Differential Equations

The term differential equation was first used by Leibniz in 1676 to denote a relationship between the differentials dx and dy of two variables x and y. Now differential equation s are understood to include any algebraical or transcendental equalities which involve either differentials or differentials coefficients. Differentials may enter in place of derivatives.

If the unknown functions depend on one argument, the differential equation is called ordinary, if they depend on several ones, then the equation is termed a partial differential equation. The general form of a differential equation in one unknown function is

Ф (x, y, у', у",… y ⁽ⁿ⁾) = 0.

The order of the differential equation is the order of the highest derivative in the equation. When an equation is polynomial in all the differential coefficient involved, the power to which the highest differential coefficient is raised is known as the degree of the equation. The equation is said to be linear when the dependent variable and its derivatives occur to the first degree only.

Examples. The equation у' = y ²/x (1)

is a first-order differential equation.

The equation of the type d²y/dx² +y= x³ (2)

is an ordinary linear equation of the second order.

Consider the following type of a differential equation.

(x + y)² = 1 (3)

is an ordinary non-linear equation of the first order and the first degree.

And at last the equation given below

x + yz = 0 (4)

is a linear partial differential equation of the firstorder in two independent variables.

One calls a function у = Ф (x) a solution of a differential equation if, substituted into the equation, it reduces the equation to an identity. The basic task of the theory of differential equations is to find all the solutions of a given differential equation. In the simplest case, this reduces to evaluating an integral. For this reason, the solution of a differential equation is also called its integral and the process of finding all the solutions is called integrating the differential equation. Generally, the integral of a given differential equation is any equation, not containing derivatives, from which the given differential equation follows as a consequence.

Questions:

1. What did the term differential equation denote when Leibniz started using it? 2. What differential equation is called partial? 3. In what way can one define the order of a differential equation? 4. What is a solution of a differential equation? 5. What does it mean to integrate a differential equation?

II. Письменно переведите 2 и 4 абзацы текста.

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

1. There is no general method to determine the limit. 2. To raise a product to a power, it is sufficient to raise each of its factors to that power. 3. To understand this phenomenon is to understand the structure of atoms. 4. Abel was not the first to make an attack on the general equation of the fifth degree. 5. He is studying mathematics in order to qualify for a better job. 6. He thought it safer to go there by train.

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

1. No one appears to have taken the implication of this idea seriously. 2. We now believe differential equations to include any algebraic equality involving either differentials or differential coefficients. 3. Two distinct points are said to be symmetric with respect to the axis of symmetry. 4. The scientists were made to fulfill a lot of experiments. 5. A line is certain to be normal to another when it meets the other at right angles. 6. At the same time we have observed the initial signals not to change with temperature.

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

1. One is to be very attentive when crossing the street. 2. We are to get a 10 per cent wage rise in June. 3. Our present concern will be to analyse the information obtained during the experiment. 4. You are to do your homework before you watch TV. 5. Our suggestion was to make use of the old equipment. 6. The problem has been to simplify the procedure under consideration.

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

1. My science advisor brought some papers for me to look them through. 2. It is impossible for a single force to produce the same effect as a couple. 3. The best decision for us to make at the moment is to wait and see. 4. For the experiment to be finished in time, we must begin to work immediately. 5. The article was provided with diagrams for the reader to understand it better. 6. His idea is for us to travel in separate cars.

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

1. Making use of these properties will help us greatly. 2. Parentheses preceded by a plus sign may be removed from an expression without changing the signs of the terms in parentheses. 3. In writing and reading numbers, the figures are separated into groups of three figures each, called periods. 4. It seems to me the case is not worth mentioning. 5. We don’t feel like discussing the problems of prime numbers any more. 6. There is no point in developing the notions concerning infinitesimals and limit any further.

VIII. Письменно переведите предложения, содержащие герундиальные обороты.

1. His having succeeded in solving the problem was quite unexpected for our scientific community. 2. There is no hope of our getting a complete analysis of the data within ten days. 3. Would you mind this student answering one more question? 4. We don’t object to your developing an alternative theory as this one seems unsatisfactory. 5. I appreciate your giving me so much of your time. 6. Newton’s having discovered the laws of mechanics determined the development of science for many years to come.

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

1. One must be very careful when formulating this statement. 2. One needs to know all rules of locating a point on the surface. 3. The technique used had some advantages over that suggested by Professor Brown. 4. One believes that the procedure mentioned above will simplify the solution of the problem. 5. The computer allows one to make calculations in a short time. 6. A very powerful method of integration is that of changing the independent variable. 7. We can advise you several procedures, but this is the most reliable one.