9/05/2014 Given y = 2/(ex + e-x), solve for x in terms of y.
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9/05/2014 Solution. Since y = 2/(ex + e-x), cross-multiplying, we get:
yex
+ ye-x
=
2
Rewriting:
yex
+ y·
1/ex
= 2
Multiplying by the LCD of ex
to
clear the fraction: ye2x
+
y
= 2ex
Rewriting and rearranging: y(ex)2
– 2ex
+
y
= 0, which is a quadratic in ex.
Using the quadratic formula: ex
= [2 ±
√(4
– 4y2)]
/ 2y
Factoring out the 4 from inside the √
and
simplifying:
ex
=
[1 ±
√(1
– y2)]/
y
so that switching to log
form:
x = ln
[(1
±
√(1
– y2))/
y]