G9 



DIFFERENTIAL EQUATIONS OF THE SECOND ORDER. 



dimensions ; it will be seen that the general solution is not expressible in a finite 

 form. 



The characteristic equation is 



say q = i ; and then u is determined by 



z + iij = xp (u) -f u, 

 where p (n) is arbitrary. The subsidiary system is 



'. ' ri + + i-o "> 



-y 7 

 = Z + np, 



We easily iind 



. 



, + y j~ ) + * 7-, yp 



- ' (/,'' ^///- * (/ 



l y 



' 









I 



and so we have 



. dw 

 2'l -- ypw 



d 



n = 



d-tv dw 



~df " y d^ 



. d~w du: . < 



= l -^ -yp^-'^; 



l=2i ^77,+P 



dxdy <'>/ 



where w is an arbitrary function of x, y, and u. But we also must have the equation 



do d:. 



- =n 

 ait du 



satisfied ; that is, we must satisfy the equation 



dy du 



(ho 

 yp - 

 Ir du 



, . ,, x> [ dho dw\ 



ywp i { 1 + xp (u)} \ -j-^ + y } . 



' ' ' [ d>f ' dx J 



If a general value of w can be deduced, then the general value of c could be 

 constructed, and conversely. 



42. Taking z + iy = *-, z l>j = s, the equation can be written in the form 



