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Юриспунденкция

ORDINARY DIFFERENTIAL AQUATIONS

Definitions. The term equation differentialis or 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 a differential equation is the order of the highest differential coefficients which is involved. 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. When, in an ordinary or partial differential equation, the dependent variable and its derivatives occur to the first degree only, and not as higher power or products, 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

d²y (1)

d x²

is an ordinary equation of the second order;

(x + y)² dy = 1(2)

d x

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

(d y)² d²y (3)

d x d x²

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

¶ z¶z (4)

dx dy

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

¶ ²y¶² y¶²y (5)

¶ x²¶ y²¶ z²

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

¶ ²z¶² z¶²z (6)

¶ x²¶ y²¶ x¶y

is a non-linear partial differential equation of the second order and the second degree in two independent variables;

Udx + Vdy + Wdz = 0 (7)

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 (8)

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 he regarded as independent and the remainder as dependent, thus, taking x as an independent variable, the equation udx + vdy + wdz = 0 may be written

dydz (9)

dx dx

or an auxiliary variable t may be introduced and the original variables regarded as functions of t, thus

dxdydz

dt dt dt

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 of 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, become the differential equation

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

The total differential equation obtained by varying x, y and c simulta- neously is

( c + y) dc + cdy – xdx = 0

or, on eliminating c.

(x² + y²– a²)^1/2 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 n^–1 involving an arbitrary constant is obtained, the latter is know 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' y'' = 2f(y) y',

and its first integral is

y'² = (c +2) f(y) dy,

where c is the arbitrary constant of integration.

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