■258 



PROFKSSOR KELLAND ON GENERAL DIFFERENTIATION. 



v = -r e : 



■Jx 



and 



>/={l + a\/ — lxS)v 



A «''t / — ^ rf*» 



= — -e +av —Ix — - 



^x dx^ 



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 

 satisfactoiy, 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 



up a function satisfying the conditions^ -»«.*■• ^=0, I shall employ the first me- 

 thod in the three following examples. 



13. Ex. 4. 



y — mx- 



dx^ 



= 0. 



let 

 then 



and 



2 A„ 



or A, =:y . — 7^=- 



A A. A„ „ 



dx^- L/l x^ /f xi 



= .(-l)^{iA.^|^.&c.} 



^o+T'+^°+"^«' 



A.=^ 



A„ = - 



A, 



.^/^I' 



A.=e 



2^ 2 1 . A„ 

 3 . 1 ■ 1 m^v 



A, — — ( 



2« 



22 2 1 . 2 A„ 



'J »?' 



2* 



V-l 



&c. 



"5.3.1-3 . Il»j8^y'3i' ^*-7.5.3.1'5.3.1-3.11 



&c. = &c. 



2-' 2^ 2l-2-3A„ 



W?* TT'' 



and 



-A fl ? 1 2^ 2 1 2^ _2f_ 2 1_l2^ 1 



i'-A„ 1^ + 1 ^^^3^ ~3. 1 • 1 ■ m'x'nr 5.3.1 " 3. 1 • \ m^ xW -v ] 



Ex. 5. 



-1 



II — mx^ r=:0. 



It is easily seen that the form of the series into which y may be expanded is this 



BCD. 



tJX X 0^1 



+ a + -^ + — +— + &c. 



