Ordinary differential equations

 

Definitions. 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. Such a relationship, in general, explicitly involves the variables x and y together with other symbols a,b,c … which represent constants.

This restricted use of the term was soon abandoned; differential equations are now understood to include any algebraical or transcendental equalities which involve either differentials or differential coefficients. It is to be understood, however, that the differential equation is not an identity.

Differential equations are classified, in the first place according to the number of variables which they involve. An ordinary differential equation expresses a relation between an independent variable, a dependent variable and one or more differential coefficients of the dependent with respect to the independent variable. A partial differential equation involves one dependent and two or more independent variables, together with partial differential coefficients of the dependent with respect to the independent variables. A total differential equation contains two or more dependent variables together with their differentials or differential coefficients with respect to a single independent variable which may, or may not, enter explicitly into the 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 , the symbol denoting .

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. When, in an ordinary or partial differential equation, the dependent variable and its derivatives occur to the first degree only, the equation is said to be linear. The coefficients of a linear equation are therefore either constants or functions of the independent variable or variables.

Thus, for example:

 

- is an ordinary equation of the second order;

(x + y)² = 1 - is an ordinary non-linear equation of the first order and the first degree;

 

- is an ordinary equation of the second order which when rationalised by squaring both members is of the second degree;

 

- is a linear partial differential equation of the first order in two independent variables;

 

+ = 0 - is a partial differential equation of the second order in three

independent variables;

 

- - is a non-linear partial differential equation of the second

order and the second degree in two independent variables;

Udx + Vdy + Wdz = 0, where U, V, W are functions of x, y and z, is a total

differential equation of the first order and the first degree and

 

x²dx² + 2xydxdy + y²dy² – z²dz² = 0 - is a total differential equation of the first

order and the second degree.

 

In the case of a total differential equation any of the variables may be regarded as independent and the remainder as dependent, thus, taking x as an independent variable, the equation Udx + Vdy + Wdz = 0 may be written

 

U +V = 0 or an auxiliary variable t may be introduced and the original variables regarded as functions of t, thus

 

U + W = 0.

 

The solutions of an ordinary differential equation.

When an ordinary differential equation is known to have been derived by the process of elimination from a primitive containing n arbitrary constants, it is evident that it admits a solution dependent upon n arbitrary constants. But since it is not evident that any ordinary differential equation of order n can be derived from such a primitive, it does not follow that if the differential equation is given a priori it possesses a general solution which depends upon n arbitrary constants. In the formation of a differential equation from a given primitive it is necessary to assume certain conditions of differentiability and continuity of derivatives. Likewise in the inverse problem of integration, or proceeding from a given differential equation to its primitive corresponding conditions must be assumed to be satisfied. From the purely theoretical point of view the first problem which arises is that of obtaining a set of conditions, as simple as possible, which when satisfied ensure the existence of a solution. This problem will be considered later, where an existence theorem, which for the moment is assumed, will be proved, namely, that when a set of conditions of a comprehensive nature is satisfied, an equation of order n does admit of a unique solution dependent upon n arbitrary initial conditions. From this theorem, it follows that the most general solution of an ordinary equation of order n involves n, and only n, arbitrary constants.

It must not, however, be concluded that no solution exists which is not a mere particular case of the general solution. To make this point clear, let us consider, for instance, the differential equation of the following form

c² + 2cy + a² – x² = 0, where c is an arbitrary, and a is a definite constant.

Let the primitive be solved for c and let this value of c be substituted into the derived equation. Then the derived equation becomes the differential equation

cdy – xdy = 0, which, on eliminating c, becomes the differential equation

(–y + (x² + y²) – a²)½ dy – xdx = 0.

The total differential equation obtained by varying x, y and c simultaneously is (c + y) dc + cdy – xdx = 0 or, on eliminating c

(x² + y² – a²)½ dc + (– y + x² + y² – a²)½ dy – xdx = 0

Thus, apart from the general solution there exists the singular solution

x² + y² = a², which obviously satisfies the differential equation. A differential equation of the first order may be regarded as being but one stage removed from its primitive. An equation of higher order is more remote from its primitive and therefore its integration is in general a step-by-step process in which the order is successively reduced, each reduction of the order by unity being accompanied by the introduction of an arbitrary constant. When the given equation is of order n, and by a process of integration an equation of order involving an arbitrary constant is obtained, the latter is known as the first integral of the given equation.

Thus, when the given equation is y'' = f(y), where f(y) is independent of x, the equation becomes integrable, when both members are multiplied by 2y', thus 2y ' (prime) y'' (double prime) = 2f(y) y', and its first integral is

y'2 = (c +2) f(y)dy, where c is the arbitrary constant of integration.

 

Post-Reading Activity

Ex. 11. Answer the following questions.

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?

Ex. 12. Match the English words and word combinations with the Russian equivalents.

1. an ordinary differential equation, 2. a partial differential equation, 3. a total differential equation, 4. an auxiliary variable, 5. a primitive, 6. differentiability and continuity, 7. point of view, 8. an existence theorem, 9. arbitrary constants, 10. simultaneously, 11. the first integral of the equation, 12. a step-by-step process, 13. differential coefficients, 14. apart from.

a. дифференцируемость и непрерывность; b. произвольные константы; c. обыкновенное дифференциальное уравнение; d. точка зрения; e. одновременно; f. дифференциальное уравнение в частных производных; g. первый интеграл уравнения; h. теорема существования; i. уравнение в полных дифференциалах; j. кроме, не говоря о, не считая; k. поэтапный процесс; l. вспомогательная переменная; m. дифференциальные коэффициенты; первообразная.

Ex. 13. 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 known 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. Let the primitive be solved for c and this value of c be substituted into the derived equation. 7. When the given equation is of order n, and by a process of integration an equation of order involving an arbitrary constant is obtained, the latter is known as the first integral of the given equation. 8. To make this point clear, let us consider, for instance, the differential equation of the following form c² + 2cy + a² - x² = 0.

 

Ex. 14. Find out whether the statements are true or false. Use introductory phrases.