1
55.4
1.￡
üGronwall??
T:
!x(t),f(t)1 uW[t0;t1]
￥dμ
L ??f
, ?μ
Lè
g? 0
P¤
x(t)? g +
Z t
t0
f(?)x(?)d?; t 2 [t0;t1];
5
x(t)? gexp(
Z t
t0
f(?)d?); t 2 [t0;t1]:
￡
ü:5￡g > 0
HGronwall??
T? ?.?L
!?
x(t)f(t)? gf(t)(1+ 1g
Z t
t0
x(?)￠f(?)d?):
7
v(t) =
Z t
t0
x(?)￠f(?)d?;
5

T?? V?
dv(t)
dt? gf(t)(1+
1
gv(t));
'
v0(t)
1+ 1gv(t)? gf(t):


T
HVt0?ts,¤:
ln(1+ 1gv(t))?
Z t
t0
f(?)d?;
'
1+ 1gv(t)? exp(
Z t
t0
f(?)d?);
?L
!¤:
x(t)? g +v(t)? g(1+ 1gv(t))? gexp(
Z t
t0
f(?)d?):
yN?g > 0
HGronwall??
T? ?.
C
!g = 0.?
Hμ
x(t)?
Z t
t0
x(?)￠f(?)d?;
1￡
üx(t) · 0.?L
!? ?B? > 0,?μ:
x(t)+
Z t
t0
x(?)￠f(?)d?:
?

￥2T?:
x(t)exp(
Z t
t0
f(?)d?):
y? > 01 ?B?
#x(t) · 0.
2
3.?¨w<Gronwall??
T,°¤￡
üGronwall??
T￥
6Bw<:
!x(t),f(t)1 uWt 2 [t0;t1]
￥dμ ??f
,C;K1dμè
. ??t 2 [t0;t1]
Hμ
x(t)? C +
Z t
t0
[f(?)x(?)+K]d?; (1)
5?t 2 [t0;t1]
H
x(t)? [C +K(t?t0)]exp(
Z t
t0
f(?)d?),(2)
￡
ü:5￡C > 0
H(2)? ?b
7
v(t) =
Z t
t0
[f(?)x(?)+K]d?;
5(2)
T V?
v0(t)
C +v(t)? f(t)+
K
C +v(t)? f(t)+
K
C +K(t?t0);


T
HVt0?ts¤
log(1+ v(t)C )?
Z t
t0
f(?)d? +log(1+ K(t?t0)C );
'
C +v(t)? [C +K(t?t0)]exp(
Z t
t0
f(?)d?);
?(1)¤:
x(t)? C +v(t)? [C +K(t?t0)]exp(
Z t
t0
f(?)d?):
yN?C > 0
H(2)? ?b
C
!C = 0b?
H(1)'1
x(t)?
Z t
t0
[f(?)x(?)+K]d?,(3)
?(3)? ?B? > 0?μ
x(t) <?+
Z t
t0
[f(?)x(?)+K]d?:
?

￥2T?
x(t)? [?+K(t?t0)]exp(
Z t
t0
f(?)d?):
y? > 01 ?B?
#Aμ
x(t)? K(t?t0)exp(
Z t
t0
f(?)d?):
yN?C > 0
H(2)9? ?b
55.5
3
4.ó?Z?
dx
dt = sin(tx);
p@x@t0(t;t0;x0)@x@x0(t;t0;x0)t0 = 0;x0 = 0)￥Vr
T,i￡
ü ?’(t;·)
^Z?
@′Hq
x(0) = ·￥3,5μ
@’
@·(t;·) > 0:
3:A ?x(t;0;0) · 0
^′ù5
dx
dt = sin(tx); x(0) = 0
￥·B3. p3′ù5
dz
dt = tz; z(0) = 0
¤
@x
@t0(t;t0;x0)jt0=0;x0=0 = 0:
p3′ù5
dz
dt = tz; z(0) = 1
¤
@x
@x0(t;t0;x0)jt0=0;x0=0 = e
t2
2,
p3′ù5
dz
dt = tcos(t’(t;·))z; z(0) = 1
¤
@’
@·(t;·) = exp(
Z t
0
cos(?’(?;·))d?):
yNμ
@’
@·(t;·) > 0: