Ex. 19. To integrate

t3du + 2txd¿du + x2d ̧3u = tTMx”.

+F1 (§)+tF2 (§) + t°F ̧ (§) + ... + t^~1 F12 (§).

1

Ex, 21. To integrate

3

Comparing this with the formula (1) Art. 30,

hence D, and A, are not commutative;

2u;

.. {de (də−1) + 2d。 — 2} u — a2ť3d ̧3u=0;

.. (do+2) (d. − 1) u — a2ť d ̧3u = 0.

For convenience write C for — a3d ̧2;

•; (do+2) (do – 1) u +Cửu=0.

We have now to diminish the factor de+2 by 2; this we do by assuming (see Art. 49)

For u write v (t+x)", and put m (m + 1) = 2 which gives m = 1 or 2. We take the latter value of m;

:. (t+x) did ̧v=dv+ d ̧v

Differentiate this with did, (Art. 39, 2);

:: (t+x) did ̧3v=0 ;

2

2

:. v = st3s,2 0 = s;2 0 + s ̧2 0

=(t+x) F, (x)+F(x) + (t + x) ƒ, (t) +ƒ(t).

1

(1).

The forms of the supernumerary arbitrary functions are to be determined by substituting this value of v in equation (1). This gives

1

. . u = v (t + x)^ = (t + x) ̃* (Fx + ft) — ' ' (t + œ) ̃1 (F'∞ +ƒ't).

Ex. 24. To integrate

(t − x) (du — du) = 4.

In this example

D1 = d1+de, A1 = d1 - d., §=x-t, §' = x+t;

SF' (') is not equal to F(§'); but it is equivalent to it in this case, because D§' (= 2) is constant, and the constant may be considered as being absorbed in the arbitrary function.

CHAPTER X.

EQUATIONS OF THE SECOND ORDER.

FUNCTIONS OF P AND q.

COEFFICIENTS

54. IN this Chapter it will be a convenience to represent du and du by p and q respectively. The following are the formulæ suitable for this class of equations. They are a mere adaptation of the general formula of Art. 7 to these equations, and no new principle is involved in them.

D1p=d1p + D. d ̧p=d}u+ D1x. did ̧u............. (1),

.......

Dıq = dıq + Dix. d«q = d¿d ̧u + D ̧x. d3u................. (2).

......

From these we immediately deduce the following forms, which it will be observed are such as can be at once adapted to those terms of an equation which are of the second order. They are the forms under which such terms generally present themselves, and will serve as guides in choosing the startingpoint in commencing the integration of any proposed equation.

Dip-Dix. Diq = d}u – (D ̧x)2. di̟u..

xx

水

.(3),

Dp+Dæ.Dq=dịu+2Dạ dầu+(2) dịu (4),

Dip+M. Dq=du + (D ̧x+ M) ddu + MD,x. du...(5).