PROFESSOR KELLAND ON GENERAL DIFFERENTIATION. 251 



Suppose j^=2 J^ e™ "^ ; then by (A) 



2 *«(»«*- a ^) e"' '=0 ; which can only be satisfied when /«=«. 

 y=Ae"'" is the solution of the equation. 

 We might have proceeded in a somewhat different manner, as follows : 

 Put Oe'"' forO, then 



mx 



m — a' 

 But —^ — 7 is finite only when m^a; and then it is constant; .-. «=A/' ', 



?n —a- 



as before. 



Ex. 2. — f-a' J'=X ; X being any function of a;. 

 dx- 



We have y={di-a^)-^ . x + ((/*-oi)-i . 0. 

 If X = 2b,.e'-' 



y=Ae'''+ 2 ,— , Z'^' (Ex.1.) ' 



r —a 



K r. 



Cor. 1 . If >■ = a, -^ — j e becomes infinite. In this case put a + a in nlace of /• ; 



j*~ — ft ^ 



then -v — i e becomes o e 



2 a* +&C. 



i« +26. a ^« , whena=0; 



of which the first term may be incorporated with A e" ' ; and the complete solu- 

 tion is 



s —a- 



CoR. 2. If X=6a; ", we have, by the well-known formula 

 jL=4/'°"e-"^a"-i da. 



1 1 r'^e'^a^-^da, 



1 /-"/I (-a)* (-a) , \ -.:. .-1 



