Generalized homogeneous equation. Homogeneous differential equations of the first order Generalized homogeneous equations of the second order

13.07.2020

def 1 type control

called homogeneous differential equation of the first order(ODE).

Th1 Let the following conditions be satisfied for the function:

1) continuous at

Then ODE (1) has a common integral, which for is given by the formula:

where is some antiderivative of the function With is an arbitrary constant.

Remark 1 If, for some, the condition is satisfied, then in the process of solving ODE (1), solutions of the form may be lost; such cases should be treated more carefully and each of them should be checked separately.

Thus from the theorem Th1 should general algorithm for solving ODE (1):

1) Make a replacement:

2) Thus, a DE with separable variables will be obtained, which should be integrated;

3) Return to the old g variables;

4) Check the values for their involvement in the solution original remote control, under which the condition

5) Write down the answer.

Example 1 Solve DE (4).

Solution: DE (4) is a homogeneous differential equation, since it has the form (1). Let's make the replacement (3), this will bring the equation (4) to the form:

Equation (5) is the general integral of DE (4).

Note that when separating variables and dividing by, solutions could be lost, but it is not a solution to DE (4), which is easily verified by direct substitution into equality (4), since this value is not included in the domain of definition of the original DE.

Answer:

Remark 2 Sometimes one can write ODEs in terms of differentials of variables X and y. It is recommended to pass from this DE notation to the expression through the derivative and only then perform the replacement (3).

Differential equations reducing to homogeneous ones.

def 2 The function is called homogeneous function of degree k in the area of, for which the equality will be fulfilled:

Here are the most common types of DE that can be reduced to the form (1) after various transformations.

1) where is the function is homogeneous, zero degree, that is, the following equality is true: DE (6) can be easily reduced to the form (1) if we put , which is further integrated using the replacement (3).

2) (7), where the functions are homogeneous of the same degree k . The DE of the form (7) is also integrated using the change (3).

Example 2 Solve DE (8).

Solution: Let us show that DE (8) is homogeneous. We divide by what is possible, since it is not a solution to the differential equation (8).

Let's make the replacement (3), this will bring the equation (9) to the form:

Equation (10) is the general integral of DE (8).

Note that when separating variables and dividing by , the solutions corresponding to the values of and could be lost. Let's check these expressions. Let's substitute them into DE (8):

Answer:

It is interesting to note that when solving this example, a function appears called the "sign" of the number X(read " signum x”), defined by the expression:

Remark 3 It is not necessary to bring DE (6) or (7) to the form (1), if it is obvious that the DE is homogeneous, then it is possible to immediately replace

3) The DE of the form (11) is integrated as an ODE if , while the substitution is initially performed:

(12), where is the solution of the system: (13), and then use the replacement (3) for the function. After obtaining the general integral, return to the variables X and at.

If , then, assuming in equation (11), we obtain a DE with separable variables.

Example 3 Solve the Cauchy problem (14).

Solution: Let us show that DE (14) is reduced to a homogeneous DE and integrated according to the above scheme:

We will solve an inhomogeneous system of linear algebraic equations(15) Cramer's method:

We make a change of variables and integrate the resulting equation:

(16) – General integral of DE (14). When dividing variables, solutions could be lost when dividing by an expression , which can be obtained explicitly after solving quadratic equation. However, they are taken into account in the general integral (16) at

Let us find a solution to the Cauchy problem: we substitute the values of and into the general integral (16) and find With.

Thus, the partial integral will be given by the formula:

Answer:

4) It is possible to lead some DEs to homogeneous ones for a new, yet unknown function, if we apply a substitution of the form:

At the same time, the number m is selected from the condition that the resulting equation, if possible, becomes homogeneous to some degree. However, if this cannot be done, then the considered DE cannot be reduced to a homogeneous one in this way.

Example 4 Solve DU. (eighteen)

Solution: Let us show that DE (18) is reduced to a homogeneous DE using substitution (17) and then integrated using replacement (3):

Let's find With:

Thus, a particular solution of DE (24) has the form

First order differential equations with separable variables.

Definition. A differential equation with separable variables is an equation of the form (3.1) or an equation of the form (3.2)

In order to separate the variables in equation (3.1), i.e. reduce this equation to the so-called equation with separated variables, perform the following actions: ;

Now we need to solve the equation g(y)=0. If it has a real solution y=a, then y=a will also be a solution of equation (3.1).

Equation (3.2) is reduced to an equation with separated variables by dividing by the product:

, which allows us to obtain the general integral of equation (3.2): . (3.3)

The integral curves (3.3) will be supplemented by the solutions if such solutions exist.

Homogeneous differential equations of the 1st order.

Definition 1. An equation of the 1st order is called homogeneous if the relation , called the homogeneity condition for a function of two variables of zero dimension.

Example 1 Show that the function is homogeneous of zero dimension.

Solution. ,

Q.E.D.

Theorem. Any function is homogeneous and, conversely, any homogeneous function of zero dimension is reduced to the form .

Proof. The first assertion of the theorem is obvious, since . Let us prove the second assertion. Let , then for a homogeneous function , which was to be proved.

Definition 2. Equation (4.1) in which M and N are homogeneous functions of the same degree, i.e. have the property for all , is called homogeneous. Obviously, this equation can always be reduced to the form (4.2) , although this may not be done to solve it. A homogeneous equation is reduced to an equation with separable variables by replacing the desired function y according to the formula y=zx, where z(x) is the new desired function. Having performed this substitution in equation (4.2), we obtain: or or .

Integrating, we obtain the general integral of the equation with respect to the function z(x) , which after repeated replacement gives the general integral of the original equation. In addition, if are the roots of the equation , then the functions are solutions of a homogeneous given equation. If , then equation (4.2) takes the form

And it becomes an equation with separable variables. Its solutions are half-lines: .

Comment. Sometimes it is advisable instead of the above substitution to use the substitution x=zy.

Generalized homogeneous equation.

The equation M(x,y)dx+N(x,y)dy=0 is called generalized homogeneous if it is possible to choose such a number k that the left side of this equation becomes a homogeneous function of some degree m relatively x, y, dx and dy provided that x is considered the value of the first measurement, y – k- th measurement , dx and dy- zero and (k-1) th measurements. For example, this would be the equation . (6.1) Indeed, under the assumption made about measurements x, y, dx and dy members of the left side and dy will have respectively dimensions -2, 2 k and k-one. Equating them, we obtain the condition that the desired number must satisfy k: -2 = 2k=k-one. This condition is satisfied when k= -1 (with such k all terms on the left side of the equation under consideration will have dimension -2). Consequently, equation (6.1) is generalized homogeneous.

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§ 7. Linear differential equations of the first order.

A linear equation of the 1st order is an equation that is linear with respect to the desired function and its derivative. It looks like:

, (7.1)

where P(x) and Q(x) are given continuous functions of x. If the function
, then equation (7.1) has the form:
(7.2)

and is called a linear homogeneous equation, otherwise
it is called a linear inhomogeneous equation.

The linear homogeneous differential equation (7.2) is an equation with separable variables:

(7.3)

Expression (7.3) is the general solution of equation (7.2). To find a general solution of equation (7.1) in which the function P(x) denotes the same function as in equation (7.2), we apply the method called the method of variation of an arbitrary constant and consists in the following: we will try to choose the function C=C(x) so that the general solution of the linear homogeneous equation (7.2) would be the solution of the inhomogeneous linear equation (7.1). Then for the derivative of function (7.3) we get:

Substituting the found derivative into equation (7.1), we will have:

or
.

Where
, where is an arbitrary constant. As a result, the general solution of the inhomogeneous linear equation (7.1) will be (7.4)

The first term in this formula represents the general solution (7.3) of the linear homogeneous differential equation (7.2), and the second term in formula (7.4) is a particular solution of the linear inhomogeneous equation (7.1) obtained from the general (7.4) with
. Let us single out this important conclusion in the form of a theorem.

Theorem. If one particular solution of a linear inhomogeneous differential equation is known
, then all other solutions have the form
, where
is the general solution of the corresponding linear homogeneous differential equation.

However, it should be noted that another method, sometimes called the Bernoulli method, is more often used to solve the linear inhomogeneous differential equation of the 1st order (7.1). We will look for a solution to equation (7.1) in the form
. Then
. We substitute the found derivative into the original equation:
.

Let us combine, for example, the second and third terms of the last expression and take out the function u(x) for brackets:
(7.5)

We require the parenthesis to vanish:
.

We solve this equation by setting an arbitrary constant C equal to zero:
. With found function v(x) back to equation (7.5):
.

Solving it, we get:
.

Consequently, the general solution of equation (7.1) has the form.

.
Differential equations.

§ 1. Basic concepts of ordinary differential equations.

Definition 1. Ordinary differential equation n-th order for the function y argument x is called a relation of the form

where F is a given function of its arguments. In the name of this class of mathematical equations, the term "differential" emphasizes that they include derivatives
(functions formed as a result of differentiation); the term - "ordinary" says that the desired function depends on only one real argument.

An ordinary differential equation may not explicitly contain an argument x, desired function
and any of its derivatives, but the highest derivative
must be included in the equation n- order. for instance

a)
is the first order equation;

b)
is a third order equation.

When writing ordinary differential equations, the notation of derivatives through differentials is often used:

v)
is a second order equation;

G)
is the first order equation,

forming after division by dx equivalent form of the equation:
.

Function
is called a solution to an ordinary differential equation if, when substituted into it, it becomes an identity.

For example, the 3rd order equation

Has a solution
.

To find by one method or another, for example, selection, one function that satisfies an equation does not mean solving it. To solve an ordinary differential equation means to find all functions that form an identity when substituted into the equation. For equation (1.1), the family of such functions is formed with the help of arbitrary constants and is called the general solution of the ordinary differential equation n th order, and the number of constants coincides with the order of the equation: y(x) : In this case, the solution is called the general integral of equation (1.1).

For example, the general solution of the differential equation
is the following expression: , and the second term can also be written as
, since an arbitrary constant divided by 2 can be replaced by a new arbitrary constant .

By setting some admissible values for all arbitrary constants in the general solution or in the general integral, we obtain a certain function that no longer contains arbitrary constants. This function is called a particular solution or a particular integral of equation (1.1). To find the values of arbitrary constants, and hence the particular solution, various additional conditions to equation (1.1) are used. For example, the so-called initial conditions for (1.2) can be given

On the right-hand sides of the initial conditions (1.2), the numerical values of the function and derivatives are given, and, moreover, total number initial conditions is equal to the number of defined arbitrary constants.

The problem of finding a particular solution to equation (1.1) from initial conditions is called the Cauchy problem.

§ 2. Ordinary differential equations of the 1st order - basic concepts.

Ordinary differential equation of the 1st order ( n=1) has the form:
or, if it can be resolved with respect to the derivative:
. Common decision y= y(x,WITH) or general integral
1st order equations contain one arbitrary constant. The only initial condition for the 1st order equation
allows you to determine the value of the constant from the general solution or from the general integral. Thus, a particular solution will be found or, which is also the Cauchy problem will be solved. The question of the existence and uniqueness of a solution to the Cauchy problem is one of the central ones in the general theory of ordinary differential equations. For a first-order equation, in particular, the theorem is valid, which is accepted here without proof.

Theorem 2.1. If in the equation the function
and its partial derivative
continuous in some area D plane XOY, and a point is given in this area
, then there exists and, moreover, a unique solution that satisfies both the equation and the initial condition
.

The geometrically general solution of the 1st order equation is a family of curves in the plane XOY, which do not have common points and differ from each other in one parameter - the value of the constant C. These curves are called integral curves for the given equation. Integral curve equations have an obvious geometric property: at each point, the tangent of the slope of the tangent to the curve is equal to the value of the right side of the equation at that point:
. In other words, the equation is given in the plane XOY field of directions of tangents to integral curves. Comment: It should be noted that for the equation
the equation and the so-called equation in symmetric form are given
.

§ 3. First order differential equations with separable variables.

Definition. A differential equation with separable variables is an equation of the form
(3.1)

or an equation of the form (3.2)

In order to separate the variables in equation (3.1), i.e. reduce this equation to the so-called equation with separated variables, perform the following actions:

;

Now we need to solve the equation g(y)= 0 . If it has a real solution y= a, then y= a will also be a solution of equation (3.1).

Equation (3.2) is reduced to a separated variable equation by dividing by the product
:

, which allows us to obtain the general integral of equation (3.2):
. (3.3)

The integral curves (3.3) will be supplemented by the solutions
if such solutions exist.

Solve the equation: .

Separating variables:

Integrating, we get

Further from the equations
and
find x=1, y=-1. These decisions are private decisions.

§ 4. Homogeneous differential equations of the first order.

Definition 1. An equation of the 1st order is called homogeneous if for its right side for any
the ratio
, called the homogeneity condition for a function of two variables of zero dimension.

Example 1 Show that function
- homogeneous zero measurement.

Solution.

Q.E.D.

Theorem. Any function
is homogeneous and, conversely, any homogeneous function
zero dimension is reduced to the form
.

Proof.

The first assertion of the theorem is obvious, since
. Let us prove the second assertion. Let's put
, then for a homogeneous function
, which was to be proved.

Definition 2. Equation (4.1)

in which M and N are homogeneous functions of the same degree, i.e. have the property for all , is called homogeneous.

Obviously, this equation can always be reduced to the form
(4.2) , although this may not be done to solve it.

A homogeneous equation is reduced to an equation with separable variables by replacing the desired function y according to the formula y= zx, where z(x) is the new desired function. After performing this substitution in equation (4.2), we obtain:
or
or
.

Integrating, we obtain the general integral of the equation with respect to the function z(x)
, which after repeated replacement
gives the general integral of the original equation. In addition, if - roots of the equation
, then the functions
- solutions of a homogeneous given equation. If
, then equation (4.2) takes the form

and becomes an equation with separable variables. Its solutions are semi-direct:
.

Comment. Sometimes it is advisable instead of the above substitution to use the substitution x= zy.

§ 5. Differential equations reducing to homogeneous ones.

Consider an equation of the form
. (5.1)

If
, then this equation is by substitution , where and are new variables, and - some constant numbers determined from the system

Reduced to a homogeneous equation

If
, then equation (5.1) takes the form

Assuming z= ax+ by, we arrive at an equation that does not contain an independent variable.

Consider examples.

Example 1

Integrate Equation

and highlight the integral curve passing through the points: a) (2;2); b) (1;-1).

Solution.

Let's put y= zx. Then dy= xdz+ zdx and

Let's shorten it by and gather members at dx and dz:

Let's separate the variables:

.

Integrating, we get ;

or
,
.

Replacing here z on the , we obtain the general integral of the given equation in the form (5.2)
or

.

This family of circles
, whose centers lie on a straight line y = x and which at the origin are tangent to the line y + x = 0. This straighty = - x in turn, a particular solution of the equation.

Now the Cauchy task mode:

A) assuming in the general integral x=2, y=2, find C=2, so the desired solution is
.

B) none of the circles (5.2) passes through the point (1;-1). But half-line y = - x,
passes through the point and gives the desired solution.

Example 2 Solve the equation: .

Solution.

The equation is a special case of equation (5.1).

Determinant
in this example
, so we need to solve the following system

Solving, we get that
. Performing in given equation substitution
, we obtain a homogeneous equation . Integrating it with a substitution
, we find
.

Returning to old variables x and y formulas
, we have .

§ 6. Generalized homogeneous equation.

The equation M(x, y) dx+ N(x, y) dy=0 is called generalized homogeneous if it is possible to choose such a number k that the left side of this equation becomes a homogeneous function of some degree m relatively x, y, dx and dy provided that x is considered the value of the first measurement, y – k th measurement , dx and dy – zero and (k-1) th measurements. For example, this would be the equation
. (6.1)

Valid under the assumption made about measurements

x, y, dx and dy members of the left side
and dy will have respectively dimensions -2, 2 k and k-one. Equating them, we obtain the condition that the desired number must satisfy k: -2 = 2k=k-one. This condition is satisfied when k= -1 (with such k all terms on the left side of the equation under consideration will have dimension -2). Consequently, equation (6.1) is generalized homogeneous.

Integrating it, we find
, where
. This is the general solution of equation (6.1).

§ 7. Linear differential equations of the first order.

A linear equation of the 1st order is an equation that is linear with respect to the desired function and its derivative. It looks like:

, (7.1)

where P(x) and Q(x) are given continuous functions of x. If the function
, then equation (7.1) has the form:
(7.2)

and is called a linear homogeneous equation, otherwise
it is called a linear inhomogeneous equation.

The linear homogeneous differential equation (7.2) is an equation with separable variables:

(7.3)

Substituting the found derivative into equation (7.1), we will have:

or
.

Where
, where is an arbitrary constant. As a result, the general solution of the inhomogeneous linear equation (7.1) will be (7.4)

Let us combine, for example, the second and third terms of the last expression and take out the function u(x) for brackets:
(7.5)

We require the parenthesis to vanish:
.

We solve this equation by setting an arbitrary constant C equal to zero:
. With found function v(x) back to equation (7.5):
.

Solving it, we get:
.

Therefore, the general solution of equation (7.1) has the form:

§ 8. Bernoulli's equation.

Definition.

Differential equation of the form
, where
, is called the Bernoulli equation.

Assuming that
, we divide both sides of the Bernoulli equation by . As a result, we get:
(8.1)

We introduce a new function
. Then
. We multiply equation (8.1) by
and pass in it to the function z(x) :
, i.e. for function z(x) obtained a linear inhomogeneous equation of the 1st order. This equation is solved by the methods discussed in the previous paragraph. Let us substitute into its general solution instead of z(x) expression
, we obtain the general integral of the Bernoulli equation, which is easily resolved with respect to y. At
solution is added y(x)=0 . The Bernoulli equation can also be solved without making the transition to linear equation by substitution
, and applying the Bernoulli method, discussed in detail in § 7. Consider the application of this method for solving the Bernoulli equation using a specific example.

Example. Find the general solution of the equation:
(8.2)

Solution.

Therefore, the general solution of this equation has the form:
, y(x)=0.

§ 9. Differential equations in total differentials.

Definition. If in the equation M(x, y) dx+ N(x, y) dy=0 (9.1) the left side is the total differential of some function U(x, y) , then it is called an equation in total differentials. This equation can be rewritten as du(x, y)=0 , therefore, its general integral is u(x, y)= c.

For example, the equation xdy+ ydx=0 is an equation in total differentials, since it can be rewritten in the form d(xy)=0. The general integral will be xy= c is an arbitrary differentiable function. We differentiate (9.3) with respect to u
§ 10. Integrating factor.

If the equation M(x, y) dx + N(x, y) dy = 0 is not an equation in total differentials and there is a function µ = µ(x, y) , such that after multiplying both sides of the equation by it, we obtain the equation

µ(Mdx + Ndy) = 0 in total differentials, i.e. µ(Mdx + Ndy)du, then the function µ(x, y) is called the integrating factor of the equation. In the case when the equation is already an equation in total differentials, we assume µ = 1.

If an integrating factor is found µ , then the integration of this equation reduces to multiplying both its parts by µ and finding the general integral of the resulting equation in total differentials.

If µ is a continuously differentiable function of x and y, then
.

It follows that the integrating factor µ satisfies the following 1st order PDE:

(10.1).

If it is known in advance that µ= µ(ω) , where ω is a given function from x and y, then equation (10.1) reduces to an ordinary (and, moreover, linear) equation with an unknown function µ from the independent variable ω :

(10.2),

where
, i.e. the fraction is a function only of ω .

Solving equation (10.2), we find the integrating factor

, With = 1.

In particular, the equation M(x, y) dx + N(x, y) dy = 0 has an integrating factor that depends only on x(ω = x) or only from y(ω = y) if the following conditions are met, respectively:

,
.

Generalized homogeneous equation. Homogeneous differential equations of the first order Generalized homogeneous equations of the second order

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§ 7. Linear differential equations of the first order.