Basic Concepts)

1. If a relation exists between a function of one or more variables and some of its derivatives, in the form of an equation in which the independent variables can also occur, then one speaks of a differential equation. Every solution of the differential equation is called a solution or an integral.

2. If the functions occurring in the differential equation depend on only one independent variable, and thus also derivatives with respect to only one variable occur, then one speaks of an ordinary differential equation. On the other hand if the required functions depend on several independent variables and accordingly partial derivatives occur, one speaks of partial differential equations.

3. The order of a differential equation is defined as the highest order of the derivatives contained in it. A differential equation of the n -th order can be expressed in the form F (x, y, у', у",… y ⁽ⁿ⁾) = 0, F denoting a function of the argument in the bracket. If F is a polynomial function or the argument у, у', … y ⁽ⁿ⁾, then its degree is equal to that of the differential equation.

4. If the equation F (x, y, у', … y ⁽ⁿ⁾) = 0, after the substitution of a function

у = φ (х) and its derivatives у', у",… y ⁽ⁿ⁾, becomes an identity in x valid for all x in an interval, then у= φ (х) is called a solution or integral, the process of obtaining it being called integration.

5. The solutions are often not elementary functions. It is often sufficient, without insisting on the complete solutions, to determine the analytic properties of a solution in the neighbourhood of a point x₀ and to investigate the uniqueness of the solution and other questions.

Questions:

I. What is a differential equation? 2. What is an integral or a differential equation? 3. What kinds of differential equations do you know? 4. How is the order of a differential equation defined? 5. What is an integration?

II. Письменно переведите 2 и 5 абзацы текста.

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

1. The language to be used here belongs to the first generation of programming languages. 2. To give a true picture of the surrounding matter is the task of natural sciences. 3. To understand the procedure, let us consider the following criteria. 4. Gauss was the first to notice a mistake in his father’s calculations. 5. She gave up work in order to have more time with the children. 6. To have made the same mistake twice was unforgivable.

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

1. The value introduced in the expression turns out to be an imaginary one. 2. All students are supposed to study Newton’s laws of mechanics in the first year. 3. We know each source of energy to present its own advantages and disadvantages. 4. These series are certain to be divergent when their terms are taken with the same sign. 5. Teacher made the pupils open their books and read the rules. 6. Scientists believed the further investigation of a number of other cases to reveal a similar complexity.

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

1. The next stage will be to publish the results of the research. 2. Students were to submit the results of their work yesterday. 4. You are to learn all the new words for the next lesson. 5. The object of the experiment is to prove the results. 6. Our suggestion was to repeat measurements. 7. When working with numerals, one is to be very careful with signs.

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

1. Two conditions must be met for the phenomenon to occur. 2. Was it interesting for you to read this book for the second time? 3. The teacher waited for the students to hand in texts. 4. For the problem to be solved, it must be stated clearly. 5. The student must concentrate on his work for the calculations to be correct. 6. There are too many people here for me to talk to all of them.

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

1. Solving mathematical puzzles is my favourite occupation. 2. We simplify equations by performing all necessary operations. 3. It is worth mentioning that Lobachevsky’s ideas influenced greatly not only the development of mathematics, but mechanics, physics, astronomy. 4. After discussing the problem in detail, they found the best solution. 5. We cannot help acknowledging the importance of this statement. 6 I am interested in finding out what she did with all that money.

VIII. Письменно переведите предложения, содержащие герундиальные обороты.

1. His having discovered this phenomenon contributed much to the world science. 2. Would you mind their showing us the whole process? 3. We objected to their initiating experiments without sufficient experimental basis. 4. I don’t remember my mother’s complaining about it. 5. The Earth’s moving around the sun is common knowledge nowadays. 6. He knew nothing of our having realized the program.

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

1. The simplest mathematical formulation for the function is one based on the use of exponential functions. 2. One should understand, however, that the problem is extremely difficult. 3. One wants to know how to find an actual expression of this function. 4. If the degree of the numerator is lower than that of the denominator, then the fraction is called proper. 5. The function of a new variable is one whose integral can easily be calculated. 6. One is to make a lot of experiments to make sure that his observation is adequate. 7. He is the one you wanted to speak to.