M. Falk, On the Integration of partial 



A) PARTIAL DIFFERENTIAL EQUATIONS, WHICH ARE LINEAR 



WITH RESPECT TO THE PARTIAL DIFFERENTIAL 



COEFFICIENTS OF z OF THE HIGHEST (n"^) ORDER. 



§■ 2. 



Derwalion of the partial differential equation from a first in- 

 tegral of given form. 



In all the following we use the notation 



daPdy'i 



Now let if and \p denote definite functions oi x , 3/ , z , Cio, r^,, 2;,c„ Ci.,, z^.^^ 

 . . . r„_i I, , -„_,,, 1 , . . . ~,i,«-i ; the first derived function of n> with respect to ^, 

 as far as x is involved in (/> both explicitely and in z and in its diiferen- 

 tial coefficients up to the (n — 2)" order, we denote by A' and by Y the 

 analogous derived function of y with respect to y\ that is 



r-«-L* f=.T 



1 



^ = ji + b b^-M ^—+1.' I 



+1 



"^ ^ k ^ ^^ ^ ^"~" ^"^' 



r=u 1=0 



• (2), 



35P 



Z„,„ representing . Substituting -^ , AT', Y' , Z' for (p , A, Y, ^ 



in (2) we have the corresponding expressions with respect to xp . 



Denoting by / an arbitrary function, we now seek, Avhat is re- 

 quired, in order that 



y =/('/') (3) 



may by dift'erentiation and elimination of /'(i/O lead to a partial differen- 

 tial equation, which is linear in the above sense and of the n"' order. 



