Scientific research results in pandemic conditions (COVID-19)
95
2. Hatami S.M. From the history of Islamic thought. - Tashkent: “Minxoj”,
2003.
3. Excerpts from the history of world philosophy. Edited by Nazarov Q.
- Tashkent: Sharq, 2004.
4. Philosophy. Edited by Akhmedova M.-Tashkent: UFMJ, 2006.
5. Jorayev N. Theoretical foundations of the philosophy of history. -
Tashkent: Manaviyat, 2008.
Nafisa Salimova, student, Termez State University, Bachelor
ANALYTICAL SOLUTION OF SIMPLE DIFFERENTIAL EQUATIONS
FIND OUT WITH THE MAPLE PROGRAM
N. Salimova
Abstract: This article deals with the solution of simple differential
equations using the Maple mathematical package using analytical methods,
demonstration of this process in specific practical problems, the creation of
algorithms and programs for solving the problem.
Keywords: Numerical solution, math package, maple, dsolve, method.
One of the areas of application of the computer remains the study of
mechanical processes and mathematical models of objects using
computational methods and computer software. The methods of
computational mathematics and the modern capabilities of computers
together serve to reveal the hitherto unknown features of mechanical
processes and objects, and at the same time to improve technological
processes.
Today, with the development of science and technology, the role of
mathematics is growing. Mathematics is used in physics, mechanics and
astronomy, as well as in solving economic problems, analyzing biological
processes, and many other fields. The mathematical model of processes in
these fields is called differential equations.
This scientific article deals with the application of computational
mathematics and computers in scientific research and is scientifically and
practically relevant (2). The paper deals with the problem of analytical and
approximate solution of simple differential equations using the Maple
program. The following is a sequential algorithm for problem solving and
problem solving. Describe the calculation methods required to solve simple
differential equations.
Scientific research results in pandemic conditions (COVID-19)
96
Object of research and methods used
The object of research is simple differential equations, boundary value
problems. Research methods: exact methods of problem solving,
approximate-exact methods and numerical methods.
The results obtained and their analysis
In practice, a chain of “elementary” calculations and substitutions that
can be performed using an arbitrary mathematical package can also solve
complex problems (for example, simple differential equations, solving
boundary value problems). The Maple software package allows you to find
solutions to many problems in specialized departments of higher
mathematics. The technology of working in the Maple environment can be
found in the specialized References: (4). The Maple math package can be
used in practical classes on "Differential Equations" and "Higher
Mathematics", in seminars, in the classes of elective disciplines on simple
differential equations and systems of equations, numerical solution of
boundary value problems.
Suppose a material point moves along the OX axis. Let the motion
function be
f (t). Moreover, at a moment
t=t
0
, let its abscissa take the value x0. Find
the law of motion of this material point (6).
This is the mathematical model of the problem
is represented by the differential equation and the initial condition. Let's
take another example. The rate of decay of radium, which is a radioactive
substance, is directly proportional to its amount. Suppose that at moment t
there is a radius
R
0
g. Determine the amount of radium Rg at an arbitrary
moment t.
If the coefficient of proportionality is equal to c (c> 0), then the problem
is reduced to the solution of this differential equation (3).
This is the solution of the equation at
R=R
0
at
t=t
0
R=R
0
e
-c(t-t0)
represented by the function.
From the above problems, it can be seen that a single differential
equation can be satisfied by several functions, so the main goal of the theory
of differential equations is to find all the solutions of a given equation and
study their properties. To achieve this goal, we now have special math
packages. These are Maple, Mathcad, MathLab, Mathematica and so on.
Using these packages, we can solve simple differential equations.
Scientific research results in pandemic conditions (COVID-19)
97
To achieve this, we first discuss the differential equations, boundary
value problems, their general and specific solutions, their analytical finding,
and in what cases mathematical packages can be used.
The following command is used to find analytical solutions of
differential equations in Maple:
dsolve (eq, var, options),
here
eq - differential equation;
var - indefinite functions;
options.
When entering a differential equation, the diff command is used to
denote the product, for example, the differential equation
y''+y=x is written
as:
diff(y(x),x$2)+y(x)=x.
This is to construct a graph of a numerical solution of a differential
equation
odeplot(dd, [x,y(x)], x=x1..x2)
can be used as a function, where
dd:=dsolve({eq,cond}, y(x), numeric)
is used as a function, followed by a variable and unknown function in square
brackets
[x,y(x)] and the graph construction interval is shown as x=x1..x2 .
To do this, we first find a numerical solution to the Cauchy problem, and
then construct a graph of the solution found:
Let's find the numerical and approximate solution of the following
Cauchy problem in the form of a 2nd order series:
,
,
.
To do this, we first find a numerical solution to the Cauchy problem, and
then construct a graph of the solution found:
> restart; ordev=6:
> eq:=diff(y(x),x$2)+x*sin(y(x))= - sin(x):
> cond:=y(0)=-1, D(y)(0)=1:
> de:=dsolve({eq,cond},y(x),numeric);
>
de:=proc(rkf45_x)...end proc
The result shows the use of the rkf45 method. If the line does not
provide the required information, this intermediate command should be
separated by a colon. If it is necessary to obtain a result for a fixed value of x
(for example, to subtract the value of the product of the solution at this
point), for example, at x = 0.5, then the following is dialed (Figure 1):
> de(0.5);
> with(plots):
Scientific research results in pandemic conditions (COVID-19)
98
>odeplot(de,[x,y(x)],-10..10,thickness=2);
Figure 1. Graph of the numerical solution of the Cauchy problem.
Cauchy problem or solution of boundary value problem. The dsolve
command can find a solution to a Cauchy problem or a boundary value
problem if the initial and boundary conditions of the indefinite function are
given for a given differential equation. A differential operator is used to
denote products under initial or boundary conditions, for example, y '' (0) =
2, or y '(1) = 0. Note that it is written as an n-th order product.
1). Let's try to solve the problem in practice and do the following, that is,
find a solution to the Cauchy problem.
:
y
(4)
+
y''=2cosx, y2,
(0)=
y'(0)=1, y''(0)=0, y'''(0)=0.
Solution:
> de:=diff(y(x),x$4)+diff(y(x),x$2)=2*cos(x);
> cond:=y(0)=-2, D(y)(0)=1, (D@@2)(y)(0)=0,
(D@@3)(y)(0)=0;
2, D(y)(0)=1, (D
cond:=y(0)=
(2)
)(y)(0)=0, (D
(3)
)(y)(0)=0
> dsolve({de,cond},y(x));
y(
x2cos(
) =
x
)
xsin(x)+x.
2). Solve another type of simple differential equation using the Maple
program using a variety of analytical methods:
Solution:
> ode_L:=sin(x)*diff(y(x),x)-cos(x)*y(x)=0;
> dsolve(ode_L,[linear],useInt);
Scientific research results in pandemic conditions (COVID-19)
99
> value(%);
> dsolve(ode_L,[separable],useInt);
> value(%);
There is no clear analytical solution to most types of differential
equations. In this case, the solution of the differential equations can be found
using approximate methods, that is, by finding the indefinite function by
spreading it on a series of levels.
To find the solution of a differential equation in the form of a power
series, you must specify the type = series (or simply series) parameter after
the variables in the dsolve command. To specify the nth order of the spread,
that is, the order of the degree before the end of the spread, place the
command Order: = n, which determines the order before the dsolve
command.
Figure 2. Graph of the solution of the Cauchy problem.
Scientific research results in pandemic conditions (COVID-19)
100
Now we find the solution of the Cauchy problem in the form of a series
of levels, and make a numerical solution and a graph of the resulting level
series for the interval at which they can fall (Figure 2).
> dsolve({eq, cond}, y(x), series);
> convert(%, polynom):p:=rhs(%):
> p1:=odeplot(de,[x,y(x)],-3..3, thickness=2,
color=black):
> p2:=plot(p,x=-3..3,thickness=2,linestyle=3,
color=blue):
> display(p1,p2);
1-The graph shows that the values of the solution that are very close to
the power series are -3 <x <1.
If it is necessary to solve such problems in a simple mathematical way,
as well as to create a schedule, it will take a lot of time and skills from
students, researchers and teachers. From the above problem, it can be easily
solved in Maple environment and at the same time create a graph of it.
References:
1.
Алексеев Е.Р., Чеснокова О.В. Решение задач вычислительной
математики в пакетах Mathcad, Mathlab, Maple (Самоучитель). М.: НТ
Пресс, 2006. – 496 с.
2.
Прохоров Г. В., Леденев М. А., Колбеев В. В. Пакет символьных
вычислений Maple V. М.: Петит., 1997.-200 с.
3.
Begmatov A. Differensial tenglamalar haqida umumiy tushunchalar.
Samarqand 2011.
4.
Голоскоков А.К. Уравнения математической физики. Решение
задач в системе Maple. Учебник для вузов. СПб.: Питер, 2004. – 448 с.
5.
Говорухин В.Н., Цибулин В.Г. Введение в Maple V.
Математический пакет для всех. М.: Мир, 1997. – 596 с.
6.
R Turgunbayev, Sh Ismailov, O Abdullayev. Differensial tenglamalar
kursidan misol va masalalar to`plami. Toshkent 2007.
