I58 PROFESSOR KELLAND ON GENERAL DIFFERENTIATION. 
1 
wok He, 
and y=(l+aV—I1ed)v 
which will be seen to coincide with the solution already given. 
This second method of solving the equation is by far the most simple and 
satisfactory, when once the principles of the calculus of operations are thoroughly 
mastered. For the purpose, however, of exhibiting the analogy amongst the dif- 
ferential equations which determine the values of the different series which make 
3 
up a function satisfying the conditions y—m x? “40, I shall employ the first me- 
x 
thod in the three following examples. 











pay 
13. Ex. 4. y—ma? —7=0. 
let y= A, +4 Bf + &e 
t Fi 2 
then a ( yi{h Ay 2 As ge 
da /L a js a 
[z 6 
and Ag+ — + > &e =m(—1)'| 2 A, + a a2 +80} 
| 
OD wee mie gcc eee 
eae mr — 1” a mae TA oa 
De eA oF a Fie Ae 
heap N er Ae ree a, 
i Soe ee eee a a gs 
25) (68.138 Lia al 688th ote 5.3.13). 1 1; m* mr 
&e. = de. 
2 1 oF * iz, 1 28 gl i 
and y=A, {1+q ma/—an 3.1 ° 1" veo 5.3.1 °3.1 1 i /ln We. } 
dy 
Rx2 5: — 2—_=(. 
It is easily seen that the form of the series into which y may be expanded is this 
Be iCe iD 
aad ee Soe &e. 

C) 
+a Bes jy Oy, &e. 
eC ee 
