Equations of the First Order. 367 



whence 



and the primitive is 



(«r 24 ) 2 -sf («r 24 )+i=o. 



Or. 8. Let *, = J, </> 2 = J (^ - j^) , R 2 = | ^ ; 



then 



Q 2 , 1 - 



fay 3 



The system gives 



|f=y(l + *»)*, d £ = \ 1C y*(\ +x *)-t-\ x {\ +S c*)\ 

 whence 



and the primitive is 



c--f ±(i+* 2 )*(f 2 -g(i+W)=°> 



or 



(Peacock, p. 342). 

 2£#. 9. Let (j>i = yfiX+f 2 x, and R=/ 3 ^. Then P is a func- 

 tion of a; only ; and iz will be found to be 



The given equation is, in fact, the product of the two linear 

 equations, 



p^yfi* +/&£/& 



both of which have the same integrating factor. 



13. Other methods of solution are obtainable by determining 

 upon what hypothesis ot or //, will have a particular form. If t 

 be an argument containing x and y, we can ascertain at once 

 whether it is possible for a function of / to satisfy the partial 

 equation ; and if it do, we can determine the form of the func- 

 tion. For we have (calling the function ^) 



n ( dH o^ <Pt .t dH n dt _ dt\ „ 



