Integration

The process of elimination by which the differential equation f (x, y, y') = 0 was obtained from the primitive equation f (x, y, C) = 0 is in general not reversible; the action of recovering the primitive, or an equivalent expression, is known as integration. More precisely, to integrate or solve a differential equation of the first order is to determine all the relations f (x, y) = 0 such that the values of y and y ' deduced from them in terms of x shall satisfy the differential equation identically.

When an infinitive set of such integrals can be grouped in one comprehensive formula, involving an arbitrary constant, say

f (x, y, C) = 0,

it is known as a general integral; it is in fact either the primitive or an expression equivalent to it. General theory proves that a differential equation of the first order has one and only one distinct general integral. If two integrals exist, each of which involves an arbitrary constant, they can be transformed into one another. Any integral that can be obtained by assigning a definite numerical value to C is a particular integral. But there may be integrals other than those that can be obtained by assigning particular values to C; there are singular integrals.

As an example, the equation

(y')² – xy' + y = 0

(which is of the first order and second degree) possesses the general integral y = C x – C²; this represents a family of straight lines, and any particular integral corresponds to a definite line in the family. But the equation is satisfied also by y = 1: 4 x²; which represents not a straight line but a parabola. This is a singular integral.

T E X T IV