I don’t think this is the case. Just the reverse.

1. The term “differential equation” was first used by Leibniz in the 16th century. 2. Differential equations are now understood to include only algebraical equalities which involve differential coefficients. 3. It is important to remember that the differential equation isn’t an identity. 4. A partial differential equation involves two dependent and three independent variables. 5. The order of the equation is that of the lowest derivative contained in it. 6. The most general solution of an ordinary equation of order n involves n, and only n, arbitrary constants. 7. A differential equation of the first order may not be regarded as being one stage removed from its primitive. 8. An equation of higher order is less remote from its primitive.

Ex. 15. 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 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.

Ex. 16. Translate into English.

1. 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 согласно числу переменных, которые они включают. 3. Из теоремы существования следует, что общее решение обыкновенного дифференциального уравнения порядка n содержит n и только n произвольных постоянных. 4. Дифференциальное уравнение в частных производных содержит одну или несколько независимых переменных с частными производными зависимых производных по независимым переменным. 5. Коэффициенты линейного уравнения являются либо постоянными, либо функциями независимых переменных. 6. При образовании дифференциального уравнения из данной первообразной необходимо допускать некоторые условия дифференцируемости и непрерывности производных. 7. Пусть первообразная решается для с и пусть это значение с подставляется в выведенное уравнение. 8. Когда множество условий выполняется, уравнение порядка n имеет единственное решение, зависящее от исходных условий.

Ex. 17. Topics for discussion.

1. The nature of the term “differential equation”.

2. The classification of differential equations according to the number of variables.

3. The classification of differential equations according to the order and the degree of the equation.

Ex. 18. Read the text and find the answers to the following questions.

1. Where do differential equations arise?

2. What is given as an example of modeling a real world problem using differential equations?

3. What does finding the velocity as a function of time involve?

4. Where is the study of differential equations a wide field?

5. What does pure mathematics focus on?

6. What does applied mathematics emphasize?

7. Where do differential equations play an important role?

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