The integral of this equation is deduced from that in

may be integrated by the same method as that in Ex. (14) of

Chap. vI., by changing k into

d

a2 and putting arbitrary dy2

functions of y instead of the arbitrary constants.

= 0

dya

is x=3{F(y+ax)+ƒ (y−ax)} − 3 ax {F' (y+ax) − ƒ' (y—ax)}

+ a2x2 {F" (y+ax) + ƒ" (y − a x)}.

1

— {p' (u + v) + y' (u − v) } − — { $ (u + v) + y (u − v) }.

P, Q, R being functions of x and y, may be transformed into linear equations by assuming

We might with advantage have applied the same transformation to the equations in examples (1), (3), and (4), as it is generally convenient to reduce the factor of ≈ to two

terms.

SECT. 3. Equations involving the differential coefficients of x in powers and products.

If the equation be of the first order make

dx

dz

= P,
dx dy

and from the given equation find q in terms of p, x, y, z, and substitute this value in the equation

which will then become an equation of the first order between four variables. The value of p found by integrating this, with the corresponding value of q will render

a complete differential, and this being integrated will give the value of x. The integral of the first equation will involve an arbitrary constant (a); and the integral of the second will introduce another (b), which is to be considered as an arbitrary function of (a); and we shall thus obtain an integral of the form

ƒ (x, y, ≈, a) = $ (a),

from which a is to be eliminated when a specific meaning is assigned to p.

Lagrange, Mémoires de Berlin, 1772, p. 353.

(1) Let p2 + q2 = 1, or q = (1 - p2)3,

Substituting these values in equation (1) it becomes

This equation is integrable if we can integrate the

system of equations

dp=0, pdxdx = 0, (1 - p) dx dy = 0.

The first gives pa, whence q = (1 − a2), and

so that

dz = a dx + (1 - a2) dy,

≈ = a x + (1 − a2)3 y + † (a).

If we differentiate this with respect to a

we obtain the

equation

a

(= x

(1 - a2)

y + $' (a),

between which and the preceding we may eliminate a when

is specified.

Y

n

a' - fa ̄ ̄da, y' = fy^dy, fd;

=

the equation becomes

= a

(da)" (dx')" = ~

=

c".

The integral of this found by the same method as in