12. Integrate the equation

dx

+

dy

0,

√(a+bx+cx2+ex3 +ƒx*) * √(a+by+cy2+ey3+fy*)

by the application of the theorem of Art. 13.

13. Deduce from that theorem the following expression for the value of a definite integral, viz.:

CHAPTER VII.

ON DIFFERENTIAL EQUATIONS OF THE FIRST ORDER, BUT NOT OF THE FIRST DEGREE.

1. REFERRING to the general type of differential equations of the first order, viz.:

dy

dx

we have now to consider those cases in which is so involved that the given equation cannot be reduced to the form

Freed from radicals the supposed equation will, however, present itself in the form

where P1, P2, ... P, are functions of x and y.

P22

An obvious preparation for the solution of such an equation, is to resolve its first member, considered as algebraic with

dy
dx'

respect to the differential coefficient into its component

factors of the first degree. If p1, Pa... Pa be the roots of (1) thus considered, we shall have

(dy - p.) (dy - p.)... (dy - p.) =

B. D. E.

dx

Pn

= 0

....

...(2),

8

...

tions of x and y.

P1, P2, P. being supposed to be determined as known funcAnd it is now manifest that any relation between x and y which makes either one or more than one of the factors of the first member to vanish, will be a solution of the equation, and that no relation between x and y not pos sessing this character will be such. Hence if we solve the separate equations

And if we

any one of the solutions obtained will be a solution of (2), since it will make one of its factors to vanish. express the different solutions thus obtained, each with its arbitrary constant annexed, in the forms

V1- C1 =0, V2- C2 = 0, ... V2- C2 = 0,

1

2

any product of two or more of these equations will also be a solution of (2), since it will cause two or more of its factors to vanish.

Either of these equations is a solution of the given equation,

and so is their product

(log y-ax —c) (log y + ax − c,) = 0 ..........

...(7).

2. And here two important questions are suggested. First, how is it that two arbitrary constants present themselves in the solution of an equation of the first order? Secondly, is it possible to express with equal generality the solution of the equation by a primitive containing a single arbitrary constant in accordance with what has been said of the genesis of differential equations of the first order, Chap. I. Art. 6? These are connected questions, and they will be answered together.

The equation (7) implies that y admits of two values each involving an arbitrary constant, but it does not imply that y admits of a value involving two arbitrary constants. The component factors of the solution separately equated to 0, as in (5) and (6), give respectively

each of which involves one arbitrary constant only, and each of which corresponds to a single factor of the given differential equation. The true canon is, not that a general solution of an equation of the first order can involve only one arbitrary constant in its expression, but that each value of y which such a solution establishes involves in its expression only a single arbitrary constant.

At the same time there is a real sense in which it remains true that every differential equation of the first order, whatever its degree may be, implies the existence of a complete primitive involving a single arbitrary constant, and there is a real sense in which such primitive constitutes the general solution of the differential equation. To reconcile these seeming contradictions I shall shew that if we suppose the arbitrary constants c, and c, in (7) identical, and accordingly replace each of them by c, we shall have an equation which will be, first the true primitive of (4), in that it will generate that equation by differentiation and the elimination of c, secondly its general solution, in that no particular relation is deducible from the solution (7) involving two arbitrary constants which may not also, by the use of a lawful freedom of interpretation, be derived from it.

whence (logy) - a2x2 - 2c log y + c2 = 0.

Or, rejecting the factor ax which does not contain p, and

replacing p by

dy

the differential equation given. Thus (9) is its complete primitive.

Again, that solution is general. The two relations between y and x which it furnishes are

and these differ in expression from (8) only in that the arbitrary constant is here supposed to be the same in one as in the other, but as it is arbitrary and admits of any value, there is no single relation implied in (8) which is not also implied in (10). And it is in this sense that the generality of the solution is affirmed.

[See the Supplementary Volume, Chapter xx. Art. 1.]