SOLUTIONS OF DIFFERENTIAL EQUATIONS. 179 



Put 



c + E = o (18) 



for the complete primitive of (17), where (c) is an arbi- 

 trary constant and (R) a function of x, y, which, however, 

 must satisfy the partial differential equation 



dR_ dR / >, 



dx ~ dy 



7. The function (R) may be taken so as to satisfy the 

 equation 



U = v + zf(w), (20) 



where v, z, w are functions of x, y. 



Substitute this value in (17) and (18); they become 

 respectively 



f" dv dz „. /\ 1 dw 



dy \dx dar^ f f (w) dx _ 

 dx J dv dz „. 1 1 dw ~ 3 ' ' * ' 



\ly + ^/ W jfJw)' Vz d^ 



c + v-\-zf(w)=o (22) 



Of course 



It is clear, therefore, that (21) is satisfied by the 



relation 



w — h — o, (23) 



where h is any arbitrary constant if f{h) —infinity. 



The differential equation (21) is, therefore, satisfied both 

 by the complete primitive (22) and by the equation (23), 

 which is a singular solution of (21), if it is not contained 

 in (22) by giving a constant value to (c). 



This form of the complete primitive, to which all primi- 

 tives may be reduced, gives the characteristic singularity 

 to the differential equation (21), which is derived from it 



