5.3 Isoperimetric Problems
335
y = c 1 −
λ
1 + y 2
(5.6)
Let y
= tan t, then
y = c 1 − λ cos t
(5.7)
Taking the derivative of y with respect to x, we get
y
= λ sin t
dt
dx
(5.8)
Therefore
λ sin t
dt
dx
= tan t
(9)
Integrating we get
x = λ sin t + c 2
(5.10)
Eliminating t in the expression (7) and expression (10), we obtain
(x − c 2 )
2
+ (y − c 1 )
2
= λ
2
(5.11)
Thus the desired curve is a circular arc, where, the constants c 1 , c 2 and λ may be
determined by the boundary conditions and the constraint condition.
Example 5.3.2 The information source variable x changes in the interval
(−∞, +∞), the density of a probability distribution for the information is p(x),
Find the optimal probability distribution density p(x), such that the information
entropy
J [ p(x)] = −
+∞
−∞
p(x) ln[kp(x)]dx
(5.1)
under the conditions
+∞
−∞
p(x)dx = 1,
+∞
−∞
x
2 p(x)dx = σ
2
(5.2)
takes the maximum, where, k and σ both are the constants.
Solution Let F = −p(x) ln[kp(x)], G 1 = p(x), G 2 = x
2 p(x). Making the
auxiliary function
H = F + λ 1 G 1 + λ 2 G 2 = −p(x) ln[kp(x)] + λ 1 p(x) + λ 1 x
2 p(x)
(5.3)