I 
236 Mr. Babbage's essay towards 
) ■ 4 
this being substituted in (E) gives 
put tyux — % then 
d. I dzlda, x 
% dx ‘ \dx{ dx 
which is a differential equation, from whose solution z or -tyux 
may be found. 
Ex. i, Given the equation 
in this case^~ =. — * 1, and the differential equation is 
% d E -j- d' % =3 0 
its integral is z = $ (0 — x) = b cos x + c sin x. 
The two constant quantities which have entered by integra- 
tion must be determined so as to satisfy the original equation. 
This condition gives 
5 / \ d 1 ]/ x 
b cos a 
■ sm a 
l 
the quantity b still remaining arbitrary ; the solution of the 
equation ^ (a — x) = is therefore 
« 7 / -\ b cos d . / \ 
= o cos ( — a: ) — « • — - — sm ( a~x ) 
\ / x — sm a s ) 
Ex . 2. Take the equation 
^ 1 d x 
in this case ax = — and = — 1 
a: f/jc 
dux 
V 
and the differential equation becomes 
± d oc* v, x d x d z d 2, % •=. 0 
whose solution is 
z 
V* 
b —p — 1 
X 
and fyx = 
2 / 7+1 
2 p+l 
■t+'+bx~ p 
+ bx 
P = 
— i±v'— 3 
