DIFFERENTIAL EQUATIONS OF THE w'" OrDER. ,21 



rential coefficients of z of the {n — 1)" order, make the arbitrary functions 

 / arbitrary constants. 



Ex. 1. 



^%,o + ?>oc'y~2,. + 3.1-3/^-1,2 + î/^'o,3 + 20r^-,.o + 2xyz,,,'+ yK;,) = 0. 



The equ. (20) has here all its roots = ■^; then (21) and (23) become. 



dti — V- dx = , 



^ X 



dz,,,-V2Uz,,,+y-^d:,,,+ 2(z,^,-\- 2y:,,, + t:^,,\'^ = 0, 

 X x' \ X x' I X 



whence by integration 



X 



The given differential equation has, therefore, only one general first 

 integral viz.: 



«'zg.o + 2xyz^^ + 3/^*0.2 + /(-) = 0, 



/ being an arbitrary function. Now integrating this equation by means of 

 the equations (20), (21) and (23), we get the auxiliary system 



dy — y^dx = 0, 



X 



y\ dx 



X \xj x^ 



whence ^ = «j and *i,o H- -'o,i ^ fv~\- + A ? ^1'"^ the 



general second integral of the given equation will be 



*'~i,o + y -0,1 



— flV 



/©+^^(1) 



ip being another arbitrary function. Applyhig again to this equation the 

 foregoing theory (which now identically coincides with that of Lagrange for 

 solving linear partial ditferential equations of the first order), we get the 

 auxiliary system 



