2.4 The Euler Equations of the Simplest Functional
123
s = −p + mp
+ n
The production costs of s products per week is
C = as
2
+ bs + c
Find the price function p(t), such that the total profits of all the year round
J [ p] =
t 1
t 0
(sp − C)dt =
t 1
t 0
[−(1 + a) p
2
+ (2an + b + n) p − am
2 p
2
+ (2am + m) pp
− (2amn + bm) p
− (an
2
+ bn + c)]dt
attains maximum. where, a, b, c, m and n are all constants and great than zero.
Solution The Euler equation of the functional is
−2(1 + a) p + 2am
2 p
+ (2an + b + n) = 0
or
2am
2 p
− 2(1 + a) p = −(2an + b + n)
The characteristic equation of the differential equation is
am
2 r
2
− (1 + a) = 0
Find out
r = ±
1
m
1 + a
a
Let the particular solution of the equation be y
∗
= A, substituting it into the
differential equation, we find
A =
2an + b + n
2(1 + a)
Therefore, the general solution of the equation is
p(t) = c 1 e
1
m
√
1+a
a t
+ c 2 e
−
1
m
√
1+a
a t
+
2an + b + n
2(1 + a)
where, the integral constants c 1 and c 2 are determined by boundary conditions. In
fact the price could not continuously change with time, it can change one time only
at intervals, but the solution of this problem has reference effect on setting the price.