4 oo 
Mr. Babbage's essay towards 
As another example, take 
4* x = — — tj/ — 
T X T X 
the particular solutions are, that of the equation itself 
and that of i|y jc = xj/ is known to be <p |x, — j , hence the 
general solution is 
which, as is readily seen, fulfills the condition. 
Problem IV. 
Required the general solution of 
i[/a: = Axxipax4-Bjr, 
having given a particular solution fx 
Assume 4'£=/.r-f-{p,r, then 4 > a x becomes f a x -f- q> « x, 
and the equation is 
f* + <p X = A X x/ aX-fA^xipaX-f Bar , 
subtracting from this the particular solution f x = A x x 
fax -f Bjc, which is given by hypothesis, there remains 
(pX=A.Xy.(pex.X, 
this equation may be solved by Prob. Ill, and we thence ob- 
tain the general solution of the given equation. 
From this may readily be deduced the solution of the fol- 
lowing equations : 
^ (* y ) ± (e~ y ) = Ay 
suppose/ is a particular solution, take \py —fy + %y, the 
equation becomes 
/ (e-0 ±f {*~ y ) 4- % M ± % {*~ y ) = Ay, 
but we have from the particular solution 
/(«*) ±/(«-0 =Ay,. 
