. Those substances that enter into the reaction are called initial, and those that are formed as a result of this process are called products. It so happens that during chemical reaction the elements that make up the starting materials change their oxidation state. That is, they can accept other people's electrons and give up their own. And in fact, and in another case, their charge changes. These reactions are called redox reactions.