D5A PROFESSOR KELLAND ON GENERAL DIFFERENTIATION. 
Di eae 
r+ar? +b 
Cor. 2. If X=6, a constant, ... ,=0 and 
aA eo Bie aS 
y= et" + Beer + x 
Cor. 3. Ifr+a74+b=0, r must be equal either to a or to @. Suppose r=a: 
=| __ becomes, by writing a’ +c in place of ,’, 
r+ar?+6 
then 6,e 

,@ +2 are x + ke.) _¢ ots b we*2a? 
Qatc+ac Qat+a 
2areen® BEE 
Qat+a “stast+b 

and y=Ae*"+Be +b, 

2 en d-ty 
Ex. 5. Ea + by=X. 
da-1 da? 4 
This gives (d~14ad~*+6).y=X. 
or (d~!—a-}) (d~*—@-4).y=X; where a~?, B-? are the roots of the equation 
#+az+6=0; 
(d~t—a-*)-1.X  (@-#-B-*)- 
C=O aa 2 
4 (3: 
=e Bee TE {@t-ah-t.ab S d: o> (a! ahi aX — 
ae 


y= ei ery: 

which is reduced to Ex. 2. 
ne 
In precisely the same manner we may solve the more general equation 2 
@ 
a Sy, 7 ma 
+ +&e. +y=X, n being a multiple of a. 
dx-* da” 2a 
+a 


Cuass Il. Hlementary Equations. 
11. The form to which more complicated equations can generally be reduced 
by 
iS y—-mar De) =X; and it is with equations of this form that we are now to be 
d x* 
occupied. The simplest case, when »=0, we have already solved. 
_dby 
Bx, y—m/ a5 =O. 
__/=D+4 
By (C) this is reduced to y—m/— = oy=0, 
