13ù
? ~???§)35n?
N￠?4?[
wg??
06/01/2022
N￠?4?[ (wg??) 13ù 06/01/2022 1 / 89
81
1 )3??5?n
2 )ò??n
3 )éD??Y5?n
4 )éD???5?n
5 ??ú?)
N￠?4?[ (wg??) 13ù 06/01/2022 2 / 89
81
1 )3??5?n
2 )ò??n
3 )éD??Y5?n
4 )éD???5?n
5 ??ú?)
N￠?4?[ (wg??) 13ù 06/01/2022 3 / 89
?? ???§D?ˉK
8>>
><
>>>:
dy
dx = f(x;y);
y(x0) = y0;
(1.1)
ù￥1?amà?êf(x;y)3Y/??
=f(x;y)
x x0j6a;jy y0j6bg
t?Y.e3~êL > 0,|é¤k(x;y1);(x;y2)2 ?k
jf(x;y1) f(x;y2)j6Ljy1 y2j
¤á,K?f(x;y)3 t''uy÷v|êF](Lipschitz)^?, L??|ê
F]~ê.
N￠?4?[ (wg??) 13ù 06/01/2022 4 / 89
?n
(3??5?n) ef(x;y)3Y/? t?Y?''uy÷v|êF]^
?,K???§D?ˉK(1.1)3??3?mjx x0j6ht??
)y = ’(x),?÷vD?^?’(x0) = y0,ù￥
h = min
(
a; bM
)
; M = max
(x;y)2
jf(x;y)j:
T?ny2,ì???±eAú:
(1)ò???§D?ˉK(1.1))3??5ˉK=z??? dè?
?§)3??5ˉK;
(2)Eè??§S??êS;
(3)y2dS??êS??5;
(4)y2dS??êS4??êò′è??§);
(5)y2)??5.
N￠?4?[ (wg??) 13ù 06/01/2022 5 / 89
?{B??,·?=y2)3?m[x0;x0 + h]t3???,3?
m[x0 h;x0]ty2L§aq. e???T?n?[y2.
y2: (1)y2) d5.
ey = y(x)′???§D?ˉK(1.1)),é???§D?ˉK(1.1)￥
1?aüàè??
y(x) = y0 +
Z x
x0
f(s;y(s))ds: (1.2)
ey = y(x)′è??§(1.2)),éè??§(1.2)üà''ux|,B?
???§D?ˉK(1.1)￥1?a.
?x = x0?,?\è??
§(1.2)?y(x0) = y0.?d, y = y(x)?′???§D?ˉK(1.1)).ù
,B?á
???§D?ˉK(1.1)?è??§(1.2)?m) d''
X.?
y2?n1,?Iy2è??§(1.2))3??5
N￠?4?[ (wg??) 13ù 06/01/2022 6 / 89
(2)E?k(Picard)S??êS.
à???Y?ê’0(x),?\è??§(1.2),?Y?ê
’1(x) = y0 +
Z x
x0
f(s;’0(s))ds;
?’1(x0) = y0.e’1(x) = ’0(x),K’0(x)ò′è??§(1.2)).?K,U
Yò’1(x)?\è??§(1.2)mà,?Y?ê
’2(x) = y0 +
Z x
x0
f(s;’1(s))ds;
?’2(x0) = y0.e’2(x) = ’1(x),K’1(x)=′è??§(1.2)).-Et?
L§,?S??êSf’i(x);i = 0;1;2; g,÷v
’n(x) = y0 +
Z x
x0
f(s;’n 1(s))ds; (1.3)
?’n(x0) = y0.ù?Y?êSf’i(x);i = 0;1;2; g???kS??
êS.
N￠?4?[ (wg??) 13ù 06/01/2022 7 / 89
eù?L§??UYe,é?,?n,|’n(x) = ’n 1(x),
′3
??4??ê’(x),|
limn!1’n(x) = ’(x):
@o,é?§(1.3)üà4??
’(x) = limn!1’n(x)= limn!1
"
y0 +
Z x
x0
f(s;’n 1(s))ds
#
=y0 + limn!1
Z x
x0
f(s;’n 1(s))ds
=y0 +
Z x
x0
limn!1f(s;’n 1(s))ds
=y0 +
Z x
x0
f(s;’(s))ds;
?’(x0) = y0.?d, ’(x)=′è??§(1.2)).
N￠?4?[ (wg??) 13ù 06/01/2022 8 / 89
ù??LES??êS|4??
|?D?ˉK)?{,???
k?úú%%C{. d?§(1.3)E??ê’n(x)?????§D?ˉ
K(1.1)11ngCq).
3’0(x)àt,·? ’0(x) = y0.?kS??êSO?úa?
8>>
<>
>:
’0(x) = y0;
’n(x) = y0 + R xx
0
f(s;’n 1(s))ds; x06x6x0 + h; n = 1;2; : (1.4)
N￠?4?[ (wg??) 13ù 06/01/2022 9 / 89
(3)y2??5.
ún
é??ên, x06x6x0 + h,K’n(x)?Y?÷v
j’n(x) y0j6b: (1.5)
y2: dê?8B{?,’0(x) = y0?, ’0(x)3?m[x0;x0 + h]tk?
?!?Y?÷v^?(1:5). b’n(x)3[x0;x0 + h]tk??!?Y?÷v
^?(1:5),d?
§(1:3)?, ’n+1(x)3[x0;x0 + h]tk??!?Y?÷v
j’n+1(x) y0j6
Z x
x0
jf(s;’n(s))jds
6M(x x0)6Mh6b;
ù￥, M = maxjf(x;y)j.
N￠?4?[ (wg??) 13ù 06/01/2022 10 / 89
ún
S??êSf’n(x)g3x06x6x0 + ht′????.
y2: du?ê
’0(x) +
1X
k=1
[’k(x) ’k 1(x)]; x06x6x0 + h (1.6)
ü?ú?
’0(x) +
nX
k=1
[’k(x) ’k 1(x)] = ’n(x);
?dS??êSf’n(x)g????5??ê(1.6)????5 d.
?d,?y2f’n(x)g3x06x6x0 + ht????,?Iy2?
ê(1.6)3x06x6x0 + ht????.
N￠?4?[ (wg??) 13ù 06/01/2022 11 / 89
dO?úa(1.4)?
j’1(x) ’0(x)j=j’1(x) y0j=
Z x
x0
f(s;’0(s))ds
6M(x x0);
j’2(x) ’1(x)j6
Z x
x0
jf(s;’1(s)) f(s;’0(s))jds
6
Z x
x0
Lj’1(s) ’0(s)jds (|êF]^?)
6L
Z x
x0
M(s x0)ds
= ML2 (x x0)2:
N￠?4?[ (wg??) 13ù 06/01/2022 12 / 89
±daí,éu??ên,ek
j’n(x) ’n 1(x)j6 ML
n 1
n! (x x0)
n
¤á,K
j’n+1(x) ’n(x)j=
Z x
x0
f(s;’n(s))ds
Z x
x0
f(s;’n 1(s))ds
6
Z x
x0
jf(s;’n(s)) f(s;’n 1(s))jds
6L
Z x
x0
j’n(s) ’n 1(s)jds
6L
Z x
x0
MLn 1
n! (s x0)
nds
= ML
n
(n + 1)!(x x0)
n+1:
N￠?4?[ (wg??) 13ù 06/01/2022 13 / 89
dê?8B{?,éu??êk,x06x6x0 + h?,
j’k(x) ’k 1(x)j6 ML
k 1
k! (x x0)
k6 MLk 1
k! h
k; (1.7)
@oòk
1X
k=1
[’k(x) ’k 1(x)]6
1X
k=1
j’k(x) ’k 1(x)j
6
1X
k=1
MLk 1
k! h
k:
N￠?4?[ (wg??) 13ù 06/01/2022 14 / 89
-ak = ML
k 1
k! h
k;du
lim
k!1
ak+1
ak = limk!1
MLk
(k + 1)!h
k+1 k!
MLk 1hk = limk!1
Lh
k + 1 = 0;
?dd?dA.d(Weierstrass)O{?,?ê
1X
k=1
MLk 1
k! h
k3?
mx06x6x0 + ht????. u′
’n(x) = ’0(x) +
nX
k=1
[’k(x) ’k 1(x)]
?????,=Sf’n(x)g3?mx06x6x0 + ht????.
N￠?4?[ (wg??) 13ù 06/01/2022 15 / 89
(4)y2S??êS4??ê′è??§).
ún
’(x)′è??§(1.2)??3?mx06x6x0 + ht?Y).
y2: d|êF]^??
jf(x;’n(x)) f(x;’(x))j6Lj’n(x) ’(x)j:
q??f’n(x)g3?mx06x6x0 + ht????,e??u’(x),K?êS
ff(x;’n(x))g3?mx06x6x0 + ht?????uf(x;’(x)).
éO?úa(1.4)￥1aüà4??
limn!1’n(x)= limn!1
"
y0 +
Z x
x0
f(s;’n 1(s))ds
#
=y0 +
Z x
x0
limn!1f(s;’n 1(s))ds;
N￠?4?[ (wg??) 13ù 06/01/2022 16 / 89
Kk
’(x) = y0 +
Z x
x0
f(s;’(s))ds:
?d, ’(x)′è??§(1.2)??u?mx06x6x0 + ht?Y).
(5)y2)??5.
dún3?, ’(x)′è??§(1.2)???Y).e?3,???Y
) (x),Kd|êF]^?
j’(x) (x)j=
Z x
x0
f(s;’(s)) f(s; (s))ds
6L
Z x
x0
j’(s) (s)jds: (1.8)
-g(x) =j’(x) (x)j,Ka(1.8)z?
g(x)6L
Z x
x0
g(s)ds:
N￠?4?[ (wg??) 13ù 06/01/2022 17 / 89
Z
x
x0
g(s)ds
!0
= g(x)6L
Z x
x0
g(s)ds;
u′-
u(x) =
Z x
x0
g(s)ds;
K
u0(x) Lu(x)60:
taüàó|e Lx,?3?m[x0;x]tè??Z
x
x0
[u0(x) Lu(x)]e Lxdx=
Z x
x0
u0(x)e Lxdx
Z x
x0
Lu(x)e Lxdx
=u(x)e Lx
x
x0
+ L
Z x
x0
u(x)e Lxdx
Z x
x0
Lu(x)e Lxdx
=u(x)e Lx u(x0)e Lx0
60;
N￠?4?[ (wg??) 13ù 06/01/2022 18 / 89
l
u(x)e Lx6u(x0)e Lx0 = e Lx0
Z x0
x0
g(s)ds = 0;
¤±u(x)60,=
Z x
x0
g(s)ds =
Z x
x0
j’(s) (s)jds60:
da(1.8)?
j’(x) (x)j60;
?d
’(x) (x) = 0;
=è??§(3-2))′??.
N￠?4?[ (wg??) 13ù 06/01/2022 19 / 89
ún
’(x); (x)t?è??§(1.2)3?m[x0;x0 + h]t?Y),K??
k’(x) (x); x2[x0;x0 + h].
d±t(1) (5)ü?y2L§?±w,è??§(1.2)3?
m[x0;x0 + h]t)3??5L÷v?n1￥^?.?d,?? d
???§D?ˉK(1.1))3??5=?n1B
y2.3í
L§￥,·?ó???
^?k?ú%C{|1ngCq)O?ú
a(1.4),???
ùCügCq)? O,§????,?°Y
???^?.aqún3.2y2L§,?ó1ngCq)’n(x)ú
???§D?ˉK(1.1)O()’(x)3?m[x0 h;x0 + h]S? O
j’n(x) ’(x)j6 ML
n
(n + 1)!h
n+1: (1.9)
N￠?4?[ (wg??) 13ù 06/01/2022 20 / 89
51:è?-????f(x;y)0u??BC1?M?B1C
? M?m.M6 ba?,X?3-1(a)¤?,
)y = ’(x)3x0 a6x6x0 + a￥k??;
M > ba?,X?3-1(b)¤?,
???????BC1;B1C¤Y??,@o)y = ’(x)òk?U??Y/
? ,|f(x;y)???.¤±,?kx0 bM 6x6x0 + bM?,aUy
)y = ’(x)3 S.?d,)3?m?jx x0j6h = min
(
a; bM
)
.??B
`2,?3-1￥:(x0;y0)??I:(0;0).
Figure:?n1￥hA???
N￠?4?[ (wg??) 13ù 06/01/2022 21 / 89
52:?n1￥?|?êf(x;y)''uy÷v|êF]^?,ù ''Ju
y.·?2~^f(x;y)3 t''uy?ê′?35?.??
ef(x;y);fy(x;y)3 t?Y,Kfy(x;y)3 tk..-jfy(x;y)j6L3 t¤
á,Kf(x;y)÷v|êF]^?
jf(x;y1) f(x;y2)j=jfy(x;y2 + (y1 y2))jjy1 y2j
6Ljy1 y2j:
ù?^??|p,du÷v|êF]^??ê???3,
Xf(x;y) =jyj,?d?êfy(x;y)?3??,?U?′?,?I^
ù|?{?1|êF]^?y.
53:XJmà?êf(x;y)éuyvk????,
= =f(x;y)
6x6 ; 1< y < +1g,Ké??D?(x0;y0)(x0 2[ ; ])¤
(?)3??m[ ; ]t?k??.???ò?n1íL§?1?
Dy.
N￠?4?[ (wg??) 13ù 06/01/2022 22 / 89
?n
éu? ?a?§
F(x;y;y0) = 0; (1.10)
XJ3:(x0;y0;y00),??￥÷v
? F(x;y;y0)é¤kCt(x;y;y0)?Y,?3?Y?ê,
? F(x0;y0;y00) = 0,
fi @F(x0;y0;y
0
0)
@y0 , 0,
K?§(1.10)3??)
y = ’(x); jx x0j6h (h?v
ê);
?÷vD?^?
’(x0) = y0; ’0(x0) = y00:
N￠?4?[ (wg??) 13ù 06/01/2022 23 / 89
y2: d??ê3?n,(ü^?? fi?,?§(1.10)￥?
êy0???L??x;y?ê
y0 = f(x;y);
ù￥, f(x;y)3(x0;y0),??S?Y,?÷vy00 = f(x0;y0):
é?§(1.10)üà''uy|
@F
@y +
@F
@y0
@y0
@y =
@F
@y +
@F
@f
@f
@y = 0;
l
k @f
@y =
@F
@y
.@F
@f :
dfi@F@f , 0,?k.,?d÷v?n1￥^?,l
??§(1.10)÷v
D?^?)3???.
N￠?4?[ (wg??) 13ù 06/01/2022 24 / 89
Example
|?§ dy
dx = 1 + y
2
÷vy(0) = 01ngCq).
): d?kS??êSO?úa(1.4)?
’0(x) = y(0) = 0;
’1(x) =
Z x
0
(1 +’0(s)2)ds = x;
’2(x) =
Z x
0
(1 +’1(s)2)ds =
Z x
0
(1 + s2)ds = x + 13x3;
’3(x) =
Z x
0
1 +’2(s)2
ds =
Z x
0
1 + (s + 13s3)2
!
ds
= x + 13x3 + 215x5 + 163x7:
N￠?4?[ (wg??) 13ù 06/01/2022 25 / 89
Example
?§ dy
dx = x
2 + y
??3Y/? : 16x61; 16y61t,á??(?2L:(0;0))
3?m,?|??O()? ??L0:05Cq)L?a.
): d?n1?, f(x;y) = x2 + y3Y/? t?Y,?@f@y = 1;|ê
F]~êL = 1,K?§3:(0;0)?3??)y = ’(x).
-a = 1;b = 1,K
M = maxjf(x;y)j= 2;
¤±)3?m?jxj6h,ù￥h = min
(
1; 12
)
= 12. d? O
a(1.9)?
j’n(x) ’(x)j6 ML
n
(n + 1)!h
n+1 = 2n
(n + 1)!
1
2
!n+1
= 12(n + 1)! 60:05:
N￠?4?[ (wg??) 13ù 06/01/2022 26 / 89
da(3-11))(n + 1)!>10,n = 3.
’0(x) = 0;
’1(x) =
Z x
0
(s2 +’0(s))ds = 13x3;
’2(x) =
Z x
0
(s2 +’1(s))ds =
Z x
0
s2 + 13s3
!
ds
=
1
3s
3 + 1
12s
4
!
x
0
= 13x3 + 112x4;
’3(x) =
Z x
0
(s2 +’2(s))ds =
Z x
0
s2 + 13s3 + 112s4
!
ds
=
1
3s
3 + 1
12s
4 + 1
60s
5
!
x
0
= 13x3 + 112x4 + 160x5:
N￠?4?[ (wg??) 13ù 06/01/2022 27 / 89
81
1 )3??5?n
2 )ò??n
3 )éD??Y5?n
4 )éD???5?n
5 ??ú?)
N￠?4?[ (wg??) 13ù 06/01/2022 28 / 89
lt?!￥)3??5?n?±w,???§D?ˉK(1.1))
3?m=?ujx x0j6h = minfa; bMg:ù`2)3??5?m=
?u?ü??S.
3?
￠SA^ˉK￥,·?F")3?m|t
??
,u′J?
e?)ò?Vg.
b???§D?ˉK(1.1)mà?êf(x;y)3,???GS?Y,éu
???:(x0;y0)2G,3 =f(x;y)
±(x0;y0)?￥%/¤4Y/
?g G,e3 tf(x;y)''uy÷v|êF]^?,K?f(x;y)''uy÷v?
ü|êF]^?.
N￠?4?[ (wg??) 13ù 06/01/2022 29 / 89
?n
()ò??n) e???§D?ˉK(1.1)mà?êf(x;y)3k.?
?GS?Y,?3GS''uy÷v?ü|êF]^?,K???§D?ˉ
K(1.1)?LGS???:(x0;y0))y = ’(x)?±ò??C??G>
..
y2: d?n1?,L:(x0;y0))y = ’(x)3?mjx x0j6htk??.
-
x1 = x0 + h; y1 = ’(x1) = ’(x0 + h);
ù￥, h?~ê.??f(x;y)3GS''uy÷v?ü|êF]^?,¤±3
?? 1 =f(x;y)
±(x1;y1)?￥%/¤4Y/?g Gt3
)y = ’1(x)?L:(x1;y1),3?mjx x1j6h1(h1?~ê)tk??,?÷
v’1(x1) = y1. l
k’1(x1) = ’(x1),3ú?mx1 h16x6x1t,d
)??5?’1(x) = ’(x),
N￠?4?[ (wg??) 13ù 06/01/2022 30 / 89
u′?ò)ò??
y =
8>>
<>
>:
’(x); x0 h6x6x0 + h;
’1(x); x0 + h6x6x0 + h + h1:
UY-Et?ò?L§,-x2 = x0 + h + h1, y2 = ’1(x2),
3 2 =f(x;y)
±(x2;y2)?￥%/¤4Y/?g G,|)y = ’2(x)?L
:(x2;y2),3jx x2j6h2(h2?~ê)tk??,?÷v’2(x2) = y2. d)
3??5?,3ú?mx2 h26x6x2t, ’2(x) = ’1(x). u′,3
?mx0 h6x6x2 + h2t,)?±ò??
y =
8>>
>>><
>>>>
>:
’(x); x0 h6x6x0 + h;
’1(x); x0 + h6x6x0 + h + h1;
’2(x); x0 + h + h16x6x0 + h + h1 + h2:
N￠?4?[ (wg??) 13ù 06/01/2022 31 / 89
dd??,)3?m?x0my
ò?. UY?g-Et?ò?L
§,B?ò)y = ’(x)??ò??Gm>..
ó/,??±?x0?y?1ò?,??ò??G?>.. l
?n
y.
3A???t,￠St′35è?-?y = ’(x)?müà?tN?
è?-??,???G>.,ù?)?????§D?ˉK(1.1)
ú).
?,????ú)??3?m7?′??m?m,?K
?UYò?.
N￠?4?[ (wg??) 13ù 06/01/2022 32 / 89
?Nò??1??±eA?(±?xmàò??~):
(1)XJG′k.??,KL:(x0;y0))y = ’(x)ò??x06x < dt,=
x!d?, (x;’(x))!??G>.;
(2)XJG′?.??,)y = ’(x)?oò??mx06x < +1,?oò?
?mx06x < d, d?k?ê,ù?x!d?, y = ’(x)!1,??
:(x;’(x))!??G>..
N￠?4?[ (wg??) 13ù 06/01/2022 33 / 89
?n
f(x;y)3xOy2?tz?:t?Y,éy÷v?ü|êF]^?,?k
~êN,|
jf(x;y)j6Njyj;
K???§D?ˉK(1.1))3( 1;+1)t3.
Example
|?§dydx = 1 y2?O?L:(0;2)ú(0;0))3?m.
):?§mà?ê1 y23?xOy2?tk??,?÷v?n19?
n3.3￥^?,?|?§?)?
y = ce
2x 1
ce2x + 1;
ù￥c???~ê.
N￠?4?[ (wg??) 13ù 06/01/2022 34 / 89
L:(0;2))?
y = 3e
2x 1
3e2x + 1:
T)3?m
1; 12 ln 13
!
[
1
2 ln
1
3;+1
!
tk??,?dL:(0;2))
3?m?
1; 12 ln 13
!
.
ó/,L:(0;0))?
y = e
2x 1
e2x + 1;
T)3?m( 1;+1)tk??,?dL:(0;0))3?m
?( 1;+1).
N￠?4?[ (wg??) 13ù 06/01/2022 35 / 89
Example
|?§dydx = lnxL:(1;1))3?m.
):?§3x > 02?tk??,?÷v?n19?n3.3￥^?,ù?
)?
y = x(lnx 1) + c;
ù￥c???~ê.L:(1;1))?
y = x(lnx 1) + 2;
ù3?m?(0;+1).
N￠?4?[ (wg??) 13ù 06/01/2022 36 / 89
81
1 )3??5?n
2 )ò??n
3 )éD??Y5?n
4 )éD???5?n
5 ??ú?)
N￠?4?[ (wg??) 13ù 06/01/2022 37 / 89
3?n3.1￥,·???
???§D?ˉK(1.1)3)3?mS)
35ú??5.eD?^?UC,KˉK)9ù3?m?ò?XU
C.X~3￥2L:(0;2)ú(0;0))?O?
y = 3e
2x 1
3e2x + 1; y =
e2x 1
e2x + 1:
ù??X???§D?ˉK(1.1))?=?6ugCtx,??6uD
?(x0;y0).
?d,D?^??3??Cz?,)y = ’(x)?±w?′''ux;x0;y0
n?ê
y = ’(x;x0;y0); (3.1)
?x = x0?, y = ’(x0;x0;y0) = y0.
N￠?4?[ (wg??) 13ù 06/01/2022 38 / 89
3a(3.1)3?mS??:x1,Kk
y1 = ’(x1;x0;y0): (3.2)
d?n1?,L:(x1;y1)-?=′L:(x0;y0)-?,u′k
y = ’(x;x1;y1):
ò(x0;y0)?\ta
y0 = ’(x0;x1;y1): (3.3)
??(x1;y1)???:,¤±da(3.3)?
y0 = ’(x0;x;y): (3.4)
da(3.1)úa(3.4)?Xe?n.
N￠?4?[ (wg??) 13ù 06/01/2022 39 / 89
?n
()''uD?é?5?n) e???§D?ˉK(1.1)3??)y =
’(x;x0;y0),K3)3?mS
y0 = ’(x0;x;y)
?¤á.
ù`23)3?mta(3.1)￥(x;y)?(x0;y0)?±pN??.
?n
e?êf(x;y)3? S?Y,?''uy÷v|êF]^?(~êL),Kéu?
§dydx = f(x;y)??ü?)’(x); (x),3§?ú3?mS,?
:x0?,k
j’(x) (x)j6j’(x0) (x0)jeLjx x0j:
N￠?4?[ (wg??) 13ù 06/01/2022 40 / 89
y2:)’(x); (x)3ú3?m[c;d]tk??,-
u(x) = [’(x) (x)]2; x2[c;d];
u′k
u0(x)=2[’(x) (x)][’0(x) 0(x)]
=2[’(x) (x)][f(x;’) f(x; )]
62L[’(x) (x)]2
=2Lu(x):
üàó|e 2Lx,k
[u0(x) 2Lu(x)]e 2Lx60;
d[u(x)e 2Lx]
dx 60:
N￠?4?[ (wg??) 13ù 06/01/2022 41 / 89
¤±,é??x2[c;d],k
u(x)e 2Lx6u(x0)e 2Lx0; x06x6d;
u′k
u(x)6u(x0)e2L(x x0): (3.5)
,???,c6x6x0?,é???§D?ˉK(1.1)￥1?a?CtC
?,-t = x,K x06t6 c; t0 = x0,???§D?ˉKC?
dy
dx = f( t;y):
u′)/a?’( t); ( t).-
v(t) = [’( t) ( t)]2;
N￠?4?[ (wg??) 13ù 06/01/2022 42 / 89
da(3.5)?
v(t)6v(t0)e2L(t t0); t06t6 c;
?t,
v( x)6v( x0)e2L( x+x0); x06 x6 c;
u′k
u(x)6u(x0)e2L(x0 x); c6x6x0: (3.6)
da(3.5)úa(3.6)
u(x)6u(x0)e2Ljx x0j; c6x6d:
ü>2??,B?
j’(x) (x)j6j’(x0) (x0)jeLjx x0j; c6x6d:
N￠?4?[ (wg??) 13ù 06/01/2022 43 / 89
?n
()éD??Y?65?n) f(x;y)3??GS?Y,?''uy÷v?
ü|êF]^?,:(x0;y0)2G,???§D?ˉK(1.1)￥1?a÷v
D?^?y(x0) = y0)?y = ’(x;x0;y0),3?m[a;b]tk??,
?x0 2[a;b],Ké????"> 0,3 = (";a;b) > 0,|
(x1 x0)2 + (y1 y0)2 < 2
?,???§D?ˉK(1.1)￥1?a÷vD?^?y(x1) = y1
)y = ’(x;x1;y1)3?m[a;b]t?k??,?
j’(x;x1;y1) ’(x;x0;y0)j<"; x2[a;b]:
y2:·??±enú?1??.
(1))y = ’(x;x0;y0)¤éAè?-?P?
S =f(x;y)
y = ’(x;x0;y0) ’(x);x2[a;b]g;
N￠?4?[ (wg??) 13ù 06/01/2022 44 / 89
S G?k.48.é??k??:Pi 2S(i = 1; ;N),3±Pi?￥%
m Ci G,|3CiSf(x;y)''uy÷v|êF]^?, L?|êF]
~ê￥???.dk?CX?n?
S
N[
i=1
Ci G;
?ò′`,ùk??m 3?m[a;b]tCX
è?-?S.
> 0L?S?SNi=1 Ci>.?l.e
= min
"; 2
;
K±Stz?:Pi?￥%!± ??? N,?ó§? ±??
¤?1Sk.4?
D =
[
Pi2S
B(Pi; ):
¤±S D G, f(x;y)3Dt''uy÷v|êF]^?, L=?ù|êF]
~ê.
N￠?4?[ (wg??) 13ù 06/01/2022 45 / 89
(2)e5y2é??"> 0,3 = (";a;b) > 0 ( < ),|
(x1 x0)2 + (y1 y0)2 < 2
?,)y = ’(x;x1;y1) (x)3?m[a;b]tk??.
d?n3ú(1)?,)y = ’(x;x0;y0)?±??ò??D>.@Dt.>
.@Dtü?:?(c; (c));(d; (d));ùpc < d;K7k[a;b] [c;d]:
?K,bc > a;d < b,dún6?
j (x) ’(x)j6j (x1) ’(x1)jeLjx x1j; c6x6d:
d)’(x)?Y5?,jx x0j6 2?,k
j’(x) ’(x0)j< 1 = 12 e L(b a):
N￠?4?[ (wg??) 13ù 06/01/2022 46 / 89
- = minf 1; 2g;(x1 x0)2 + (y1 y0)26 2?,k
j (x) ’(x)j26j (x1) ’(x1)j2e2Ljx x1j
=j (x1) ’(x0) +’(x0) ’(x1)j2e2Ljx x1j
6(j (x1) ’(x0)j+j’(x0) ’(x1)j)2e2Ljx x1j
62(j (x1) ’(x0)j2 +j’(x0) ’(x1)j2)e2Ljx x1j
=2(jy1 y0j2 +j’(x0) ’(x1)j2)e2Ljx x1j
62( 2 + 12)e2Ljx x1j
64 21e2L(b a)
= 2; c6x6d; (3.7)
Kk
j (c) ’(c)j< ; j (d) ’(d)j< :
ù`2:(c; (c))ú(d; (d))tá3?DSü,
?3>.@Dt,?c?
bg?.?d,) (x)3?m[a;b]tk??.
N￠?4?[ (wg??) 13ù 06/01/2022 47 / 89
(3)òt?y2L§￥?m[c;d]??[a;b],-Ea(3.7)íL§,B
?,(x1 x0)2 + (y1 y0)26 2?,k
j’(x;x1;y1) ’(x;x0;y0)j< 6"; x2[a;b]:
N￠?4?[ (wg??) 13ù 06/01/2022 48 / 89
í?
()éD??Y5?n) ?êf(x;y)3??GS?Y,?''uy÷v?
ü|êF]^?,K???§D?ˉK(1.1))y = ’(x;x0;y0)?
?x;x0;y0?ê3ù3??S′?Y.
y2: d?n7ú?n3?,)’(x;x0;y0)???gCtxúD?(x0;y0)
n?Y?ê.§?ú)???3?m′
(x0;y0) < x < (x0;y0):
-
V =f(x;x0;y0)
(x0;y0) < x < (x0;y0);(x0;y0)2Gg;
K)y = ’(x;x0;y0)3Vtk??.
L???:(x;x1;y1))’(x;x1;y1)??3?m7,?1x;x1,3
?m[a;b],|a < x;x1 < b;?’(x;x1;y1)3?m[a;b]tk??,?éx?
Y.
N￠?4?[ (wg??) 13ù 06/01/2022 49 / 89
?d,é??"> 0,3 1 > 0;|jx xj< 1?,kx2[a;b],?
j’(x;x1;y1) ’(x;x1;y1)j< "2:
d?n7?,é??"> 0,3 2 = 2(";a;b) > 0,|
(x1 x0)2 + (y1 y0)2 < 22
?,)y = ’(x;x0;y0)3[a;b]t?k??,?
j’(x;x0;y0) ’(x;x1;y1)j< "2:
= minf 1; 2g,K
(x x)2 + (x1 x0)2 + (y1 y0)2 < 2
?,k
j’(x;x0;y0) ’(x;x1;y1)j
6j’(x;x0;y0) ’(x;x1;y1)j+j’(x;x1;y1) ’(x;x1;y1)j
6"2 + "2
=";
y = ’(x;x0;y0)3??:(x;x1;y1)??Y,?ò′3ù3??S?Y.
N￠?4?[ (wg??) 13ù 06/01/2022 50 / 89
??
1k?ê ???§ dy
dx = f(x;y; ); (3.8)
??3??G =f(x;y)2G; < < gt.?êf(x;y; )3G Sk??,
eé??(x;y; )2G ;3±§?￥%￥B G ú? ?''~
êL > 0,|é??(x;y1; );(x;y2; )2B,k
jf(x;y1; ) f(x;y2; )j6Ljy1 y2j;
K?f(x;y; )3G S''uy??/÷v?ü|êF]^?.
d?n1?,é?? 0 2( ; ),???§D?ˉK(1.1)?L
:(x0;y0)2G)y = ’(x;x0;y0; )??(?,?÷vy0 = ’(x0;x0;y0; 0).
N￠?4?[ (wg??) 13ù 06/01/2022 51 / 89
?n
()éD?ú?ê?Y?65?n) ?êf(x;y; )3G S?Y?''
uy??/÷v?ü|êF]^?,??(x0;y0; 0)2G ,L
:(x0;y0; 0)2G ?÷vD?^?y(x0) = y0)y = ’(x;x0;y0; 0)3?
m[a;b]tk??, a6x06b,Ké??"> 0,3 = (";a;b) > 0;|
(x1 x0)2 + (y1 y0)2 + ( 0)2 < 2
?,?§(3.8)÷vD?^?y(x1) = y1)y = ’(x;x1;y1; )3?m[a;b]t
?k??,?
j’(x;x1;y1; ) ’(x;x0;y0; 0)j<"; x2[a;b]:
?n
()éD?ú?ê?Y5?n) ?êf(x;y; )3G S?Y,?3G S
''uy??/÷v?ü|êF]^?,K?§(3.8))y = ’(x;x0;y0; )?
?x;x0;y0; ?ê3§3??S′?Y.
N￠?4?[ (wg??) 13ù 06/01/2022 52 / 89
81
1 )3??5?n
2 )ò??n
3 )éD??Y5?n
4 )éD???5?n
5 ??ú?)
N￠?4?[ (wg??) 13ù 06/01/2022 53 / 89
?n
()éD???5?n) XJ?êf(x;y)9@f(x;y)@y ?3??GS?Y,
K???§D?ˉK(1.1))y = ’(x;x0;y0)??x;x0;y0?ê3§
3??S′?Y??.
y2:±e?4ú?1y2.
(1)??@f@y(P?f0y)3??GS?Y,¤±f(x;y)3GS''uy÷v?ü|ê
F]^?.d)éD??Y5?n?,???§D?ˉK(1.1)
)y = ’(x;x0;y0)??x;x0;y0?ê3§3??S3??Y.
(2)??y = ’(x;x0;y0)′???§D?ˉK(1.1)),@’@x3?k
@’
@x = f(x;’(x;x0;y0)):
N￠?4?[ (wg??) 13ù 06/01/2022 54 / 89
df(x;y)9’(x;x0;y0)?Y?, @’@x?Y.¤±,)y = ’(x;x0;y0)3§
3??S''ux′?Y??.
(3)PL:(x0;y0)ú(x0 + x0;y0)(j x0j??,? x0 , 0))?O?
y = ’(x;x0;y0) ’; y = ’(x;x0 + x0;y0) :
d?n1￥?????§D?ˉK(1.1)úè??§(1.2) d5
’ = y0 +
Z x
x0
f(x;’)dx; = y0 +
Z x
x0+ x0
f(x; )dx: (4.1)
N￠?4?[ (wg??) 13ù 06/01/2022 55 / 89
ü??~
’=
Z x
x0+ x0
f(x; )dx
Z x
x0
f(x;’)dx
=
Z x0
x0+ x0
f(x; )dx +
Z x
x0
f(x; )dx
Z x
x0
f(x;’)dx
=
Z x0+ x0
x0
f(x; )dx +
Z x
x0
[f(x; ) f(x;’)]dx
=
Z x0+ x0
x0
f(x; )dx +
Z x
x0
f0y(x;’+ ( ’))( ’)dx; (4.2)
ù￥, 0 < < 1.?§(4.2)??? ad??￥??n.
df0y;’; ?Y59è?￥??n?
Z x0+ x0
x0
f(x; )dx = x0f(x0;y0) + x0 1; (4.3)
N￠?4?[ (wg??) 13ù 06/01/2022 56 / 89
f0y(x;’+ ( ’)) = f0y(x;’) + 2; (4.4)
ù￥, x0 !0?, 1 !0; 2 !0. x0 = 0?,da(4.1)?’ = ,
K 2 = 0,?da(4.3)?, 1 = 0.
?§(4.2)üàó?± x0,?
’
x0 =
1
x0
Z x0+ x
x0
f(x; )dx +
Z x
x0
f0y(x;’+ ( ’)) ’ x
0
dx
=
Z x
x0
(f0y(x;’) + 2) ’ x
0
dx f(x0;y0) 1: (4.5)
-z = ’ x
0
,KtaC?
z =
Z x
x0
(f0y(x;’) + 2)zdx f(x0;y0) 1:
üà''ux|,?? d???§D?ˉK
N￠?4?[ (wg??) 13ù 06/01/2022 57 / 89
8>>
><
>>>:
dz
dx = (f
0
y(x;’) + 2)z;
z(x0) = f(x0;y0) 1; x0 , 0:
(4.6)
x0 = 0?,du 1 = 2 = 0,D?ˉK(4.6)?z?D?ˉK
8>>
><
>>>:
dz
dx = f
0
y(x;’)z;
z(x0) = f(x0;y0):
(4.7)
??’ú ′a(4.1)?Y),Kz = ’ x
0
′D?ˉK(4.6)).
x0 = 0?,￠St’ ,¤±z?′D?ˉK(4.7)).?d,=?D
?ˉK(4.6)=?.
N￠?4?[ (wg??) 13ù 06/01/2022 58 / 89
duD?ˉK(4.6)￥mà?ê(f0y(x;’) + 2)z''ux;y; x0′?Y,?
''uz??/÷v?ü|êF]^?,?dz′x;x0;z0; x0?Y?ê,u
′
lim
x0!0
z = lim
x0!0
’
x0 =
@’
@x0
3,?÷vD?ˉK(4.6). dCt?l{?)
z = (f(x0;y0) + 1)e
R x
x0(f
0y(x;’) 2)dx;
?d @’
@x0 = f(x0;y0)e
R x
x0 f
0y(x;’)dx:
(4.8)
N￠?4?[ (wg??) 13ù 06/01/2022 59 / 89
(4)ó/,?y@’@y
0
3??Y.
y = ’(x;x0;y0 + y0) (j y0j??)′???§D?ˉK(1.1)?L
:(x0;y0 + y0)). aq(3)￥íL§,?u = ’ y
0
′D?ˉK
8>>
><
>>>:
du
dx = (f
0
y(x;’) + 3)u;
u(x0) = 1;
).ù￥, y0 !0?, 3 !0,? y0 = 0?, 3 = 0:d?, u?D?
ˉK 8
>>><
>>>:
du
dx = f
0
y(x;’)u;
u(x0) = 1;
(4.9)
).
N￠?4?[ (wg??) 13ù 06/01/2022 60 / 89
?d, u?x;x0;u0; y0?Y?ê,?k
@’
@y0 = lim y0!0
’
y0 = e
R x
x0 f
0y(x;’)dx:
(4.10)
d)éD???5?n, @’@x
0
ú@’@y
0
?O′D?ˉK(4.7)úD?ˉ
K(4.9)),?da(4.8)úa(4.10)L??5.
N￠?4?[ (wg??) 13ù 06/01/2022 61 / 89
Example
D?ˉK 8
>>><
>>>:
dy
dx = sin(xy
2);
y(x0) = y0;
)?y = ’(x;x0;y0),á|@’(x;x0;y0)@x
0
x0=1;y0=0
ú@’(x;x0;y0)@y
0
x0=1;y0=0
.
):-
f(x;y) = sin(xy2);
Kk
@f
@y = cos(xy
2) 2yx:
duf(x;y)?@f@yt3xOy2?t?Y,?÷v?n10^?.
?d?
§?, y = 0?′÷vD?^?y(1) = 0),k’(x;1;0) = 0.
N￠?4?[ (wg??) 13ù 06/01/2022 62 / 89
?d,da(4.8)úa(4.10)?
@’(x;x0;y0)
@x0
x0=1;y0=0
= f(1;0)e
R x
1 2x’cos(x’
2(x;1;0))dx = 0;
@’(x;x0;y0)
@y0
x0=1;y0=0
= e
R x
1 2x’cos(x’
2(x;1;0))dx = 1:
N￠?4?[ (wg??) 13ù 06/01/2022 63 / 89
81
1 )3??5?n
2 )ò??n
3 )éD??Y5?n
4 )éD???5?n
5 ??ú?)
N￠?4?[ (wg??) 13ù 06/01/2022 64 / 89
? ???§ dy
dx = f(x;y) (5.1)
?) ′1k????~ê),§3A?tL??xè?-?
’(x;y;c) = 0: (5.2)
~X:
(1) x2 + y2 = c; c > 0;
(2) y2 + (x c)2 = 1; 1< c < +1;
(3) y (x c)2 = 2; 1< c < +1:
′, (1)?±:?￥%?xó% , (2)?±(c;0)?￥%!1???
?x , (3)??x?-?,X?3-2¤?.
N￠?4?[ (wg??) 13ù 06/01/2022 65 / 89
Figure:è?-??
N￠?4?[ (wg??) 13ù 06/01/2022 66 / 89
éu?3-2(b),·?uyü^??y = 1z?:o?(2)¤L?è?-
?x￥ 3T:??;?3-2(c)?′Xd,??y = 2tz?:o
?(3)¤L?ù￥?^??3T:??.u′,J?-?????.
??
32?tk?^?Y??-? .XJéu??:P2 ,3:P,
??S,è?-?x(5.2)￥?k?^-??L:P,?3:P??-
? ??,K?-? ?è?-?x(5.2)?|??.
¤±,3?3-2(b)￥,ü^??y = 1′è?-?xy2 + (x c)2 = 1??;
3
?3-2(c)￥,??y = 2′è?-?xy (x c)2 = 2??.′,??′
z?xè?-??k??. (1)￥¤L?ó% òvk??. e?·?
??è?-?x3??7?^?.
N￠?4?[ (wg??) 13ù 06/01/2022 67 / 89
?n
(c-O{) e-? ′è?-?x(5.2)?|??,K§ó?÷v
8>>
<>
>:
’(x;y;c) = 0;
’0c(x;y;c) = 0; (5.3)
?c,
V(x;y) = 0: (5.4)
y2: ?? ??Xe?ê/a:
x = f(c); y = g(c);
ù￥, c?è?-?x(5.2)￥?ê.è?-?x(5.2)?,?
’(f(c);g(c);c) = 0: (5.5)
du??′?Y??,?df(c)úg(c)éc?′?Y??,é?
§(5.5)''uc|?
’0xf0(c) +’0yg0(c) +’0c = 0; (5.6)
N￠?4?[ (wg??) 13ù 06/01/2022 68 / 89
ù￥
’0x = ’0x(f(c);g(c);c); ’0y = ’0y(f(c);g(c);c):
é??~êc,
(f0(c);g0(c)) = (0;0)?(’0x;’0y) = (0;0) (5.7)
?,da(5.6)?
’0c(f(c);g(c);c) = 0: (5.8)
a(5.7)?¤á?,=
(f0(c);g0(c)) , (0;0)?(’0x;’0y) , (0;0);
ù??? 3:P(c) = (f(c);g(c))???t(f0(c);g0(c))??L:P(c)-
?’(x;y;c) = 03P(c):??t( ’0y;’0x)?′?òz. q??ùü??
?t3:P(c)?′?,¤±
f0(c)’0x + g0(c)’0y = 0:
u′,da(5.6)óa(5.8)¤á.
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?d,e??è?-?x(5.2)??,éu??~êc,a(5.5)úa(5.8)7
,ó?¤á. u′, c-O{′??37?^?.
Example
|-?x(x c)2 + y2 = 4c??.
): dc-O{? (
(x c)2 + y2 = 4c;
2(x c) = 4;
?c,-?
y2 = 4x + 4:
2u,T-??¤|-?x??,X?3-3¤?.
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Figure: ??y2 = 4x + 4
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dc-O{?,÷v^?(5.3)?^?(5.4)-?????ò′è?-?
x(5.2)
??,?I???úy. ~X,è?-?xy2 + cx = 0dc-O{?
:(0;0),T:?′ù?è?-?x??. e??n??
??3
??^?.
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?n
dè?-?x(5.2)c-O{(5.4)(?
?^?Y??-
?V(x;y) = 0,?¤?ê/a
: x = u(c); y = v(c);
?÷v^?
(u0(c);v0(c)) , (0;0); (’0x;’0y) , (0;0); (5.9)
ù￥’0x = ’0x(u(c);v(c);c), ’0y = ’0y(u(c);v(c);c),K ′è?-?x(5.2)?
|??.
y2:3-? t??:P(c) = (u(c);v(c));Kk
’(u(c);v(c);c) = 0;’0c(u(c);v(c);c) = 0:
du(’0x;’0y) , (0;0),|^??ê3?n?,?§(5.5)3:P(c)??(
??^?Y??-?
c : y = h(x); ?x = m(y):
T-?3:P(c)??
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k = ’
0x(u(c);v(c);c)
’0y(u(c);v(c);c);
???L??-? c3P(c)?k??t
t1 = ( ’0y;’0x):
-? 3:P(c)??t?
t2 = (u0(c);v0(c)):
da(5.6)?
u0(c)’0x + v0(c)’0y = 0;
ù??X??tt1út23P(c):?′?.
q??éu-? ,c??, ???:,¤±§??1uè?-?
x(5.2)S.?ò′`,-? ?è?-?x(5.2)￥-??ó,3
:P(c)???t?.u′,yù-? ′è?-?x(5.2)?|
??.
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Example
|^?n3.11u-?y2 = 4x + 4′~3.6￥-?x??.
):-
’(x;y;c) = (x c)2 + y2 4c:
-?y2 = 4x + 4???ê/a
( x = u(c) = c 2;
y = v(c) = 2pc 1; (c > 1):
??
(u0(c);v0(c)) =
1; 1pc 1
!
, (0;0);
?
(’0x;’0y) = ( 4; 4pc 1) , (0;0);
d?n3.11?, y2 = 4x + 4′-?x’(x;y;c) = 0?|??.
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Example
|)?§
14
dy
dx
!2
+ xdydx y = 0:
):-p = dydx;K?§z?
14p2 + xp y = 0; (5.10)
üàó?''ux|,?
12pdpdx + xdpdx = 0;
=
dp
dx
12p + x
!
= 0:
Kk dp
dx = 0; p c;
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ù￥, c???~ê. ?\?§(5.10)??)
y = cx 14c2: (5.11)
12p + x = 0?,=x = 12p;d?§(5.10)???ê/aA)
8>>
>>><
>>>>
>:
x = 12p;
y = 14p2:
(5.12)
?3-4??
~8?)úA)?/.·?uy?)(5.11)??1A
)(5.12),
?3A)z?:t?k?)￥,?)3T:?ù??.
éuùA?A???,·?ú\?)Vg.
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Figure: ~3.8?)úA)?/
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??
? ???§
F(x;y;y0) = 0 (5.13)
k??A)
: y = ’(x): (5.14)
XJéz?:P2 ,3:P???S???§(5.13)k???ó
u )3T:? ??,K?y = ’(x)′???§(5.13)?).
~8￥A)(5.12)B′?§?).e?·????)37?^
?.
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?n
(p-O{) ?êF(x;y;y0)é(x;y;y0)2G′?Y(-y0 = p),?''
uyúp3?Y?êF0y;F0p.e?êy = ’(x)′???§(5.13)??
?),?
(x;’(x);’0(x))2G;
K?)÷v 8
>><
>>:F(x;y;p) = 0;F0
p(x;y;p) = 0;
(5.15)
l￥?p,
W(x;y) = 0: (5.16)
y2: duy = ’(x)′???§(5.13)),ù÷va(5.15)￥1?a.
·??^?y{5y21a¤á. b3,?:(x0;y0;p0)|
F0p(x0;y0;p0) , 0;
ù￥, y0 = ’(x0); p0 = ’0(x0).
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duF(x0;y0;p0) = 0;(x0;y0;p0)2G;?a??ê3?n?,???
§(5.13)3:(x0;y0)?3??)
y0 = f(x;y); (5.17)
?÷vf(x0;y0) = p0.
???§(5.13)¤k÷vy(x0) = y0;y0(x0) = p0
)7?′???§(5.17)).
q???êf(x;y)3:(x0;y0),?S?Y,?éyk?Y?ê
f0y(x0;y0) = F
0y(x;y;f(x;y))
F0p(x;y;f(x;y));
Kd?n1?,???§(5.13)÷vD?^?y(x0) = y0)′3??
?.ù??)??g?.?
y = ’(x)7,÷va(5.15)￥1a.
T?nó????
?)?U3??.·??±la(5.15)￥?é
?).,,?é)′?′?),?I??úy.
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éu~8￥?),?±yù÷vp-O{.′,·?I?rN′:
dp-O{???′???§(5.13));e′),????′?
).
~X?§y02 4y2 = 0,dp-O{?
8>>
<>
>:
p2 4y2 = 0;
2p = 0: (5.18)
?p,y = 0.
y = 0′?§),%?′?).???§?)
?y = ce 2x,
y = 01u?)S.
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???1e,eU|??§?),?±(üA??/!? ??
y;e?U|??§?),·???O?)????^?.
?n
?êF(x;y;p)é(x;y;p)2G′ ?Y??.e???
§(5.13)dp-O{
F(x;y;p) = 0; F0p(x;y;p) = 0;
?p?êy = ’(x)′???§(5.13)).XJ§÷v
F0y(x;’(x);’0(x)) , 0;
F00pp(x;’(x);’0(x)) , 0;
F0p(x;’(x);’0(x)) = 0;
Ky = ’(x)′???§(5.13)?).
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Example
?§
yy02 = (y + 1)ex
′?3?)?
):-y0 = p,K
F(x;y;p) = yp2 (y + 1)ex:
dp-O{
yp2 (y + 1)ex = 0; (5.19)
2yp = 0: (5.20)
da(5.20))
(1) y = 0?′?§);
(2) p = 0?,?\a(5.19)
y = ’(x) = 1:
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??
F0y(x; 1;0)=(p2 ex)
(x; 1;0)
= ex , 0;
F00pp(x; 1;0)=2y
(x; 1;0)
= 2 , 0;
F0p(x; 1;0)=2yp
(x; 1;0)
= 0;
d?n14?, y = 1′?§?).
Example
|?§2y(y0 1) xy02 = 0?).
):-p = y0,K?§?
2y(p 1) xp2 = 0: (5.21)
éta''up|?
2y 2px = 0; (5.22)
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dp-O{,òa(5.22)?\a(5.21)?p,?
y2 2xy = 0; (5.23)
)
y = 0; y = 2x:
§?t??§). d?n14N′y, y = 2x=??§?).
ˉ￠t, y = 2x?′è?-?x
2cxy = (1 + cx2)2
??,X?3-5¤?.
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?4?(Clairaut)?§
y = xp + f(p); p = dydx; (5.24)
ù￥, f(p)′''up?Y???ê.
Figure:?)y = 2x
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é?§(5.24)?1|),üà''ux?O|?
p = xdpdx + p + f0(p)dpdx;
z{ dp
dx(x + f
0(p)) = 0:
edpdx = 0,Kp c,?\?§(5.24)￥,
y = cx + f(c); (5.25)
=??§(5.24)?),ù￥c???~ê.
ex + f0(p) = 0,(ü?§??ê/a??A)
( x + f0(p) = 0;
y = xp + f(p): (5.26)
T)T?l?)(5.25)L?è?-?￥|??c-O{|)L§?
?.2y,T)(??)(5.25)??.?d,?4??§?)???
?x,?Iò?§￥p?±??~êc=?.
A)(5.26)=??4
??§?).
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Example
|)~3.8.
):-p = dydx,K?§?z??§(5.10),n
y = xp 14p2;
T?§??4??§. u′dt????ù?)?
y = cx 14c2:
dA)(5.26)? 8
>>>>
<>
>>>:
x 12p = 0;
y = xp 14p2:
(5.27)
?p,y = x2=??§?).
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