344 On Differential Equations of the First Order. 



Now the first of these taken along with the equations (/3) leads to the 

 equivalent system 



/iSSjB*— 1=0; 



f 2 ~ q - z -r + y = 0, 



f 3 ^(^-z) 2 —y-z=o. 



Tor this system the conditions 



[A./J-0, [/;,/j=o, [/ 1( / 2 ]=o 



are satisfied ; and the values of p, q, r drawn from it being 

 p=-, q=z±*/y + z, r*=y±*/y + z, 



CG 



we find an integral 



u=lQga?+y*+§(y+*)+°« 



The second equation, 



q + r—y—z=0, 

 does not lead to any solution. 



Example 4. Find the possible solution of the two simultaneous equa- 

 tions 



f 2 =a?py 2 +a?yp~xy*^zr 2 ~-xy=0. 



Here the condition [/ l5 / 2 ]=0 is satisfied ; we have then to find a f auction 

 f 3 such that 



[/,,/.]-o, [/ 2 ,/ 3 ]=o, 



and it will be easily found that 



is, a common integral. "We have then 



q' 2 + y = a; 



andp, r 



must be found from 









x 2 p — 



a?4-5fr 2 -=. 



-a, 





Mp 7 



-atya—zr* 



= 0; 



whence 













a 



, 1 





p=- 



(l + a)x 



2 07 





2= V 



a —v> 





V(i+«> ; 



and we get for a complete primitive 



-TOs; + *'-*<-»> , +Viq£ +ix 



