OF SIMULTANEOUS ORDINARY DIFFERENTIAL EQUATIONS. 563 
We thus find 
pq (x —1P=0, 
*=+1, ory=+1, orz=+1. 
whence 
Any one of these is found to be a singular first integral, and tc reduce the equations 
to one of the form 
p (a? —1)=y?—1, 
which we have seen how to integrate. 
The singular solution of this is again 
(x* —1)(¥° — 1) =0. 
System of Plane Curves. Another extension of Cuairaut’s Form.  (§§ 53-55.) 
§ 53. There is an extension of CLA1RAvT’s form to higher orders, with one dependent 
variable. 
Write p, for d’y/dx’, so that p, will mean y. 
Then integration by parts gives (7 being Kin), 
(Pes a“ dx =x; Wires TP OI Des ar r (7 a 1) BB Orr o 0 + ( Fe Vy ue ! Pr 
Call this expression q,, and take the equation, 
d (Yo > Oh) Riches! Qu) ='(0) 
This may be solved at once by differentiating ; since 
dq,/dx = 2" r+ 
) 0 .6 A 
Pasi {ge + oge tas De + unt | = 0, 
Odo Gn 
we have 
The first factor gives the complete primitive, which consists of the equations, 
Oy Ch Ui Chin oo o Op = Cy 
where Mp, @,... G, are constants, connected by the relation 
Ol (Ch Chip 6 0 o Wp) = O; 
but otherwise arbitrary. 
The value of y may be found as follows—in the equation 
Uy 0” = py — Tp" > + 1 (7 — 1) pp_2X"” 
take differences with respect to 7; thus 
[eG PE es (oN pects (A FE Sh ann (Ge eae ern en st olo 6 
AN Cre? 
