Find particular solutions that satisfy the given initial conditions. Differential equations

6.1. BASIC CONCEPTS AND DEFINITIONS

When solving various problems in mathematics and physics, biology and medicine, quite often it is not possible to immediately establish a functional relationship in the form of a formula connecting the variables that describe the process under study. Usually you have to use equations that contain, in addition to the independent variable and the unknown function, also its derivatives.

Definition. An equation connecting an independent variable, an unknown function and its derivatives of various orders is called differential.

An unknown function is usually denoted y(x) or simply y, and its derivatives - y", y" etc.

Other designations are also possible, for example: if y= x(t), then x"(t), x""(t)- its derivatives, and t- independent variable.

Definition. If a function depends on one variable, then the differential equation is called ordinary. General form ordinary differential equation:

or

Functions F And f may not contain some arguments, but for the equations to be differential, the presence of a derivative is essential.

Definition.The order of the differential equation is called the order of the highest derivative included in it.

For example, x 2 y"- y= 0, y" + sin x= 0 are first order equations, and y"+ 2 y"+ 5 y= x- second order equation.

When solving differential equations, the integration operation is used, which is associated with the appearance of an arbitrary constant. If the integration action is applied n times, then, obviously, the solution will contain n arbitrary constants.

6.2. DIFFERENTIAL EQUATIONS OF THE FIRST ORDER

General form first order differential equation is determined by the expression

The equation may not explicitly contain x And y, but necessarily contains y".

If the equation can be written as

then we obtain a first-order differential equation resolved with respect to the derivative.

Definition. The general solution of the first order differential equation (6.3) (or (6.4)) is the set of solutions , Where WITH- arbitrary constant.

The graph of the solution to a differential equation is called integral curve.

Giving an arbitrary constant WITH different values, partial solutions can be obtained. On surface xOycommon decision represents a family of integral curves corresponding to each particular solution.

If you set a point A (x 0 , y 0), through which the integral curve must pass, then, as a rule, from a set of functions One can single out one - a private solution.

Definition.Private decision of a differential equation is its solution that does not contain arbitrary constants.

If is a general solution, then from the condition

you can find a constant WITH. The condition is called initial condition.

The problem of finding a particular solution to the differential equation (6.3) or (6.4) satisfying the initial condition at called Cauchy problem. Does this problem always have a solution? The answer is contained in the following theorem.

Cauchy's theorem(theorem of existence and uniqueness of a solution). Let in the differential equation y"= f(x,y) function f(x,y) and her

partial derivative defined and continuous in some

region D, containing a point Then in the area D exists

the only solution to the equation that satisfies the initial condition at

Cauchy's theorem states that under certain conditions there is a unique integral curve y= f(x), passing through a point Points at which the conditions of the theorem are not met

Cauchies are called special. At these points it breaks f(x, y) or.

Either several integral curves or none pass through a singular point.

Definition. If the solution (6.3), (6.4) is found in the form f(x, y, C)= 0, not allowed relative to y, then it is called general integral differential equation.

Cauchy's theorem only guarantees that a solution exists. Since there is no single method for finding a solution, we will consider only some types of first-order differential equations that can be integrated into quadratures

Definition. The differential equation is called integrable in quadratures, if finding its solution comes down to integrating functions.

6.2.1. First order differential equations with separable variables

Definition. A first order differential equation is called an equation with separable variables,

The right side of equation (6.5) is the product of two functions, each of which depends on only one variable.

For example, the equation is an equation with separating

mixed with variables
and the equation

cannot be represented in the form (6.5).

Considering that , we rewrite (6.5) in the form

From this equation we obtain a differential equation with separated variables, in which the differentials are functions that depend only on the corresponding variable:

Integrating term by term, we have

where C = C 2 - C 1 - arbitrary constant. Expression (6.6) is the general integral of equation (6.5).

By dividing both sides of equation (6.5) by, we can lose those solutions for which, Indeed, if at

That obviously is a solution to equation (6.5).

Example 1. Find a solution to the equation that satisfies

condition: y= 6 at x= 2 (y(2) = 6).

Solution. We will replace y" then . Multiply both sides by

dx, since during further integration it is impossible to leave dx in the denominator:

and then dividing both parts by we get the equation,

which can be integrated. Let's integrate:

Then ; potentiating, we get y = C. (x + 1) - ob-

general solution.

Using the initial data, we determine an arbitrary constant, substituting them into the general solution

Finally we get y= 2(x + 1) is a particular solution. Let's look at a few more examples of solving equations with separable variables.

Example 2. Find the solution to the equation

Solution. Considering that , we get .

Integrating both sides of the equation, we have

where

Example 3. Find the solution to the equation Solution. We divide both sides of the equation into those factors that depend on a variable that does not coincide with the variable under the differential sign, i.e. and integrate. Then we get

and finally

Example 4. Find the solution to the equation

Solution. Knowing what we will get. Section

lim variables. Then

Integrating, we get

Comment. In examples 1 and 2, the desired function y expressed explicitly (general solution). In examples 3 and 4 - implicitly (general integral). In the future, the form of the decision will not be specified.

Example 5. Find the solution to the equation Solution.

Example 6. Find the solution to the equation , satisfying

condition y(e)= 1.

Solution. Let's write the equation in the form

Multiplying both sides of the equation by dx and on, we get

Integrating both sides of the equation (the integral on the right side is taken by parts), we obtain

But according to the condition y= 1 at x= e. Then

Let's substitute the found values WITH to the general solution:

The resulting expression is called a partial solution of the differential equation.

6.2.2. Homogeneous differential equations of the first order

Definition. The first order differential equation is called homogeneous, if it can be represented in the form

Let us present the solution algorithm homogeneous equation.

1.Instead y let's introduce a new functionThen and therefore

2.In terms of function u equation (6.7) takes the form

that is, the replacement reduces a homogeneous equation to an equation with separable variables.

3. Solving equation (6.8), we first find u and then y= ux.

Example 1. Solve the equation Solution. Let's write the equation in the form

We make the substitution:
Then

We will replace

Multiply by dx: Divide by x and on Then

Having integrated both sides of the equation over the corresponding variables, we have

or, returning to the old variables, we finally get

Example 2.Solve the equation Solution.Let Then

Let's divide both sides of the equation by x2: Let's open the brackets and rearrange the terms:

Moving on to the old variables, we arrive at the final result:

Example 3.Find the solution to the equation given that

Solution.Performing a standard replacement we get

or

or

This means that the particular solution has the form Example 4. Find the solution to the equation

Solution.

Example 5.Find the solution to the equation Solution.

Independent work

Find solutions to differential equations with separable variables (1-9).

Find a solution to homogeneous differential equations (9-18).

6.2.3. Some applications of first order differential equations

Radioactive decay problem

The rate of decay of Ra (radium) at each moment of time is proportional to its available mass. Find the law radioactive decay Ra, if it is known that at the initial moment there was Ra and the half-life of Ra is 1590 years.

Solution. Let at the instant the mass Ra be x= x(t) g, and Then the decay rate Ra is equal to

According to the conditions of the problem

Where k

Separating the variables in the last equation and integrating, we get

where

For determining C we use the initial condition: when .

Then and, therefore,

Proportionality factor k determined from the additional condition:

We have

From here and the required formula

Bacterial reproduction rate problem

The rate of reproduction of bacteria is proportional to their number. At the beginning there were 100 bacteria. Within 3 hours their number doubled. Find the dependence of the number of bacteria on time. How many times will the number of bacteria increase within 9 hours?

Solution. Let x- number of bacteria at a time t. Then, according to the condition,

Where k- proportionality coefficient.

From here From the condition it is known that . Means,

From the additional condition . Then

The function you are looking for:

So, when t= 9 x= 800, i.e. within 9 hours the number of bacteria increased 8 times.

The problem of increasing the amount of enzyme

In a brewer's yeast culture, the rate of growth of the active enzyme is proportional to its initial amount x. Initial amount of enzyme a doubled within an hour. Find dependency

x(t).

Solution. By condition, the differential equation of the process has the form

from here

But . Means, C= a and then

It is also known that

Hence,

6.3. SECOND ORDER DIFFERENTIAL EQUATIONS

6.3.1. Basic Concepts

Definition.Second order differential equation is called a relation connecting the independent variable, the desired function and its first and second derivatives.

In special cases, x may be missing from the equation, at or y". However, a second-order equation must necessarily contain y." In the general case, a second-order differential equation is written as:

or, if possible, in the form resolved with respect to the second derivative:

As in the case of a first-order equation, for a second-order equation there can be general and particular solutions. The general solution is:

Finding a Particular Solution

under initial conditions - given

numbers) is called Cauchy problem. Geometrically, this means that we need to find the integral curve at= y(x), passing through a given point and having a tangent at this point which is

aligns with the positive axis direction Ox specified angle. e. (Fig. 6.1). The Cauchy problem has a unique solution if right part equations (6.10), incessant

is discontinuous and has continuous partial derivatives with respect to uh, uh" in some neighborhood of the starting point

To find constants included in a private solution, the system must be resolved

Rice. 6.1. Integral curve

Let us consider a linear homogeneous equation of the second order, i.e. the equation

and establish some properties of its solutions.

Property 1
If is a solution to a linear homogeneous equation, then C, Where C- an arbitrary constant, is a solution to the same equation.
Proof.
Substituting into the left side of the equation under consideration C, we get: ,
but, because is a solution to the original equation.
Hence,

and the validity of this property has been proven.

Property 2
The sum of two solutions to a linear homogeneous equation is a solution to the same equation.
Proof.
Let and be solutions of the equation under consideration, then
And .
Now substituting + into the equation under consideration we will have:
, i.e. + is the solution to the original equation.
From the proven properties it follows that, knowing two particular solutions of a linear homogeneous second-order equation, we can obtain the solution , depending on two arbitrary constants, i.e. from the number of constants that the second order equation must contain a general solution. But will this decision be general, i.e. Is it possible to satisfy arbitrarily given initial conditions by choosing arbitrary constants?
When answering this question, we will use the concept of linear independence of functions, which can be defined as follows.

The two functions are called linearly independent on a certain interval, if their ratio on this interval is not constant, i.e. If
.
Otherwise the functions are called linearly dependent.
In other words, two functions are said to be linearly dependent on a certain interval if on the entire interval.

Examples

1. Functions y 1 = e x and y 2 = e -x are linearly independent for all values ​​of x, because
.
2. Functions y 1 = e x and y 2 = 5 e x linearly dependent, because
.

Theorem 1.

If the functions and are linearly dependent on a certain interval, then the determinant is called Vronsky's determinant given functions is identically equal to zero on this interval.

Proof.

If
,
where , then and .
Hence,
.
The theorem has been proven.

Comment.
The Wronski determinant, which appears in the theorem considered, is usually denoted by the letter W or symbols.
If the functions are solutions of a linear homogeneous equation of the second order, then the following converse and, moreover, stronger theorem is valid for them.

Theorem 2.

If the Wronski determinant, compiled for solutions and a linear homogeneous equation of the second order, vanishes at least at one point, then these solutions are linearly dependent.

Proof.

Let the Wronski determinant vanish at the point , i.e. =0,
and let and .
Consider a linear homogeneous system

relatively unknown and .
The determinant of this system coincides with the value of the Wronski determinant at
x=, i.e. coincides with , and therefore equals zero. Therefore, the system has a non-zero solution and ( and are not equal to zero). Using these values ​​and , consider the function . This function is a solution to the same equation as the and functions. In addition, this function satisfies zero initial conditions: , because And .
On the other hand, it is obvious that the solution to the equation satisfying the zero initial conditions is the function y=0.
Due to the uniqueness of the solution, we have: . Whence it follows that
,
those. functions and are linearly dependent. The theorem has been proven.

Consequences.

1. If the Wronski determinant appearing in the theorems is equal to zero for some value x=, then it is equal to zero for any value xfrom the considered interval.

2. If the solutions are linearly independent, then the Wronski determinant does not vanish at any point in the interval under consideration.

3. If the Wronski determinant is nonzero at least at one point, then the solutions are linearly independent.

Theorem 3.

If and are two linearly independent solutions of a homogeneous second-order equation, then the function , where and are arbitrary constants, is a general solution to this equation.

Proof.

As is known, the function is a solution to the equation under consideration for any values ​​of and . Let us now prove that whatever the initial conditions
And ,
it is possible to select the values ​​of arbitrary constants and so that the corresponding particular solution satisfies the given initial conditions.
Substituting the initial conditions into the equalities, we obtain a system of equations
.
From this system it is possible to determine and , since determinant of this system

there is a Wronski determinant for x= and, therefore, is not equal to zero (due to the linear independence of the solutions and ).

; .

A particular solution with the obtained values ​​and satisfies the given initial conditions. Thus, the theorem is proven.

Examples

Example 1.

The general solution to the equation is the solution .
Really,
.

Therefore, the functions sinx and cosx are linearly independent. This can be verified by considering the relationship of these functions:

Example 2.

Solution y = C 1 e x +C 2 e -x equation is general, because .

Example 3.

The equation , whose coefficients and
continuous on any interval not containing the point x = 0, admits partial solutions

(easy to check by substitution). Therefore, its general solution has the form:
.

Comment

We have established that the general solution of a linear homogeneous second-order equation can be obtained by knowing any two linearly independent partial solutions of this equation. However, there are no general methods for finding such partial solutions in final form for equations with variable coefficients. For equations with constant coefficients, such a method exists and will be discussed later.

Today, one of the most important skills for any specialist is the ability to solve differential equations. Solving differential equations - not a single applied task can do without this, be it calculating any physical parameter or modeling changes as a result of an accepted decision. 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 of different equations in algebra, but these are, for the most part, educational examples, not applicable in practice. The truly interesting mathematics begins when we want to describe the processes occurring in real life. But how can we reflect the time factor that governs real processes—inflation, output, or demographic indicators?

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

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

The simplest differential equation has the form $y’(x)=f(x)$, where $f(x)$ is a certain function, and $y’(x)$ is the derivative or rate of change of the desired function. It can be solved by ordinary integration: $$y(x)=\int f(x)dx.$$

Second simplest type is called a differential equation with separable variables. Such an equation looks like this: $y’(x)=f(x)\cdot g(y)$. It can be seen that the dependent variable $y$ is also part of the constructed function. The equation can be 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 to the differential equation of separable type.

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, special methods are used (Lagrange’s method of variation of an arbitrary constant, Bernoulli’s 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 systems of differential equations. Solving them requires preliminary preparation and experience in solving simpler problems.

The so-called partial differential equations are of great importance for physics and, unexpectedly, finance. This means that the desired 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 securities) depending on its profitability, the size of payments, as well as the start and end dates of payments. Solving a partial differential equation is quite complex, 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. First, let's set the task.

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

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

Let's start with the solution. As is known from microeconomics, marginal revenue is a derivative of total revenue, and revenue is equal to zero at zero level sales

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

Let's integrate the equation by taking antiderivative function from both parts, we 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 $R(0)=0$. Let's 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 is solved.

Other examples by different types Remote controls are collected on the page:

Often just a mention 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 further study of difurs becomes simply torture. It’s not clear what to do, how to decide, where to start?

However, we will try to show you that difurs are 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 you need to find a function in them y(x) , which will turn the equation into an identity.

D differential equations are of great practical importance. This is not abstract mathematics that has no relation to the world around us. Differential equations are used to describe many real natural processes. For example, the vibrations of a string, the movement of a harmonic oscillator, using differential equations in problems of mechanics, find the speed and acceleration of a body. Also DU find wide application in biology, chemistry, economics and many other sciences.

Differential equation (DU) is an equation containing 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 nonlinear, homogeneous and inhomogeneous, first and higher order differential equations, 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 the remote control.

A general solution to a differential equation is a general set of solutions that transform the equation into an identity. A partial 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 its derivatives.

Ordinary differential equations

Ordinary differential equations are equations containing one independent variable.

Let's consider the simplest ordinary differential equation of the first order. It looks like:

Such an equation can be solved by simply integrating its right-hand side.

Examples of such equations:

Separable equations

In general, this type of equation looks like this:

Here's an example:

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

After this, it remains to integrate both parts and obtain a solution.

Linear differential equations of the first order

Such equations look like:

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 varying 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, certain preparation is required and it will be quite difficult to take them “at a glance”.

An example of solving a differential equation with separable variables

So we looked at the simplest types of remote control. Now let's look at the solution to one of them. Let this be an equation with separable variables.

First, let's rewrite the derivative in a more familiar form:

Then we divide the variables, that is, in one part of the equation we collect all the “I’s”, and in the other - the “X’s”:

Now it remains to integrate both parts:

We integrate and obtain a general solution to this equation:

Of course, solving differential equations is a kind of art. You need to be able to understand what type of equation it is, and also learn to see what transformations need to be made with it in order to lead to one form or another, not to mention just the ability to differentiate and integrate. And to succeed in solving DE, you need practice (as in everything). And if you have this moment you don’t have time to figure out how differential equations are solved, or the Cauchy problem has stuck like a bone in your throat, or you don’t know, 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”: