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 arbitrary constants, it is evident that it admits of a solution dependent upon arbitrary constants. But since it is not evident that any ordinary differential equation of order 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 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 in Chapter III., 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 does admit of a unique solution dependent upon arbitrary initial conditions. From this theorem it follows that the most general solution of an ordinary equation of order involves , and only , 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, consider the differential equation obtained by eliminating the constant from between the primitive,

and the derived equation,

The derived equation in general involves ; let the primitive be solved for and let this value of be substituted in the derived equation. The derived equation then becomes the differential equation

where the brackets indicate the fact of the elimination of . In its total form, this equation can be written

Now let and vary simultaneously, then

When is eliminated as before this equation becomes

and therefore, in view of the previous equation,

There are thus two alternatives: either is a constant, which leads back to the primitive,

or else

The latter relation between and may or may not be a solution of the differential equation; if it is a solution, and is not a particular case of the general solution, it is known as a singular solution.

Consider, for instance, the primitive

where is an arbitrary, and a definite, constant. The derived equation is

which, on eliminating , becomes the differential equation

The total differential equation obtained by varying and simultaneously is

or, on eliminating ,

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

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 , 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

where is independent of , the equation becomes integrable when both members are multiplied by , thus

and its first integral is

where is the arbitrary constant of integration.