1
55.1
1.
!x(t) = ’(t)
^′ù5
dx
dt = f(t;x); x(t0) = x0
 uW[t0?h;t0 +h]
￥ ??3, ?f(t;x) ? u×
R = f(t;x) 2R2, jt?t0j? a;jx?x0j? bg
 ??,R
1?x
@LipschitzHq,Lipschitzè
1L,h = minfa; bMg,M = maxfjf(t;x)j,
(t;x) 2 Rg.
!’n(t)
^PicardY}?
??nQY}¤?￥f
,￡
üμ ?/￥μ9:
j’n(t)?’(t)j? ML
n
(n+1)!h
n+1:
￡
ü:?^
!t 2 [t0;t0 +h],t 2 [t0?h;t0]￥￡
ü? ? ?
,’(t) uW[t0;t0 +h]

@sZ
?
’(t) = x0 +
Z t
t0
f(?;’(?))d?;
?f’n(t)g￥/A ?μ
j’0(t)?’(t)j?
Z t
t0
jf(?;’(?))jd M(t?t0):
?N#LipschitzHq¤
j’1(t)?’(t)j?
Z t
t0
jf(?;’0(?))?f(?;’(?))jd?
L
Z t
t0
j’0(?)?’(?)jd LM
Z t
t0
(t0)d?
= ML2! (t?t0)2:
B?1L
!?k = m+1
H,
j’m(t)?’(t)j? ML
m
(m+1)!(t?t0)
m+1;
5?k = m+1
H
j’m+1(t)?’(t)j?
Z t
t0
jf(?;’m(?))?f(?;’(?))jd?
L
Z t
t0
j’m(?)?’(?)jd ML
m+1
(m+1)!
Z t
t0
(t0)m+1d?
= ML
m+1
(m+2)!(t?t0)
m+2:
yN?
DB
,E? ?i?
n?μ?t 2 [t0;t0 +h]
H
j’n(t)?’(t)j? ML
n
(n+1)!(t?t0)
n+1? MLn
(n+1)!h
n+1:
'
óμ9? ?.
2
3. pZ?dxdt = x2V?(0;1)￥? ?Qí
3.
3:
ó′ù5￥PicardY}?
 ?/
’0(t) = 1;
’1(t) = 1+
Z t
0
’20(?)d? = 1+t;
’2(t) = 1+
Z t
0
’21(?)d? = 1+t+t2 + 13t3;
?N¤? ?Qí
31
’3(t) = 1+
Z t
0
’22(?)d?
= 1+t+t2 +t3 + 23t4 + 13t5 + 19t6 + 163t7:
5. ?¨Picardi?B?? ? p?l ? u×R = f(t;x) 2R2, jtj? 1;jxj? 1g
￥Z?
dx
dt = x
2 +t
V?(0;0)￥3￥i uW,i pN uW
D??￥3￥μ??V0:05￥í
3.
3:? út0 = 0;x0 = 0;a = 1;b = 1;M = maxfjx2 + tj, (t;x) 2 Rg = 2;f(t;x) = x2 + t,
h = minfa; bMg = 12b?Picardi·B?? ?
ó′ù5 uWt 2 [?12; 12]
i·
Bb?
4 p¤Lipschitzè
L = maxfj@f@xj, (t;x) 2 Rg = 2b?μ9(2.4) V¤3￥i u
Wt 2 [?12; 12]
μ
j’n(t)?’(t)j? 2￠2
n
(n+1)!(
1
2)
n+1 = 1
(n+1)!:
?’(t)1
ó′ù5￥??3,’n(t)1
ó′ù5￥?nQí
3bA ??n = 3
H,’3(t)1
ó′ù5 uWt 2 [?12; 12]
D??￥3￥μ??V0:05￥í
3,y1?
Hμ
j’3(t)?’(t)j? 124 < 0:05:
ó′ù5￥PicardY}?
 -
1[1
’0(t) = 0;
’1(t) =
Z t
0
(’20(?)+?)d? = t
2
2 ;
’2(t) =
Z t
0
(’21(?)+?)d? = t
2
2 +
t5
20;
’3(t) =
Z t
0
(’21(?)+?)d? = t
2
2 +
t5
20 +
t8
160 +
t11
4400;
#
pí
31
’3(t) = t
2
2 +
t5
20 +
t8
160 +
t11
4400:
8.
k p′ù5
dx
dt = P(t)x+Q(t); x(t0) = x0
3
￥PicardY}?
,iYV pY}?
￥K p′ù5￥3,? úP(t),Q(t) (1 ??f
.
3:
ó′ù5￥PicardY}?
 ?/
’0(t) = x0;
’1(t) = x0 +
Z t
t0
(P(?)’0(?)+Q(?))d?
= x0(1+
Z t
t0
P(?)d?)+
Z t
t0
Q(?)d?;
’2(t) = x0 +
Z t
t0
(P(?)’1(?)+Q(?))d?
= x0(1+
Z t
t0
P(?)d? +
Z t
t0
P(?)(
Z?
t0
P(s)ds)d?)
+
Z t
t0
P(?)(
Z?
t0
Q(s)ds)d? +
Z t
t0
Q(?)d?:
?s?s?
4¤?
Z t
t0
P(?)(
Z?
t0
P(s)ds)d? = 12!(
Z t
t0
P(?)d?)2;
Z t
t0
P(?)(
Z?
t0
Q(s)ds)d? =
Z t
t0
Q(?)(
Z t
P(s)ds)d?:
yN
’2(t) = x0f1+
Z t
t0
P(?)d? + 12!(
Z t
t0
P(?)d?)2g
+
Z t
t0
Q(?)(1+
Z t
P(s)ds)d?:
?
1 V¤
’3(t) = x0f1+
Z t
t0
P(?)d? + 12!(
Z t
t0
P(?)d?)2
+ 13!(
Z t
t0
P(?)d?)3g
+
Z t
t0
Q(?)(1+
Z t
P(s)ds+ 12!(
Z t
P(s)ds)2)d?:
B?1?
DB
,E V￡
ü ?i￥n? 2?μ
’n(t) =
nX
k=1
x0
k!(
Z t
t0
P(?)d?)k +
Z t
t0
Q(?)
n?1X
k=1
1
k!(
Z t
P(s)ds)kd?:
A ??
f’n(t)gBá
l ??
x(t) = x0 exp(
Z t
t0
P(?)d?)+
Z t
t0
Q(?)exp(
Z t
P(s)ds)d?
= exp(
Z t
t0
P(?)d?)fx0 +
Z t
t0
Q(?)exp(?
Z?
t0
P(s)ds)d?g:
?
^
ó′ù5￥3.