24 M. Falk, On the Integration of partial 



whence by integration 



*.- - / (D + ^f-f") - _ 



/ and F being arbitrary functions. Now putting k Ç Ç-^A — F \p\ instead 

 of/ (yy^ and applying Taylors theorem on F i'^-—-\ , we get 



h^- = k <t) il\ ^ ^- F (l -\. ^ ^1] {L)<^ <l) . 



X \X/ X \X XI 



Now dividing by k and then putting a; = , the general solution of 

 the given equation becomes 



■\^ being put for F' . 



Ex. 4. Before explaining the general theory of non-linear partial 

 differential equations of the n"' order, we will by a simple example shew, 

 how the foregoing theory may be used to integrate such equations. Sup- 

 pose we have 



f' being a given function. Ditferentiating partially with respect to x , we get 



Applying to this equation, which is linear, the foregoing theory, 

 the equation (20) gives the roots m^ = , m^ = ^J--^- . The former root re- 



produces the given equation (with an arbitrary function of y instead of /') 

 and is, tlierefore, of no use; the latter reduces the equations (21) and (23) to 



dij — ^J^ dx = and dz^ ^, = . 



"0,1 



The latter of these gives Ci „ = «,, wliich, in connection with the 

 equation dz = z^j^dx -\- Zf^^ dy (or, in this case, (iv = 2ii „ c?a^), gives 



z = 2^^1,0 X -\- ß 

 as a complete first integral. This, in connection with the given equation, 



z — /3 2x J.,, , 



gives ,-,,„ = --^ , ^0.1 = ^ f'[y) ; 



2« z — /Ö 



and , as these values are obtained from two distinct first integrals of the 



same partial diff. equ. of the second order, they will render the equation 



d: ~ z^^ dx + ~ 1 ^hl integrablc. Thus we get 



