Theory of differential equations. By Andrew Russell Forsyth.

305.]

HAVING INTERMEDIATE INTEGRALS

459

A GENERAL CLASS OF EQUATIONS.
305. To illustrate the working by a particular case, consider
the equation
E (ry - /32) + F (a -/37y) + G (/38 - 2)
+ Aa + B/3 + C+ D + H= 0,
which may have an intermediate integral.
According to the preceding argument, we take
= -,9 — 7- -,
Ut,. U,, Ut
u8 =_ it's _UY
=- t 3 Ut Ut
we substitute these values of a and 8 in the equation, and we
then make the resulting equation evanescent so far as regards the
determination of /3 and 7.
The terms in /32, /, 72 disappear in virtue of a single relation
Eut - Fus + Gu, = 0.
The term in / disappears in virtue of the relation
E- + F -A s - - D B = 0,
U212 Ut utr Ut
the preceding relation being used to simplify the form. Similarly,
from the disappearance of the term in y, we have
F y+ G A UtD + C
and lastly, the aggregate of terms independent of 3 and y gives
the relation
F uxy - A ux _ D Uy + H = 0.
Ur Yt Utr Ut
There are apparently four equations.
From the second and the third of these relations, we find (also
using the first relation)
(EG - F2) (Fu - Dur) = (- AFG + BF2 - OFE + DE2) Ut,
(EG - F2) (Fuy - At) = (A G2 - BFG + CF2 - DEF) u,,;