32 M. Falk, On the Integration of partial 



Ex. 1. ^2,0 — 4,-0,2 = Oû^y . 



Differentiating this equ. three times with respect to «, we get 



the general solution of which is 



As the two last terms by the theorems I and III also belong to 

 the general solution of the given equation, it is only the functions (p , tp^ 

 and (p2 that have to be determined by the condition, that the given equa- 

 tion must be satisfied. This condition will be fulfilled, if 



cp, iy) — 2Ç"(y) = , <p, "(3/) = , AC, "(y) = — y ; 



we may, therefore, put 



(p,{y) = — I" , (piO/) = and (p(y) ^' 



24 ^'^'' ^'-'^ 960 



Hence the general solution of the given equation will be 



_|_ ^(y-2œ) + ^//l(^/+2.^0 . 



y^ x^y^ 



960 Tr 



If the given equation had been differentiated twice with respect to 

 y , it would have given 



-2,2 ^"0.4 — " • 



Now putting Zo^2 — ^' 1 "^^'G get 



•^2,0 '*-0,2 ^= O , 



the solution of which is 



z' = <p{y — 2.v) -h •^(y+2.iO 



or ro,, = <p{y—2x) + v// (y+2.^•) , 



and, if this last be integrated twice with respect to ?/, it gives 



Determining here x and %i by substitution in the given equation, 

 its general solution becomes 



^ = ^ + <p,{y-2cc) + x^.(y+2.r), 

 Avhich is somewhat simpler than that before given. 



I 



