252 PROFESSOR KELLAND ON GENERAL DIFFERENTIATION. 



7. The solution of the foregoing examples might have been obtained very 

 differently, thus : 



ji J- 



If d- y-a- y = \; y=—,^ ^ = ^^3^ . X 



a- —a- 



Now -; — X is the solution of the ordinary differential equation -T--av=X; its 

 a — a "'* 



value is, consequently, «« « ^y e - "-^ X rf j- + C^ . Hence 

 For instance, if X=0, the solution of the equation is 



y=2aiCe'"' ; 



which is the same as that given above. 



8. Ex.3. ^+^4-6y=0 



ax rf.r* 



This may be wi-itten {d+aS + b). y=0 ; or {di-a^)(cl'-^') .y=0; where 

 a' + (3' = -a, and (aj3y-=b, or a^, /S* are the roots of the equation 22 + a« + ^-=o. 

 _j^==A(rfi-ai)-i.O + B(<^i-/3i)-i.O 

 = Ae"-^ + Be''^ (Ex. 1.) 

 Cor. 1. If a=|9, we must write a + e instead of /3, and proceed as in similar 



cases. 



The result is y^Ae" +Ba^e" 



Cor. 2. In precisely the same way we may find the solution of the equation 



diy dy , diy „ 



dx^ "'^ dx- 



