Equations Reducing to Homogeneous Equations. Homogeneous equations. An exhaustive guide (2019). Generalized homogeneous equation
Basic concepts of the theory of differential equations
A differential equation is an equation that relates an independent variable, the desired function and its derivatives. A solution is a function that, when substituted into an equation, turns it into an identity.
If the desired function depends on one variable, the DE is called ordinary, otherwise, the DE in partial derivatives. highest order
Differential equations of the 1st order. Cauchy problem, existence and uniqueness theorem for its solution. General, particular solution (integral), special solution.
F(x;y;y ’ )=0 – DE of the 1st order (1)
y ’ =f(x;y) DE allowed with respect to derivative(2)
P(x;y)dx+Q(x;y)dy=0 – differential form(3)
The problem of finding a solution to the 1st order DE that satisfies a given initial condition (y(x 0)=y 0) is called the Cauchy problem.
T. If in equation (2) the function f (x; y) and ...
its partial derivative f y ’
(x;y) are continuous in some domain D containing the point (x 0 ;y 0), then there is a unique solution y=φ(x) of this equation that satisfies the initial condition.
The general solution is a function y=φ(x;с) containing an arbitrary constant.
A particular solution is a function y=φ(x;с 0) obtained from a general solution with a constant value с=с 0 .
If the general solution is found in an implicit form Ф(x;y;c)=0, then it is called the general integral of the DE. And Ф (x; y; c 0) \u003d 0 is a partial integral of the equation.
The function φ(x;c) is called a special solution of the differential equation F(x,y,y’) = 0 if the uniqueness of the solution is violated at each point of this function in the domain of the differential equation.
Geometric interpretation of DE of the 1st order. Isocline method
Equation y ’ =f(x;y) establishes a relationship between the coordinates of a point and slope factor y ’ tangent to the integral curve. The DE gives a field of directions on the Oxy plane. A curve in which the direction of the field is the same at all points is called an isocline. Isoclines can be used for approximate construction of integral curves. Isocline equation f(x;y)=c.
Separable Variable Equations
Separated variable equation: P(x)dx+Q(y)dy=0
General integral of DE:
Separated variable equation: P 1 (x)Q 1 (y)dx+P 2 (x)Q 2 (y)dy=0
Homogeneous DU. Equations Reducing to Homogeneous
A function f(x;y) is called a homogeneous function of the nth order if, when multiplying each of its arguments by an arbitrary factor λ, the entire function is multiplied by λ n , i.e. f(λ x; λ y)= λ n f(x; y). remote control y ’ =f(x;y) is called homogeneous if the function f(x;y) is homogeneous f-i zero order
P(x;y)dx+Q(x;y)dy=0 differential form of homogeneous DE
The type equation can be reduced to a homogeneous type. You need to create a system like this:
Let the solution of this system be:
Then, to reduce the equation to a homogeneous type, it is necessary to make a substitution of the form
If the system does not have a solution, a replacement should be made.
Stop! Let's all the same try to understand this cumbersome formula.
In the first place should be the first variable in the degree with some coefficient. In our case, this
In our case it is. As we found out, it means that here the degree for the first variable converges. And the second variable in the first degree is in place. Coefficient.
We have it.
The first variable is exponential, and the second variable is squared, with a coefficient. This is the last term in the equation.
As you can see, our equation fits the definition in the form of a formula.
Let's look at the second (verbal) part of the definition.
We have two unknowns and. It converges here.
Let's consider all terms. In them, the sum of the degrees of the unknowns must be the same.
The sum of the powers is equal.
The sum of the powers is equal to (at and at).
The sum of the powers is equal.
As you can see, everything fits!
Now let's practice defining homogeneous equations.
Determine which of the equations are homogeneous:
Homogeneous equations - equations with numbers:
Let's consider the equation separately.
If we divide each term by expanding each term, we get
And this equation completely falls under the definition of homogeneous equations.
How to solve homogeneous equations?
Example 2
Let's divide the equation by.
According to our condition, y cannot be equal. Therefore, we can safely divide by
By substituting, we get a simple quadratic equation:
Since this is a reduced quadratic equation, we use the Vieta theorem:
Making the reverse substitution, we get the answer
Answer:
Example 3
Divide the equation by (by condition).
Answer:
Example 4
Find if.
Here you need not to divide, but to multiply. Multiply the whole equation by:
Let's make a replacement and solve the quadratic equation:
Making the reverse substitution, we get the answer:
Answer:
Solution of homogeneous trigonometric equations.
The solution of homogeneous trigonometric equations is no different from the solution methods described above. Only here, among other things, you need to know a little trigonometry. And be able to solve trigonometric equations (for this you can read the section).
Let's consider such equations on examples.
Example 5
Solve the equation.
We see a typical homogeneous equation: and are unknowns, and the sum of their powers in each term is equal.
Similar homogeneous equations are not difficult to solve, but before dividing the equations into, consider the case when
In this case, the equation will take the form: But the sine and cosine cannot be equal at the same time, because according to the basic trigonometric identity. Therefore, we can safely divide it into:
Since the equation is reduced, then according to the Vieta theorem:
Answer:
Example 6
Solve the equation.
As in the example, you need to divide the equation by. Consider the case when:
But the sine and cosine cannot be equal at the same time, because according to the basic trigonometric identity. That's why.
Let's make a substitution and solve the quadratic equation:
Let us make the reverse substitution and find and:
Answer:
Solution of homogeneous exponential equations.
Homogeneous equations are solved in the same way as those considered above. If you forgot how to decide exponential equations- see the relevant section ()!
Let's look at a few examples.
Example 7
Solve the Equation
Imagine how:
We see a typical homogeneous equation, with two variables and a sum of powers. Let's divide the equation into:
As you can see, after making the replacement, we get the given quadratic equation (in this case, there is no need to be afraid of dividing by zero - it is always strictly greater than zero):
According to Vieta's theorem:
Answer: .
Example 8
Solve the Equation
Imagine how:
Let's divide the equation into:
Let's make a replacement and solve the quadratic equation:
The root does not satisfy the condition. We make the reverse substitution and find:
Answer:
HOMOGENEOUS EQUATIONS. AVERAGE LEVEL
First, using an example of one problem, let me remind you what are homogeneous equations and what is the solution of homogeneous equations.
Solve the problem:
Find if.
Here you can notice a curious thing: if we divide each term by, we get:
That is, now there are no separate and, - now the desired value is the variable in the equation. And this is an ordinary quadratic equation, which is easy to solve using Vieta's theorem: the product of the roots is equal, and the sum is the numbers and.
Answer:
Equations of the form
called homogeneous. That is, this is an equation with two unknowns, in each term of which there is the same sum of the powers of these unknowns. For example, in the example above, this amount is equal to. The solution of homogeneous equations is carried out by dividing by one of the unknowns in this degree:
And the subsequent change of variables: . Thus, we obtain an equation of degree with one unknown:
Most often, we will encounter equations of the second degree (that is, quadratic), and we can solve them:
Note that dividing (and multiplying) the whole equation by a variable is possible only if we are convinced that this variable cannot be equal to zero! For example, if we are asked to find, we immediately understand that, since it is impossible to divide. In cases where this is not so obvious, it is necessary to separately check the case when this variable is equal to zero. For example:
Solve the equation.
Solution:
We see here a typical homogeneous equation: and are unknowns, and the sum of their powers in each term is equal.
But, before dividing by and getting the quadratic equation with respect, we must consider the case when. In this case, the equation will take the form: , hence, . But the sine and cosine cannot be equal to zero at the same time, because according to the basic trigonometric identity:. Therefore, we can safely divide it into:
I hope this solution is completely clear? If not, read the section. If it is not clear where it came from, you need to return even earlier - to the section.
Decide for yourself:
- Find if.
- Find if.
- Solve the equation.
Here I will briefly write directly the solution of homogeneous equations:
Solutions:
Answer: .
And here it is necessary not to divide, but to multiply:
Answer:
If you haven't gone through this yet, you can skip this example.
Since here we need to divide by, we first make sure that one hundred is not equal to zero:
And this is impossible.
Answer: .
HOMOGENEOUS EQUATIONS. BRIEFLY ABOUT THE MAIN
The solution of all homogeneous equations is reduced to division by one of the unknowns in the degree and further change of variables.
Algorithm:
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If the equation can be converted to the form , then this equation is called homogeneous. It is easy to show that the equation in differential form M(x, y) dx + N(x, y) dy = 0 is homogeneous if and only if the functions M(x, y) And N(x, y) homogeneous functions of the same degree. Recall that a function F(x 1 ,x 2 ,..,x n) is called homogeneous of degree k if it satisfies the relation F(tx 1 ,tx 2 ,..,tx n)=t k F(x 1 ,x 2 ,..,xn).
A homogeneous differential equation is reduced to an equation with separable variables by replacing y = xu, or, which is the same, , where u the new desired function. Indeed, then y" = u + u"x and the original equation can be rewritten as u + u"x= f(u), or u"x= f(u)u. From the last f(u)u we can write down.
Example. Solve the equation (y 2 - 2xy)dx + x 2 dy = 0. This is a homogeneous equation, since y 2 - 2xy and x 2 are homogeneous functions of the second degree. We make the change y = xu, dy = udx + xdu. Substituting into the equation, we have
(x 2 u 2 - 2x 2 u)dx + x 2 (udx + xdu) = 0.
Opening the brackets, giving similar ones and reducing by x 2, we get an equation with separable variables
(u 2 - u)dx + xdu = 0
Separating the variables, we get 
or, which is the same, 
Integrating the last relation, we have lnu - ln(u-1) = lnx + lnC. By potentiating (going from logarithmic function to e x), we can write or, making the reverse substitution, we obtain common integral equations
Equations of the form 
are reduced to a uniform transfer of the origin to the point of intersection of the lines a 1 x + b 1 y + c 1 = 0, a 2 x + b 2 y + c 2 = 0, if the determinant is nonzero, and by replacing a 1 x + b 1 y = z if this determinant is equal to zero.
Decide homogeneous equations online possible with the help of a special service
The function f(x, y) is called a homogeneous function of the n-th dimension with respect to the variables x and y, if for any the identity
The first order differential equation is called homogeneous relatively X And at, if the function is a homogeneous function of zero dimension with respect to X And y.
Solution of a homogeneous differential equation.
Since according to the condition 
. Let's get 
, i.e. the homogeneous zero dimension function depends only on the ratio of the arguments. And the equation itself in this case will take the form .
Let's make a substitution; those. , then , substitute into the original equation 
is a differential equation with separable variables 
Type equation 
(1)
can be reduced to a homogeneous type.
General form transformations.
In order to reduce equation (1) to a homogeneous type of differential equations, it is necessary to compose a system of the form: 
First case.
This system has a solution.
Let the solution of this system be:
.
Then, to reduce equation (1) to a homogeneous type, it is necessary to make a substitution of the form
Second case.
Recall. The equation
We bring to a homogeneous type, made up the system ,
and this system has no solutions.
In this case, you should replace 
.
6. Inhomogeneous linear differential equations of the first order. Solution of an inhomogeneous linear differential equation of the first order by the Bernoulli method. Bernoulli's equations.
An inhomogeneous differential equation is a differential equation (ordinary or in partial derivatives), which contains an identically non-zero free term - a term that does not depend on unknown functions.
Linear Equation of the first order in standard notation has the form
An ordinary differential equation of the form:
called Bernoulli equation(for or we obtain an inhomogeneous or homogeneous linear equation).
Let's choose so that
for this it is sufficient to solve the equation with separable variables of the 1st order. After that, to determine, we obtain the equation 
- equation with separable variables.
7. Homogeneous and non-homogeneous linear differential equations of the first order by the method of variation of an arbitrary constant.
A differential equation is homogeneous if it does not contain free member- a term that does not depend on the unknown function. So, we can say that the equation is homogeneous if .
If , one speaks of an inhomogeneous differential equation
Type equation 
is called a linear inhomogeneous equation.
Type equation
is called a linear homogeneous equation.