DIFFERENTIAL EQUATIONS OF THE n"' ORDER. 33 



Ex. 2. -3,1 — 2 --J 2 + Co 3 = a-f-ô.c + cy. 



Integrating this witli respect to ?/ , it gives 



'2 — 2c-j,, H- r„_2 = ay + bxy + -^- + (p"{x) , 



(p" representing an arbitrary function. Differentiating with respect to x , 

 we get 



^z.o — 2.-2,1 + -'1,2 = kf + <P"'Cr), 

 the general solution of which is 



7 , 3 



z = (p{x) + 4^ + cPjC?/) + My^^) + '^'z(2/+^^) . 



Now determining <piQ/) so, that the given equation may be satis- 

 fied, its general solution becomes 



z = <p[x) + {a^bx) |- + (25 fc) |j + My +00) + .t';t(3/+*') • 



B) TO INTEGRATE PARTIAL DIFFERENTIAL EQUATIONS, 

 WHICH ARE NOT LINEAR WITH RESPECT TO THE DIFFE- 

 RENTIAL COEFFICIENTS OF THE HIGHEST ORDER. 



§• s. 



Ill §. 6 we have briefly exemplified, how the foregoing theory ot 

 linear equations may be used also to integrate those, that are not linear. 

 We shall now deduce the general formulée, that are to be used for the in- 

 tegration of non-linear partial differential equations of the n"" order with 

 one dependent and two independent variables, and also shew, how these 

 formulai in special cases coincide with known auxiliary systems. 



The result of the differentiation with respect to x and y of the 

 equation 



T{^X,y,Z, -1,01 -0,M •••1 •'n.O) ^n-l.n •••) *0,ij = U . . \p'±) 



Nova Acta Reg. Soc. Sc. Ups. Ser. HI. 5 



