Or, substituting for u and v their values in x and y,

≈ = — { (1 + y2)} + y} − * { (1+y°)1 −y} log x+ ƒ

2

Equations of the second and higher orders may sometimes be reduced by transformations similar to those employed in Chap. IV. Sect. 2.

By means of the same transformation as in the last example we find

the integral of which is v= (x + y)a p(x),

Assume da = xdu, dy = ydv; then by Ex. (6) of Chap.

III. Sect. 1, of the Diff. Calc. we have generally

But by a known theorem of Vandermonde if

[x] = x(x-1)... (x − r + 1),

Therefore, as the symbols of differentiation are subject to the same laws of combination as the algebraical symbols, the differential equation may be written

≈ = P(v − u) + €" P1 (v − u) + &c. + €(n−1)" Pn-1 (v − u) ;

y

or x = f. (-) + xƒ (-) + x2ƒ: (~2) + &c. + x2-1ƒ.-1 (~~);

fo, fi, &c. being arbitrary functions.

the integral of which (see Ex. (11) of the preceding section) is

-av

bu

≈ = e− (av+bu) fdv ε"" [du e1"V + e¬"" $ (u) + e−1u ↓ (v) ;

or ≈ =

fdy y-1 fdx x-1V +

ƒ (x) + /1, F (y).

By the same

this may be put

process as in Ex. (9) of Chap. IV. Sect. 2, under the form

and thence by the same process as in Ex. (10) of Chap. IV.

and therefore

1

x=

= = { p' (x + a y) + \' ( x − ay) } − = {p(x+ay) + 4 (x −ay)}.

This equation occurs in the Theory of Sound. Airy's Tracts, p. 271.

See

This equation is of the same form as that in Ex. (6) of Chap. v., and its integral will be found from that given

there by putting a

d

dy

for c, and changing the arbitrary con

stants into arbitrary functions of y. Hence we find

Z=X

F

a

- + {† (v + 2) + ƒ ( y − 9 )}·

(20) The integral of the equation

may in the same way be deduced from that of Ex. (8) of the same Chapter: the result is

The integral of this equation may be deduced from that

in Ex. (10) of Chap. v. by putting a2

d2

for q3. This

dy

gives us

1

x= {F (y − a x) − ƒ (y + ax) } + F' (y − a x) + ƒ′(y + ax).