The equation M(x, y) dx+ N(x, y) dy=0 is called a generalized homogeneous if it is possible to select 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 dimension, y – k‑ th measurements , dx And dy – respectively zero and (k-1) th measurements. For example, this would be the equation. (6.1)
Valid under the assumptions made regarding measurements
x,
y,
dx
And dy
members of the left side 
And dy
will have dimensions -2, 2 respectively k And
k-1. Equating them, we obtain a condition that the required number must satisfy k:
-2 = 2k
=
k-1. This condition is satisfied when k
= -1 (with this k all terms on the left side of the equation under consideration will have a dimension of -2). Consequently, equation (6.1) is generalized homogeneous.
A generalized homogeneous equation is reduced to an equation with separable variables using substitution 
, Where z– new unknown function. Let us integrate equation (6.1) using the indicated method. Because k
= -1, then 
, after which we get the equation.
Integrating it, we find 
, where 
. This is a general solution to equation (6.1).
§ 7. Linear differential equations of the 1st order.
A 1st order linear equation 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)
– 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 to equation (7.1), in which the function P(x) denotes the same function as in equation (7.2), we apply a technique called the method of variation of an arbitrary constant and consists of the following: we will try to select the function C=C(x) so that the general solution to the linear homogeneous equation (7.2) would be a solution to the inhomogeneous linear equation (7.1). Then for the derivative of function (7.3) we obtain:
.
Substituting the found derivative into equation (7.1), we will have:
or 
.
Where 
, Where 
- arbitrary constant. As a result, the general solution to 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 of formula (7.4) is a particular solution of the linear inhomogeneous equation (7.1), obtained from the general (7.4) with 
. We highlight 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 
- general solution of the corresponding linear homogeneous differential equation.
However, it should be noted that to solve the linear inhomogeneous differential equation of the 1st order (7.1), another method is more often used, sometimes called the Bernoulli method. We will look for a solution to equation (7.1) in the form 
. Then 
. Let's substitute the found derivative into the original equation: 
.
Let us combine, for example, the second and third terms of the last expression and extract the function u(x)
behind the bracket: 
(7.5)
We require the parenthesis to be nullified: 
.
Let us solve this equation by setting an arbitrary constant C
equal to zero: 
. With the found function v(x)
Let's return to equation (7.5): 
.
Solving it, we get: 
.
Consequently, the general solution to equation (7.1) has the form.
Differential equations in generalized functions
Let there be an equation. If is an ordinary function, then its solution is an antiderivative, that is. Let now be a generalized function.
Definition. A generalized function is called a primitive generalized function if. If is a singular generalized function, then there are possible cases when its antiderivative is a regular generalized function. For example, an antiderivative is; the antiderivative is a function, and the solution to the equation can be written in the form: , where.
There is a linear equation of the th order with constant coefficients
where is a generalized function. Let be a differential polynomial of the th order.
Definition. A generalized solution of the differential equation (8) is a generalized function for which the following relation holds:
If is a continuous function, then the only solution to equation (8) is the classical solution.
Definition. A fundamental solution to equation (8) is any generalized function such that.
Green's function is a fundamental solution that satisfies a boundary, initial, or asymptotic condition.
Theorem. A solution to equation (8) exists and has the form:
unless convolution is defined.
Proof. Really, . According to the convolution property it follows: .
It is easy to see that the fundamental solution to this equation is, since
Properties of generalized derivatives
The operation of differentiation is linear and continuous from to:
in, if in;
Every generalized function is infinitely differentiable. Indeed, if, then; in turn, etc.;
The result of differentiation does not depend on the order of differentiation. For example, ;
If and, then Leibniz’s formula for differentiation of a product is valid. For example, ;
If it is a generalized function, then;
If a series composed of locally integrable functions converges uniformly on each compact set, then it can be differentiated term-by-term any number of times (as a generalized function), and the resulting series will converge in.
Example. Let
The function is called the Heaviside function or unit function. It is locally integrable and therefore can be considered as a generalized function. You can find its derivative. According to the definition, i.e. .
Generalized functions corresponding to quadratic forms with complex coefficients
So far, only quadratic forms with real coefficients have been considered. In this section we study the space of all quadratic forms with complex coefficients.
The task is to determine the generalized function, where is a complex number. However, in the general case there will not be a unique analytical function of. Therefore, in the space of all quadratic forms, the “upper half-plane” of quadratic forms with a positive definite imaginary part is isolated and a function is determined for them. Namely, if a quadratic form belongs to this “half-plane”, then it is assumed that where. Such a function is a unique analytic function of.
We can now associate the function with a generalized function:
where integration is carried out over the entire space. Integral (13) converges at and is an analytical function of in this half-plane. Continuing this function analytically, the functional for other values is determined.
For quadratic forms with a positive definite imaginary part, the singular points of the functions are found and the residues of these functions at the singular points are calculated.
The generalized function analytically depends not only on, but also on the coefficients of the quadratic form. Thus, it is an analytical function in the upper “half-plane” of all quadratic forms of the form where there is a positive definite form. Consequently, it is uniquely determined by its values on the “imaginary semi-axis,” i.e., on the set of quadratic forms of the form, where is a positive definite form.
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1st 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 separated variable equation, do the following: 
;
Now we need to solve the equation g(y)= 0. If it has a real solution y=a, That y=a will also be a solution to equation (3.1).
Equation (3.2) is reduced to a separated equation by dividing by the product:
, which allows us to obtain the general integral of equation (3.2): 
. (3.3)
Integral curves (3.3) will be supplemented with solutions 
, if such solutions exist.
Homogeneous differential equations of the 1st order.
Definition 1. A first-order equation is called homogeneous if its right-hand side satisfies the relation 
, called the condition of homogeneity of 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 statement of the theorem is obvious, because . Let's prove the second statement. Let us put then for a homogeneous function 
, which was what needed to be proven.
Definition 2. Equation (4.1) in which M And N– homogeneous functions of the same degree, i.e. have the property for all , called homogeneous. Obviously, this equation can always be reduced to the form (4.2), although this may not be necessary 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)– new required 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 to a homogeneous given equation. If , then equation (4.2) takes the form
And it becomes an equation with separable variables. Its solutions are half-direct: .
Comment. Sometimes it is advisable to use the substitution instead of the above substitution x=zy.
Generalized homogeneous equation.
The equation M(x,y)dx+N(x,y)dy=0 is called a generalized homogeneous if it is possible to select 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 dimension, y – k‑ th measurements ,dx And dy – respectively zero and (k-1) th measurements. For example, this would be the equation 
. (6.1) Valid under the assumption made regarding measurements x, y, dx And dy members of the left side and dy will have dimensions -2, 2 respectively k And k-1. Equating them, we obtain a condition that the required number must satisfy k: -2 = 2k=k-1. This condition is satisfied when k= -1 (with this k all terms on the left side of the equation under consideration will have a dimension of -2). Consequently, equation (6.1) is generalized homogeneous.
def 1 DU type
called homogeneous differential equation of the first order(ODU).
Th 1 Let the following conditions be met for the function:
1) continuous at
Then ODE (1) has a general integral, which is given by the formula:
where is some antiderivative of the function With is an arbitrary constant.
Note 1 If for some the condition is met, then in the process of solving ODE (1) solutions of the form may be lost; such cases must be treated more carefully and each of them must be checked separately.
Thus from the theorem Th1 should general algorithm for solving ODE (1):
1) Make a replacement:
2) Thus, a differential equation with separable variables will be obtained, which should be integrated;
3) Return to old gvariables;
4) Check the values for their involvement in the solution original remote control, under which the condition will be satisfied
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 a change (3), this will bring 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 this 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:
Note 2 Sometimes you can write ODEs in terms of differentials of variables X And u. It is recommended to move from this notation of the remote control to the expression through the derivative and only then carry out the replacement (3).
Differential equations reduced to homogeneous ones.
def 2 The function is called homogeneous function of degree k in area, for which the equality will be satisfied:
Here are the most common types of differential equations that can be reduced to form (1) after various transformations.
1) where is the function is homogeneous, degree zero, that is, the equality is valid: DE (6) is easily reduced to the form (1), if we put , which is further integrated using replacement (3).
2) (7), where the functions are homogeneous of the same degree k . DE of the form (7) is also integrated using substitution (3).
Example 2 Solve DE (8).
Solution: Let us show that DE (8) is homogeneous. Let us divide by what is possible, since it is not a solution to DE (8).
Let's make a change (3), this will bring equation (9) to the form:
Equation (10) is the general integral of DE (8).
Note that when separating variables and dividing by, 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(reads " signum x"), defined by the expression:
Note 3 Reducing the DE (6) or (7) to the form (1) is not necessary; if it is obvious that the DE is homogeneous, then you can immediately make the replacement
3) A DE of the form (11) is integrated as an ODE if , and the substitution is initially performed:
(12), where is the solution of the system: (13), and then use replacement (3) for the function. After receiving the general integral, they return to the variables X And at.
If , then, assuming in equation (11), we obtain a differential equation 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:
Let us solve the inhomogeneous system of linear algebraic equations (15) using the Cramer method:
Let's make a change of variables and integrate the resulting equation:
(16) – General integral of DE (14). When separating variables, solutions could be lost when dividing by an expression, which could be obtained explicitly after solving the quadratic equation. However, they are taken into account in the general integral (16) at
Let's find a solution to the Cauchy problem: substitute the values 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 reduce some differential equations to homogeneous ones for a new, as yet unknown function if we apply a substitution of the form:
In this case 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 DE under consideration cannot be reduced to a homogeneous one in this way.
Example 4 Solve DE. (18)
Solution: Let us show that DE (18) is reduced to a homogeneous DE using substitution (17) and is further integrated using substitution (3):
Let's find With:
Thus, a particular solution of DE (24) has the form
