A particular solution to a differential equation satisfying the initial conditions. Differential equations

Application

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Educational institution "Belarusian State

Agricultural Academy"

Department of Higher Mathematics

DIFFERENTIAL EQUATIONS OF THE FIRST ORDER

Lecture notes 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 relationship 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 desired function. In this case, relation (1) must have at least one derivative.

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

Consider the differential equation

. (2)

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

If equation (2) can be resolved 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 a first order differential equation written in differential form.

Because
, then equation (3) can be written in the form
or
, where we can count
And
. This means that equation (3) is converted to equation (4).

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

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

or
.

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

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

General solution 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 a 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 a 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 an infinite number of solutions. In order to select one solution from this set, which is called a private one, you need to set some additional conditions.

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 the given numeric value , if the independent variable x takes the given numeric value , i.e.

,
, (5)

Where D– domain of definition 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)

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 . Let's write the equation in the form
or
. Let's integrate both sides of the resulting equation:
,
. We'll finally write it down
.

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

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

Equation

(8)

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

Example . Find the general solution to the equation
.

Solution . Let's write this equation in the form:
or
. Then
,
,
,
. Thus,
is the general solution of this equation.

Equation of the form

(9)

integrates 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 – the function of at and differential dy. To do this, both sides of the equation need to be multiplied by dx and divide by
. As a result, we obtain the equation

, (10)

in which the variables X And at separated. Let's 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 is the general integral of this equation.
.

Let the equation be given in the form

This equation is called first order differential equation with separable variables in a symmetrical form.

To separate the variables, you need to divide both sides of the equation by
:

. (12)

The resulting equation is called separated differential equation . Let's integrate equation (12):

.(13)

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

Example 4 . Integrate a differential equation.

Solution . Let's write the equation in the form

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

,
,

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

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

Solution . Considering 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
. Let's substitute it into the general integral and find WITH:
,WITH=1. Then the expression
is a partial solution of a given differential equation, written as a partial integral.

Linear differential equations of the first order

Equation

(14)

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

If
, then the equation

(15)

called linear homogeneous . If
, 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.

We will look for a solution to equation (14) in the form of a product of two functions

, (16)

Where
And
- some continuous functions. Let's substitute
and derivative
into equation (14):

Function v we will select in such a way that the condition is satisfied
. 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
you can take one of the partial solutions of the homogeneous equation, i.e. at WITH=1:
. Let's substitute into the second equation of the system:
or
.Then
. Thus, the general solution to a first-order linear differential equation has the form
.

Example 6 . Solve the equation
.

Solution . We will look for a solution to the equation in the form
. Then
. Let's substitute into the equation:

or
. Function v choose in such a way that the equality holds
. Then
. Let's solve the first of these equations using the method of separation of variables:
,
,
,
,. Function v Let's 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 to a differential equation?

What is an integral curve?

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

What is called a partial 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 to write a differential equation with separable variables 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?

Tasks for independent work

Solve differential equations with separable variables:

A)
; b)
;

V)
; G)
.

2. Solve first order linear differential equations:

A)
; b)
; V)
;

G)
; d)
.

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

The order of the differential equation is called 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, for the sake of brevity, we will omit the word “ordinary”.

Examples of differential equations:

(1) ;

(3) ;

(4) ;

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

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

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

Solving a differential equation every function is called y = f(x), when substituted 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 the solution to the differential equation.

Solution. Let's write this equation in the form . The solution is to find the function from its derivative. The original function, as is known from integral calculus, is an antiderivative for, i.e.

This is it solution to this differential equation . Changing in it C, we will obtain different solutions. We found out that there is 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 to the differential equation in Example 1 is general.

Partial solution of the differential equation a solution in which arbitrary constants are given specific numerical values is called.

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

Solution. Let's integrate both sides of the equation a number of times equal to the order of the differential equation.

As a result, we received a general solution -

of a given third order differential equation.

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

If, in addition to the differential equation, the initial condition is given in the form , then such a problem is called Cauchy problem . Substitute the values and into the general solution of the equation and find the value of an arbitrary constant 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 subject to .

Solution. Let us substitute the values from the initial condition into the general solution y = 3, x= 1. We get

We write down the solution to the Cauchy problem for this first-order differential equation:

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

Example 4. Find the general solution to the differential equation.

Solution. The equation is written in such a form that you can immediately integrate both sides.

We apply the method of integration by change of variable (substitution). Let it be then.

Required to take dx and now - attention - we do this according to the rules of differentiation of a complex function, since x and there is a complex function (“apple” is the extraction of a square root or, which is the same thing, raising to the power “one-half”, and “minced meat” is the very expression under the root):

We find the integral:

Returning to the variable x, we get:

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

Not only skills from 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. Knowledge about proportions from school that has not been forgotten (however, depending on who) from school will help solve this problem. This is the next example.

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

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.