clearing the result of fractions and arranging it according to the powers of y, we have

dQ_dR

dx dx

(2P + 12 - d) y2 + (PR + R 22 - 9-de):

dQ dR
Q
dx

dx

= 0..

Whence, equating separately to 0 the coefficients of the different powers of y, we have the ternary system

The last equation gives S-cQ, c being an arbitrary constant. Substituting this value of S in the equation obtained by eliminating P from the first two equations of the system, we find

(2c-R) dQ+2QdR = RdR,

or, regarding therein R as the independent and Q as the dependent variable,

and from the substitution of the value of Q in the first equation of the ternary system,

These values of S, Q, and P, in which R is arbitrary, reduce the given differential equation to the form

{R-c+c' (R-2c)2 + y} dy - c'y (R- 2c) dR=0...(32), and present its integrating factor in the form

1

y3 + Ry* + {c (R − c) + cc′ (R − 2c)2} y'

R being an arbitrary function of x.

For other examples the student is referred to Lacroix (Traité du Calcul Diff. et du Calcul Int. Vol. II. Chap. IV.). The results of this method are usually of a very complex character, while their generality is limited by the restrictions which must be imposed in order to render the system of reducing equations solvable. Thus Euler's equation above considered is virtually only a limited case of the general equation (21). If we assume

it becomes

y+c=s, R-2c=t,

(s+t) ds + cc'tdt + c ́t (tds — sdt),

which evidently falls under that equation.

[The Jacobian theory of the Last Multiplier, which is connected with the subject of the present Chapter, is discussed in the Supplementary Volume, Chapter XXXI.]

EXERCISES.

1. The following equations admit of integrating factors of the form (x), viz.

(1) (x2 + y2+ 2x) dx + 2y dy = 0.

(2) (x2+ y2) dx-2xy dy = 0.

Determine these factors and integrate the equations.

2. The equation 2xy dx + (y3-3x2) dy = 0 has an integrating factor which is a function of y. Determine it, and integrate the equation.

3. Find those integrating factors of the equation

ydx+(2y-x) dy = 0

which are homogeneous functions of x and y of the respective degrees 0 and -2, and from the consideration of those factors deduce the complete primitive of the equation.

4. For each of the following equations examine whether there exists an integrating factor μ satisfying the particular condition specified, and if so determine the factor, and integrate the equation.

(1) y (x2 + y2) dx + x (xdy — ydx) = 0, μ a homogeneous function of the degree - 3.

(2) (y+axy) dy-ay3dx + (x + y) (xdy - ydx) = 0, μ as in the previous example.

(3) (y-x) dy + ydx - xd

degree - 1.

=

0,μ homogeneous of the

(4) (x2+ y2+1) dx - 2xydy = 0, μ a function of y3- x2.

(5) (y-3x2y3 - 2x3) dx + (2y3 + 3x3y3— x) dy = 0, μ a function of x+y.

(6) (x2+x2y+2xy — y3—y3) dx+(y2+xy2+2xy—x2—x3)dy=0, μ a function of the product (1+x) (1+ y).

(7) (3y2 - x) dx + (2y3 — 6xy) dy = 0, μ a function of x + y2.

5. The equation y (x2 + y2) dx + x (xdy — ydx) = 0 has an integrating factor of the form ep (x+y). Determine it, and, from the comparison of the result with that of (1) Ex. 4, deduce the complete primitive.

dy

6. The linear equation

dx

+Py = Q having an integrating

factor of the form ea, deduce a corresponding expression for an integrating factor of the equation

where P is any function of x, has an integrating factor of the €-2f Pila

form

(y — P)* *

Lacroix, Tom. II. p. 278.

where P and Q are functions of x, can be made integrable

by a factor of the form

y

{y+(~)]=, and determine the form

{y+f(x)}}"

of f(x).

CHAPTER VI.

OF SOME REMARKABLE EQUATIONS OF THE FIRST ORDER AND DEGREE.

1. THERE are certain differential equations of the first order and degree, to which, in addition to their intrinsic claims upon our notice, some degree of historical interest belongs. Among such, a prominent place is due to two equations which, having been first discussed by the Italian mathematician Riccati and by Euler respectively, have from this circumstance derived their names. To these equations, and to some other allied forms, the present Chapter will be devoted. Riccati's equation is usually expressed in the form

But as both it and some other equations closely related to it and possessing a distinct interest, may, either immediately or after a slight reduction, be referred to the more general equation

the discussion of which happens to be much more easy than that of the special equations which are included under it, we shall consider this equation first.

To reduce Riccati's equation under the general form (2),

which is seen to be a particular case of (2).