PROFESSOR KELLAND ON GENERAL DIFFERENTIATION. 285 



= j^^d(ae~ "'/{y + ^ua^x) 



which is the solution of the equation in the form of a definite integral. 

 Tj- q di z dz 



dxi ^2/ 



The first form of the solution is evidently 



which is reduced to z^e"'" e"'''^''''/(i/ + 2acx) 



~ —T= I ^ dMe~"' /(i/ + 2acj: + 2u)ajx) 



as in Example 2. 



V A dz „ di d „d^z o 



dx dx^ dy dy^ 



This equation may be written {d—2ad^^ + a^h'^—c^)z—{i, 

 which is of the form of Ex. 3, Class 1, and the solution is 



gC X « CO ^^ 



=— T- / dijie""' {f{y-'r2acx-\-2ijia^x)-v^{y — 2acx + 2oia^x)\ 



■cc dz ^ d^ d^ „ d z ^ 



Ex.5. ;3 — 2a— ^ — TZ + a^-i-=0. 



dx dx^ dy^' dy 



This equation gives (rf- 2 a rf* ^^ + «^ 5) ^; = 

 or «=e»^'^ V(y) + «e<'''''' (|) (y) (Class 1, Ex. 3. Cor. 1.) 



=f{y + <^ x) + X (p (y + a^ x) 



■c a dz d^ di ,d z 



Ex. D. J- + a r ■ r « + o 3— = c s. 



This equation may be written {d+adi di + b8—c)z=0 

 which coincides with Ex. 3, Class 1 ; and the solution is ' 



z = e''-f{y) + e^^(f>i2f) 



where a*, (3^ are the roots of the equation in v 



V + a 8^ v^ + b 8 — c = : 

 1 «^* IJa^ Xa- 





VOL. XVI. PART III. 4 C 



