v;7ti PROFESSOR KELLAND ON GENERAL DIFFERENTIATION, 



rentl.v without any sacrifice of generality, by the introduction of a relation be- 

 tween A and B. The relation is, A + B = — -^j^— 

 flence the complete solution of the equation 



y + axii + bx f + 2 a .r^ ^^^ = -7- , IS T 



19. Ex. 3. zy + ax---^,-{a + bx)-~^, + 2b 



ax a X - 



Multiply by ' -^ and the result will be 



d X 



of which the symbolical form is 



' dx'^ Ace-- dx"^ 



^,-.._„,-./i;^,_(a.-^ + 6)'-:^^ 3,-24^^^2^=0 



:D-1 , _. ,J-D-2 o J -D- 3 



= — y — iae ' + b)' — — _ » — 26- — ==^ 



[-b " ^ ' /-D /-D 



which is equivalent to 



ye--'+ ««-'(i):^i-fD + i)(D + 2)) ^~* ((D + l)(D + 2)-(D + l)(D + 2)(D + 3))^ = ° 



or ^«"'+'^«''dT2^-*(D + 2)(D + 3)^=® 



Hence, by multiplication, 



y-|(D + 2)(D + 3)e-'^2^-^(D + 2)(D + 3).-2'^=0 



Qj. 3,_»(D + 2)«-'y-|(D + 2)<'-'(D + 2).-S = by (B) 



which is of the form {} '"b'^ ~ b"^') ^=^ ' 



which, being put under the form 



{l + a(j>) (l + (3(p)y=0 gives 



!/ = A(l+a(|))-i .0 + B(l + ^(/))-' .0 

 Now (1 + a 0) ^ ^ . is the solution of the equation 



A -- 

 of which the result is ^, = -7 « ' • 



