Q t
ð Þ ¼ lim
Δ!0
1
Δ
Z tþΔ
t 0
P t þ Δ, s
ð
Þ ds À
Z t
t 0
P t, s
ð Þds
!
¼ lim
Δ!0
1
Δ
Z tþΔ
t 0
P t þ Δ, s
ð
Þ ds À
Z t
t 0
P t þ Δ, s
ð
Þ ds þ
Z t
t 0
P t þ Δ, s
ð
Þ ds
À
Z t
t 0
P t, s
ð Þds
!
¼ lim
Δ!0
1
Δ
Z tþΔ
t
P t þ Δ, s
ð
Þ ds þ
Z t
t 0
P t þ Δ, s
ð
Þ ds À
Z t
t 0
P t, s
ð Þds
!
% lim
Δ!0
1
Δ
P t, t
ð ÞΔ þ lim
Δ!0
1
Δ
Z t
t 0
P t þ Δ, s
ð
Þ ds À
Z t
t 0
P t, s
ð Þds
!
¼ P t, t
ð Þ þ
Z t
t 0
∂P t, s
ð Þ
∂t
ds:
ð15:92Þ
Replacing P(t, s) of (15.92) with b s
ð ÞR s, t
ð Þ, we have
d
dt
Z t
t 0
b s
ð Þℜ s, t
ð Þds ¼ b t
ð Þℜ t, t
ð Þ þ
Z t
t 0
b s
ð Þ
∂ℜ s, t
ð Þ
∂t
ds
¼ b t
ð Þ þ
Z t
t 0
b s
ð Þ
∂ℜ s, t
ð Þ
∂t
ds,
ð15:93Þ
where with the last equality we used (15.85). Considering (15.83) and (15.93) and
differentiating (15.90) with respect to t, we get
_
x t
ð Þ ¼ x t 0
ð ÞF t 0
ð Þ
À1 _
F t
ð Þ þ
Z t
t 0
ds b s
ð ÞF s
ð Þ
À1 _
F t
ð Þ
h
i
þ b t
ð Þ
¼ x t 0
ð ÞF t 0
ð Þ
À1 F t
ð ÞA þ
Z t
t 0
ds b s
ð ÞF s
ð Þ
À1 F t
ð ÞA
h
i
þ b t
ð Þ
¼ x t 0
ð Þℜ t 0 , t
ð ÞA þ
Z t
t 0
ds b s
ð Þℜ s, t
ð ÞA
½
Š þ b t
ð Þ
¼
("
x t 0
ð Þℜ t 0 , t
ð Þþ
Z t
t 0
ds b s
ð Þℜ s, t
ð Þ
½
Š
)
A þ b t
ð Þ
¼ x t
ð ÞA þ b t
ð Þ,
ð15:94Þ
where with the second equality we used (15.70) and with the last equality we used
(15.90). Thus, we have certainly recovered the original differential equation of
(15.76). Consequently, (15.90) is the proper solution for the given inhomogeneous
equation with inhomogeneous BCs.
15.3 System of Differential Equations
599