254 



PROFESSOR KELLAND ON GENERAL DIFFERENTIATION. 



by 



^Ae-' + B/'+S. 



r + ar^ + b 



Cor. 2. If X=6,. a constant, .-. ^=0 and 



br 



Cor. 3. If /■ + «/•* + 4=0, /• must be equal either to a or to ^. Suppose r=a\ 



then b/ 



r + aA + b 



becomes, by writing a' + c in place of /-, 



Or r = O t 



2a^c+ac 2ai + a 



and y = Ae"'' + Be '° + b, 



la^xe"^ 



+ 2 6, 



2 a- + a ' *■ + o «i + 6 



Ex. 5. 



This gives 



^ +a— ^+6^ = X. 



{d-'^ + ad-i + b).!/ = X. 



or (rf-i-a-i)(rf-i-|S-*) .3^ = X;where a-^ /?"- are the roots of the equation 



z" + az + b = ; 



= Ae- + B/%4^f(.i-«^)-.«^f^-(<^^-/3i)-Vif4} 

 which is reduced to Ex. 2. 



d" V 



In precisely the same manner we may solve the more general equation — ^ 



+ a 



d"-^y 

 dx"-' 



+ b '^- — ^ +&C. +i^=X, n being a multiple of a. 



dx 



Class II. Ulementm'y EqvMions. 



11. The form to which more complicated equations can generally be reduced 



is u-mx" — =X; and it is with equations of this form that we are now to be 



•^ dx^ 



occupied. The simplest case, when »=0, we have already solved. 



._ d^y _f^ 

 Ex. 1. ^~"''^'^rf^~ 



By (C) this is reduced to y-m\/ -\ — j, • ^ - 0, 



