Vocabulary and grammar exercises. I. Give the Russian equivalents for

I. Give the Russian equivalents for:

an ordinary differential equation, a partial differential equation, a total differential equation, an auxiliary variable, a primitive, differentiability, point of view, a priori, an existence theorem, arbitrary constants, simultaneously, the first integral of the equation, a step-by-step process, differential coefficients, apart from.

II. Give the verbs corresponding to the following nouns and translate them into Russian.

differentiation, definition, use, variable, representation, identity, classification, expression, dependence, occurrence, remainder, solution, derivation, elimination, formation, integration, satisfaction, existence, consideration, substitution, introduction.

III. State the part of speech of the following words.

dependent, equation, powerful, differ, contain, order, variable, explicitly, degree, ordinary, regard, solution, possess, primitive, involve, constant, theoretical, mere, particularly, value, exist, unity.

IV. Translate the following sentences into Russian.

1. Differential equations are now understood to include any algebraical or transcendental equalities which involve either differentials or differential coefficients. 2. When an equation is polynomial in all the differential coefficients involved, the power to which the highest differential coefficient is raised is know as the degree of the equation. 3. The coefficients of the linear equation are either constants or functions of the independent variable or variables. 4. In the formation of a differential equation from a given primitive it is necessary to assume certain conditions of differentiability and continuity of derivatives. 5. From this theorem, it follows that the most general solution of an ordinary equation of order n involves n, and only n, arbitrary constants. 6. To make this point clear, let us consider, for instance, the differential equation of the following form c² + 2cy + a² - x² = 0. 7. Let the primitive be solved for c and this value of c be substituted into the derived equation. 8. When the given equation is of order n, and by a process of inte- gration an equation of order n-1 involving an arbitrary constant is obtained, the latter is known as the first integral of the given equation.

V. State the function of the gerund and translate into Russian.

1. Instead of representing the position of a point in a plane in terms of its horizontal and vertical distances along two standard lines of reference, it is sometimes more convenient to define the position of the point by length and direction. 2. The new parametric net is therefore the net whose equation in the old parameters u, v is written by setting the right member of this equation equal to zero. 3. Certain notions from analytic projective geometry are quite useful in interpreting some of the formulas of metric differential geometry. 4. In writing “the sine of the angle PON” in an equation or formula, it would be abbreviated sin PON. 5. By solving a triangle we mean that we have some of the sides and angles given and proceed to calculate the rest. 6. We found the solutions of this system of equations by eliminating unknowns, that is, by multiplying equations by scalars and then adding to produce equations in which some of the xy were not present. 7. Then we indicate that g is an element of G by writing g Î G. 8. The method of solving by successive eliminations may perhaps be known to the reader. 9. For the purpose of formulating a precise definition of the angle between two tangents at a point of a surface, a positive sense of rotation in the tangent plane of the surface at the point is assigned by this convention. 10. Our purpose in considering two separate problems is one of convenience. 11.The method of illustrating the variation of functions by the use of graphs is well-known to the reader. 12. The problem of subtracting a number from a smaller number is considered impossible in arithmetic.

VI. Answer the questions on the text.

1. When was the term “differential equation” first used? 2. What did this term denote at that time? 3. What does this term denote now? 4. In what ways are differential equations classified? 5. What kinds of differential equations do you know? 6. What does an ordinary differential equation express? 7. What does a partial differential equation involve? 8. What does a total differential equation contain? 9. What is the order of a differential equation? 10. What is the degree of a differential equation? 11. When is an ordinary or a partial differential equation linear? 12. What does the most general solution of an ordinary equation of order n involve?

VII. Translate into English.

I. Обыкновенное дифференциальное уравнение выражает связь между зависимой и независимой переменными, а также между одной или несколькими производными зависимой переменной и независимой переменной. 2. Дифференциальные уравнения классифицируются со- гласно числу переменных, которые они включают. 3. Дифференциальное уравнение в частных производных содержит одну или несколько независимых переменных с частными производными зависимых производных по независимым переменным. 4. Коэффициенты линейного уравнения являются либо постоянными, либо функциями независимых переменных. 5. При образовании дифференциального уравнения из дан- ной первообразной необходимо допускать некоторые условия дифференцируемости и непрерывности производных. 6. Из теоремы существования следует, что общее решение обыкновенного дифференциального уравнения порядка n содержит n и только n произвольных постоянных. 7. Пусть первообразная решена для с и пусть это значение с подставлено в выведенное уравнение. 8. Когда множество условий выполняется, уравнение порядка n имеет единственное решение, зависящее от исходных условий.