Total differential method. Equations in total differentials
The solution method is considered differential equations with separable variables. An example of a detailed solution of a differential equation with separable variables is given.
ContentDefinition
Let s (x), q (x)- functions of the variable x;
p (y), r (y)- functions of the variable y.
A differential equation with separable variables is an equation of the form
Method for solving a differential equation with separable variables
Consider the equation:
(i) .
Let us express the derivative y′ in terms of differentials.
;
.
Let's multiply by dx.
(ii)
Divide the equation by s (x)r(y). This can be done if s(x) r(y) ≠ 0 This can be done if s.
.
When s
we have
Integrating, we obtain the general integral in quadratures (x)r(y)(iii) . Since we divided by s, then we obtained the integral of the equation for s (x) ≠ 0 and r
(y) ≠ 0 ..
Next you need to solve the equation . r (y) = 0 If this equation has roots, then they are also solutions to equation (i). Let the equation r . has n roots a i, r
(a i ) = 0
, i = 1, 2, ... , n.
. Then the constants y = a i are solutions to equation (i). Some of these solutions may already be contained in the general integral (iii). Note that if the original equation is given in form (ii), then we must also solve the equation s(x) = 0
Its roots b j, s
(b j ) = 0
, j =
1, 2, ... , m
.
give solutions x = b j .
An example of solving a differential equation with separable variables
.
Solve the equation 0
.
Let's express the derivative through differentials: 0
Multiply by dx and divide by .
For y ≠ 0 we have:
Let's integrate. 0 .
We calculate the integrals using the formula.
Substituting, we obtain the general integral of the equation
Now consider the case, y =
Obviously y =
is a solution to the original equation. It is not included in the general integral. Therefore, we will add it to the final result..
It is known that the displacement of a material point during uniformly accelerated motion is a function of time and is expressed by the formula:
In turn, acceleration a is derivative with respect to time t from speed V, which is also time derivative t from moving S. Those.
Then we get: 
- the equation connects the function f(t) with the independent variable t and the second-order derivative of the function f(t).
Definition. Differential equation is an equation that relates independent variables, their functions, and derivatives (or differentials) of this function.
Definition. If a differential equation has one independent variable, then it is called ordinary differential equation, if there are two or more independent variables, then such a differential equation is called partial differential equation.
Definition. The highest order of derivatives appearing in an equation is called order of the differential equation.
Example.
- ordinary differential equation of the 1st order. IN general view is recorded 
.
- ordinary differential equation of the 2nd order. In general it is written 
- first order partial differential equation.
Definition. General solution differential equation is such a differentiable function y = (x, C), which, when substituted into the original equation instead of an unknown function, turns the equation into an identity.
Properties of the general solution.
1) Because constant C is an arbitrary value, then generally speaking a differential equation has an infinite number of solutions.
2) Under any initial conditions x = x 0, y(x 0) = y 0, there is a value C = C 0 at which the solution to the differential equation is the function y = (x, C 0).
Definition. A solution of the form y = (x, C 0) is called private solution differential equation.
Definition. Cauchy problem(Augustin Louis Cauchy (1789-1857) - French mathematician) is the finding of any particular solution to a differential equation of the form y = (x, C 0), satisfying the initial conditions y(x 0) = y 0.
Cauchy's theorem. (theorem on the existence and uniqueness of a solution to a 1st order differential equation)
If the functionf(x,
y) is continuous in some regionDin the planeXOYand has a continuous partial derivative in this region 
, then whatever the point (x 0
, y 0
) in areaD, there is only one solution 
equations 
, defined in some interval containing point x 0
, taking at x = x 0
meaning
(X 0
) = y 0
, i.e. there is a unique solution to the differential equation.
Definition.
Integral A differential equation is any equation that does not contain derivatives and for which the given differential equation is a consequence. 
Example. Find common decision differential equation 
.
The general solution of the differential equation is sought by integrating the left and right sides of the equation, which is previously transformed as follows:
Now let's integrate: 
is the general solution of the original differential equation.
Suppose we are given some initial conditions: x 0 = 1; y 0 = 2, then we have
By substituting the resulting value of the constant into the general solution, we obtain a particular solution for the given initial conditions (solution to the Cauchy problem).
Definition. Integral curve is called the graph y = (x) of the solution to a differential equation on the XOY plane.
Definition. By special decision of a differential equation is such a solution at all points of which the Cauchy uniqueness condition is called (see. Cauchy's theorem.) is not fulfilled, i.e. in the neighborhood of some point (x, y) there are at least two integral curves.
Special solutions do not depend on the constant C.
Special solutions cannot be obtained from the general solution for any value of the constant C. If we construct a family of integral curves of a differential equation, then the special solution will be represented by a line that touches at least one integral curve at each point.
Note that not every differential equation has special solutions.
Example.
Find a special solution if it exists.
This differential equation also has a special solution at= 0. This solution cannot be obtained from the general one, but when substituting into the original equation we obtain an identity. The opinion that the solution y = 0 can be obtained from the general solution with WITH 1 = 0 wrong, because C 1 = e C 0.
First order differential equations.
Definition. First order differential equation is called a relation connecting a function, its first derivative and an independent variable, i.e. ratio of the form:
If we transform this relation to the form 
then this first order differential equation will be called the equation, resolved with respect to the derivative.
Let's represent the function f(x,y) as: 
then, when substituting into the above equation, we have:
this is the so-called differential form first order equations.
Equations of the formy ’ = f ( x ).
Let the function f(x) be defined and continuous on some interval
a< x
< b.
В таком случае все решения данного
дифференциального уравнения находятся
как
. If the initial conditions x 0 and y 0 are given, then the constant C can be determined.
Separable equations
Definition.
Differential equation 
called separable equation, if it can be written in the form
.
This equation can also be represented as:
Let's move on to new notations 
We get:
After finding the corresponding integrals, a general solution to the differential equation with separable variables is obtained.
If the initial conditions are given, then when they are substituted into the general solution, a constant value C is found, and, accordingly, a particular solution is found.
Example. Find the general solution to the differential equation: 
The integral on the left side is taken by parts (see. Integration by parts.):
this is the general integral of the original differential equation, since the desired function and is not expressed through an independent variable. This is what it's all about difference general (private) integral from general (private)
solutions.
To check the correctness of the received answer, we differentiate it with respect to the variable x.
Example.- right 
Find the solution to the differential equation
provided y(2) = 1.
for y(2) = 1 we get 
Total: 
or
- private solution;
Examination:
, total
Example.- right. 
Solve the equation
- general integral
Example.- right. 
Example.- right. 
- common decision
provided y(1) = 0. We will take the integral on the left side by parts (see.).
Integration by parts.
If y(1) = 0, then 
.
Example. Total, partial integral:
Solve the equation. To find the integral on the left side of the equation, see Table of basic integrals.
Example.- right. 
clause 16. We obtain the general integral:
Let's transform the given equation:
Example.- right. 
.
;
;
We obtained the general integral of this differential equation. If we express the desired function y from this relation, we obtain a general solution.
Let's say some initial conditions x 0 and y 0 are given. Then: 
We obtain a particular solution
Definition. Homogeneous equations. The function f(x, y) is calledhomogeneousn– th measurements
Example. with respect to its arguments x and y, if for any value of the parameter t (except zero) the identity holds: 
Is the function homogeneous?
Definition.
Thus, the function f(x, y) is homogeneous of the 3rd order. 
called Differential equation of the form homogeneous
, if its right-hand side f(x, y) is a homogeneous function of zero dimension with respect to its arguments. Any equation of the form is homogeneous if the functions(x, y) P And(x, y) Q
– homogeneous functions of the same dimension. Any solution homogeneous equation
is based on reducing this equation to an equation with separable variables. 
Consider the homogeneous equation
Because function f(x, y) is homogeneous of zero dimension, then we can write: 
Because the parameter t is generally arbitrary, let us assume that
. We get: 
, i.e.
The original differential equation can thus be written as:
Thus, we obtained an equation with separable variables for the unknown function u.
Example.- right. 
.
Let's introduce an auxiliary function u.
.
Note that the function we introduced u is always positive, because otherwise, the original differential equation containing 
.
Substitute into the original equation:
We separate the variables: 
Integrating, we get:
Passing from the auxiliary function back to the y function, we obtain the general solution:
Equations reduced to homogeneous.
In addition to the equations described above, there is a class of equations that, using certain substitutions, can be reduced to homogeneous ones.
These are equations of the form 
.
If the determinant 
then the variables can be separated by substitution
where and are solutions to the system of equations 
Example.- right.
We get
Finding the value of the determinant 
.
Solving a system of equations
We apply substitution into the original equation:
Replace the variable 
when substituting into the expression written above, we have:
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