Moreover it contains virtually only one arbitrary constant, for the relation F(a, b) = 0 permits us to determine b as a function of a. Hence it will constitute the complete primitive of (3). See also Chap. I. Art. 10.

This result may be expressed in the following theorem.

If any differential equation of the first order be expressible in the form

F(4, 4) = 0.......

(4),

dy

such that the dif

where and are functions of x, y, dx'

are derivable from a single primitive involving a and b as arbitrary constants, the solution of the given differential equation will be found by limiting that primitive by the condition

F (a, b) = 0,

so as actually or virtually to eliminate one of the arbitrary

are derivable from a common primitive; for, on differentiating them, we have respectively

d'y +y

= 0,

dx

and these agree as differential equations of the second order, Chap. I. Art. 9. That common primitive, found by eliminating dy between (2) and (3), is

dx

We might also proceed as in the solution of Clairaut's equation. Differentiating the given equation, we have

as will be seen in Chap. x. Art. 1, in which the relation between b and a remains to be determined as before. The first factor equated to 0 constitutes the differential equation of the

singular solution, which will be obtained by eliminating

between that equation and the equation given.

dy

dx

Clairaut's equation belongs to the above class. We may express it in the form

10. Well-chosen transformations facilitate much the solution of differential equations of the first order.

Ex. 1. Given -p = f(x2+ y2). Lacroix, Tom. II. y-xp =ƒ(x2+y2)*. √(1+p3)

p. 292.

Assuming xr cos 0, y=r sin 0, we have

As y-xp is the expression for the length of the per√(1+p2)

pendicular let fall from the origin upon the tangent to a curve, the above is the solution of the problem which proposes to determine the equation of a curve in which that perpendicular is a given function of the distance of the point of contact from the origin.

To render the above equation homogeneous if possible let y=2"; we find

This will be homogeneous with respect to z and x, if we have

the former of which expresses a condition between the indices of the given equation, the latter the value which must be given to n when that condition is satisfied.

It appears then that the equation

can be rendered homogeneous by the assumption y = 28.

If the more general transformation y=2", x=t", which seems at first sight to put us in possession of two disposable constants, be employed, the necessity for the fulfilment of the same condition between a, B, and k, will not be evaded, the ratio of the constants m and n, not their absolute values, proving to be alone available.

Ex. 3. The equation of the projection on the plane xy of the lines of curvature of the ellipsoid is

dy

2

Axy + (x2 - Ay — B) dy — xy = ...... (1)~

dx

dx

Assuming xs, y=t, the equation is reduced to one of Clairaut's form, Art. 6. Its solution is

=

The equation may also, without preliminary transformation, be integrated by Lagrange's method, Art. 9. We may express it in the form

are derived from a common primitive y-ax2=b. Thẹ solution of (2) will therefore be,

y2 - ax2 = b

with the connecting relation between the constants,

Aab+ Ba+b=0.

And this will be found to agree with the previous result.

EXERCISES.

The following examples are chiefly in illustration of Arts. 1, 2, 3, 5.