Equations of the First Order. 365 



is a solution of the partial, since all the other terms vanish on 

 that assumption. The value of ix thus obtained is je~^ v dx 

 {^r Y x hemgftyxdx), and fi is then derived by the fundamental 

 system. The constants of these two integrations are immaterial, 



as they merge in the arbitrary constant. Secondly, if -j- is a 



function of y only (which call tyy), we have a similar value of -or, 

 viz. J e - ^ dy 3 and corresponding results. 



Ex. I. Let £j= 1 , <f> 2 = — , R 2 = \-\', then P= — > 



ri x r2 x& x 2 xe'J' x 



and *r = log#. The fundamental system then gives 



whence 



4* - A - .* V* ^'- - i f 1 - — V 2 



and the complete primitive is 



cV-2c#(^-l) + l = 0. 



j&v. 2. Lett's*, <£ 2 =2^-a 2 , R 2 =^-2Z^+g 2 ; 



then P= -, and w== log a?. The fundamental system then gives 

 a? 



whence 



/a = log (y—bx + sfif — 2bxy 2 + A 2 ) ; 



and the complete primitive is 



cW-Zcse (| -i) + Z> 2 ~« 2 =0 (Boole, p. 185). 

 JEa?. 3. Let 

 *i = loga?, 2 =loga?0 + l\ R 2 ==loga^logtf-|--lJ; 



then P = — G - , and ar= I (— — y dx—2%x suppose. 

 The system then gives 



^==0*loga?-y-a?)-4, £ = -log^(^log^-?y-^)-^ 



whence 



^= -~*2(#log# — ?/ — a?)*; 



