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(357-359 беÑлаÑ)
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Tulakova Ziyodaxon Rivojidinovna, Senior Lecturer of the Department of
Natural Sciences, TUIT FF
Shokirov Asrorjon Murodjonovich , Assistant of the Department of Natural
Sciences, TUIT FF
POSSIBLE DIFFERENTIAL EQUATIONS THAT CAN REDUCE THE ORDER
Z. Tulakova, A. Shokirov
Abstract: This article explains how to reduce the order of the main
integration method for all types of high-order equations or to bring this
equation into a low-order equation by substituting variables into it.
Key words: simple differential equation, differential equation, integral,
product, Cauchy problem, function.
What are the differential equations before we consider the differential
equations that can be reduced in order? we can find the answer to the
question. In the equations studied so far, the unknowns consisted of
numbers. In mathematics and its various applications, we encounter the
study of equations involving functions and their derivatives (or
differentials) in an unknown place. Such equations are called differential
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equations. An equation is called a simple differential equation if the
unknown function sought in the differential equation depends on only one
arbitrary variable (argument).
The first textbook on differential equations in Uzbek was written by
academician T.N. It was written by Qori-Niyazi in the 1940s. The textbook,
which also describes the application of the theory of differential equations
to the solution of practical problems that meet modern requirements, was
written by academician MS Saloxiddinov and prof. Published by GN
Nasriddinov (Tashkent, "Uzbekistan", 1994).
symbolically the n-order differential equation
F (x, y, y ', y (n-1), y (n)) = 0 (1)
apparently or if this equation is solved with respect to an n-order
product,
y (n) = / (W, y (n-1)) (2)
can be written in the view.
The general solution of an n-order differential equation depends on x
and n arbitrary variables:
u = g (x, C l, C 2, ..., Cn).
Therefore, in order to distinguish a particular solution from a general
solution, some additional conditions must also be given that allow the
identification of arbitrary variables. These conditions determine the values
of the function in question and all its derivatives up to (n-1) -order
(including y) at a point, ie at x = xo
y (X0) = U0, y '(xQ) = y1, ..., yn-1) (xQ) = y n-1 (3)
can be created by giving. (3) The system is called the system of initial
conditions. The problem of finding a special solution of the given differential
equation (3) satisfying the system of initial conditions (3) is called the
Cauchy problem. The problem of integrating a higher-order differential
equation is more difficult than the problem of integrating a first-order
equation and does not always lead to the integration of the first-order
equation. However, for all types of higher-order equations except linear
equations, the main method of integration is to reduce the order, that is, to
reduce the given equation to a lower order equation by replacing the
variables in it. However, it is not always possible to reduce the order of the
equation. We consider below the simplest types of n-order equations that
allow us to reduce the order of the equations.
1. Reducing the order of this equation y = (/ x) (4) is done by serial
integration:
v "M / (x) dx + Ci; y (n" 2) = / (/ (f (x) dx + C1) dx + c2 = / dx / (f (x)
dx + Cix + c2
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j.n-1
y = / dx / dx ... J / (x) dx + Ci ^ - + C2 â-â + ... + Cn;
2. The function y and its derivatives y ', y' '... y (k-1) are not explicitly
involved.
F (x, y (k), y (k + 1) y (n-1), y (n)) = 0 (5)
the order of the differential equation
y (k) = z; y (k + 1) = z '; ... y (n) = z (n-k)
using substitutions, k is reduced to a unit: F (x, z, z '... z (n-k)) = 0
3. The free x variable did not participate in the disclosure
F (y, y ', y' ', ... y (n)) = 0 (6)
The order of the equation:
is reduced to one unit by switching.
4. The function F (x, y, y ', y' ', ... y (n)) is homogeneous with respect to
y, y', y '', ... y (n),
F (x, y, y ', y ", ... y (n)) = 0 (7)
the order of the equation is reduced to one by replacing y = e-.
5. The left side of the equation is a definite product. In this case, the
reduction of the order of the equation to one unit is done by direct
integration.
In conclusion, of course, such a case is rare. In some cases, such an
appearance of the equation is achieved through some artificial form
substitutions.
References:
1. R. Turgunbaev "Collection of examples and problems from the course
of differential equations". Tashkent. 2007 y.
2. J. Oramov "Differential equations" textbook Tashkent. 2013 y.
