Контрольное задание № 2. I. Прочитайте текст, переведите его и письменно ответьте на вопросы, следующие за текстом

I. Прочитайте текст, переведите его и письменно ответьте на вопросы, следующие за текстом.

Differential Equations (Introductory)

Equations in which the unknown function or the vector function appears under the sign of the derivative or the differential are called differential equations. An ordinary differential equation is an equation involving one or more derivatives of the dependent variable у with respect to a single independent variable x. But if the unknown function appearing in the differential equation is a function of two or more independent variables, the differential equation is called a partial differential equation.

The order of the equation is that of the highest derivative contained in it, so that the general differential equation of order n can be written in the form

F(y⁽ⁿ⁾, y^(n-1), …y⁽¹⁾, y,)=0 (1)

the symbol y^r denoting d^ry/dx^r *. The degree of the equation is defined mathematically to be that of its highest order derivative, when the equation has been made rational as far as the derivatives are concerned. The equation

yⁿ = (xy^5/2+y')^1/2 (2)

for example, is of order 2 and degree 2, since the equation must be squared to rationalize the contributions from the derivatives.

We shall refer to an equation of the form

yⁿ+f_n-1(x)y^(n-1)+f_n-2(x)y^(n-2)+…+f₁(x)y'+f₀(x)y=g(x) (3)

in which the f_r(x) are functions of x only, as linear, and any other type of equation will be called non-linear. Any ordinary or partial differential equation is said to be linear when the dependent variable and its derivatives occur to the first degree only, and not as higher power or products.

The solution of a differential equation is a function which, when substituted into the differential equation, reduces it to an identity. It can be proved that the most general solution of an ordinary differential equation of order n contains n arbitrary constants. This general solution is called the Complete Primitive, and a Particular Integral is obtained by giving specific values to these arbitrary constants.

The procedure of finding the solutions of a differential equation is called integration of the differential equation. In some cases it is easy to find an exact solution, but in more complicated cases it is very often necessary to apply approximate methods of integrating differential equations. Obtaining an exact or approximate solution of initial-value problems is the principal task of the theory of differential equations, however it is often required to determine only certain properties of solutions. For instance, one often has to establish whether periodic or oscillating solutions exist, to estimate the rate of increase or decrease of solutions, and to find out whether a solution changes appreciably for small changes in the initial values.

Questions:

1. What is a differential equation? 2. What differential equation is called ordinary? 3. What determines the order of a differential equation? 4. What is the procedure of finding the solution of a differential equation? 5. What is the principal task of the theory of differential equations?

II. Письменно переведите 2 и 5 абзацы текста.

III. Переведите письменно следующие предложения, обращая внимание на функции инфинитива.

1. The first point to notice is that the above definition does not give a unique value for the integral. 2. Lobachevsky was the first to publish a paper on non-Euclidean geometry. 3. To prove Euclid’s fifth postulate became senseless after the discovery of new geometries. 4. To be able to state what the number represents, one has to recall the special definition for a number raised to an irrational power. 5. She learnt typing in order to help her husband with his work.

6. He found it easy to earn extra money.

IV. Переведите письменно следующие предложения, принимая во внимание особенности перевода инфинитивных оборотов Complex Subject и Complex Object.

1. The interpretation of these symbols is thought to be not correct. 2. This student is unlikely to tell a basic difference between linear and non-linear systems. 3. We expect the latest developments in the field of science to stimulate experiments on a more professional level. 4. The regulations require me to wear this uniform. 5. His method proved to be the only possible one. 6. No one heard her open the door and go out.

V. Переведите письменно следующие предложения, обращая внимание на перевод глагола to be с последующим инфинитивом.

1. A very complicated problem is to be solved to achieve good results. 2. We were to meet and discuss the problem on Friday. 3. Our plan was to begin the experimental part of the research by the end of the month. 4. His idea is to develop a new technology of the process. 5. I was to have started the work last week, but I changed my mind. 6. Our aim is to improve the results of the previous experiments.

VI. Переведите письменно следующие предложения, содержащие “ for-phrase”.

1. It is not advisable for children to go to bed late. 2. Much experimental work is needed for these phenomena to be explained. 3. For a force to exist, there must be two objects involved. 4. Here are the computations for them to use in their work.

5. Everybody waited for the new data of the experiment to be published. 6. This decision was for her to be made.

VII. Переведите письменно следующие предложения. Обратите внимание на функции герундия в предложении.

1. They evaluated the equation by having substituted the numerical values for x and y. 2. A graphical method of verifying formulae in the algebra of sets is called Vienn diagrams. 3. After obtaining the equation it is often possible to determine further geometric characteristics for these conditions. 4. We have succeeded in maintaining productivity level over the whole period. 5. We don’t feel like analyzing the compatibility of the arithmetical axioms right now. 6. Children couldn’t get used to solving such difficult problems.

VIII. Письменно переведите предложения, содержащие герундиальные обороты.

1. My friend’s becoming a prize winner of the contest was the first good news for all of us. 2. His having presented the problem in every detail helped us greatly to grasp the idea. 3. They insist on our applying this theory to a wider range of phenomena. 4. The manager objects to our making private calls on the office phone. 5. You can’t prevent your son’s spending his own money. 6. Two scientists’ doing research independently made it possible to create two essentially different ways of solving the same problem.

IX. Письменно переведите предложения, обращая внимание на функции слов one и that.

1. There were a lot of books in the library, but I couldn’t find the one I wanted. 2. The similarity one finds in this case is not accidental. 3. One must think of another approach to solving the problem. 4. The newly developed technique has certain advantages over the old one. 5. The theory of integration of transcendental functions is less systematic than that of the integration of rational and algebraical functions. 6. One should know the history of one’s country. 7. These integrals can be reduced to those of rational functions by substitution.