Page 192 - Foundations Of Differential Calculus

9. On Differential Equations 175
I. Let
x
e + e −x
y = .
e − e −x
x
x
When both numerator and denominator are multiplied by e ,wehave
e 2x +1
y = ,
e 2x − 1
so that
y +1 y +1
2x
e = and 2x =ln ,
y − 1 y − 1
whose differential is
dy dy
dx = − = .
2
y − 1 1 − y 2
II. Let
x −x
e + e
y =ln .
2
By the first differentiation
x
e − e −x
dy = dx,
x
e + e −x
or
dy e 2x − 1 2x dy + dx
= and e = .
dx e 2x +1 dx − dy
Hence

dy + dx
2x =ln .
dx − dy
If we take dx to be constant, then
2
dx d y
dx = ,
2
dx − dy 2
or
2 2 2
dx = d y + dy .

295. In a similar way trigonometric quantities can be removed from an
equation by differentiation, as can be understood from the following exam-
ples.
I. Let
x
y = a arcsin .
a
Then
adx
dy = √ .
2
a − x 2