!"#$%&' &()"*+,-*+. /-0*'"12 3"2&1 4%&-12.
56/ 7&#%-'8
!" #$%&'() (*+ ,-*%&'() .$)/&)0(/-1* .)10'$2,!
! "#$% %&'"$(! )& )$** '(!%$+&, -(.!+/,0 */0&, /!+ 123 "#&(,0 4(, (-"/$!$!5
/%067"("$' %(*."$(!% "( +$8&,&!"$/* &9./"$(!% )#(%& #$5#&%" +&,$:/"$:&% /,& 6.*;
"$7*$&+ -0 / %6/** 7/,/6&"&,
ǫ.
1& )$** <!+ "#/" "#& %(*."$(!% '#/!5& ,/7$+*0 $!
ǫ → 0= > !"#$%&' ()'*$'+&*!," 7,(-*&6 $% '#/,/'"&,$%&+ -0 "#&
4/'" "#/" "#& ǫ = 0 7,(-*&6 #/% 9.$"& +$8&,&!" %(*."$(! 7,(7&,"$&% /% '(67/,&+
"( "#& 0 < ǫ << 1 7,(-*&6= ! / ')#$%&' ()'*$'+&*!," 7,(-*&6 /% ǫ → 0 "#&
%(*."$(! "&!+% "( "#& %(*."$(! 4(, ǫ = 0. ?#$% $% -&%" $**.%",/"&+ -0 *((@$!5 /"
"#$! ,&5$(!% /%
%(6& %$67*& &A/67*&%=
-.&/(%)
B(!%$+&,
y ′′ + 2ǫy ′ − y = 0,
/!+
0 < ǫ << 1.
)#&,&
m1 = −ǫ +
ǫ→0
y(1) = 1
CD=EF
?#& 5&!&,/* %(*."$(! $%
y(x, ǫ) =
>%
y(0) = 0,
√
e m1 x − e m2 x
,
e m1 − e m2
1 + ǫ2 ,
m2 = −ǫ −
)& #/:&
y(x) →
√
1 + ǫ2 .
sinh(x)
,
sinh(1)
/!+ &:&,0"#$!5 %&&6% (@=
1& '/! /*%( (-"/$! / %(*."$(! /% 4(**()%G 1,$"&
y = Y0 + ǫY1 + ǫ2 Y2 + . . . .
H.-%"$"."$(! $!"( "#& &9./"$(! CD=EF /!+ &9./"$!5 '(&I'&!"% (4 *$@& 7()&,% (4
ǫ
"( J&,( 5$:&%
Y0′′ − Y0 = 0, Y0 (0) = 0, Y0 (1) = 1
Y1′′ − Y1 = −2Y0′ , Y1 (0) = 0, Y1 (1) = 0.
H(*:$!5 CD=KF 5$:&%
Y0 =
sinh(x)
,
sinh(1)
Y1 = (1 − x)
sinh(x)
.
sinh(1)
>5/$! "#&,& /,& !( 7,(-*&6% ; )& #/:& / ,&5.*/, 7&,".,-/"$(! 7,(-*&6=
-.&/(%)
LM
CD=KF
!"#$%&'
ǫy ′′ + 2y ′ − y = 0,
y(0) = 0,
()*+,
y(1) = 1,
-!' 0 < ǫ << 1. ./& #!012$!" $# 3# 4&-!'&
y(x, ǫ) =
5/&'& "!5
√
1
m1 = (−1 + 1 + ǫ),
ǫ
6# ǫ → 0 5& /37&
y∼
√
1
m2 = (−1 − 1 + ǫ).
ǫ
2
m2 ∼ − .
ǫ
1
m1 → ,
2
8!2& 2/32 3# ǫ → 0
1
1
2
(e − e
2x
x
− 2ǫ
)
1
2x
x
(e 2 − e− ǫ ) ∼ e− 2 (e 2 − e− ǫ ).
0&3'09 2/&'& 3'& 25! %$#2$":2 '&;$!"#<
• xǫ = O(1), 3"% 2/&"
• x >> ǫ
e m1 x − e m2 x
,
e m1 − e m2
1
2x
x
y ∼ e− 2 (e 2 − e− ǫ ).
3"% 2/&"
1
x
y ∼ e− 2 e 2 .
./& 3"3092$: #!012$!" -!' %$=&'&"2 7301&# !- ǫ $# #/!5" $" >$;* )*?* 8!2& 2/32
1
e
x−1
2
0.8
0.6
1.0
0.01
0.4
0.1
0.2
0
0
0.2
0.4
0.6
0.8
>$;1'& )< @!012$!" y(x, ǫ) -!' %$=&'&"2 7301&# !- ǫ*
AB
1
!" #$%& '$( )!*(+"# ,*-'.%/ '(
ǫ → 0.
#'(+&%*, %'3' *#
x = O(ǫ). 0" !*1" *( "2*3-%"
x = O(ǫ) '# )*%%". * !"#$%&' (%')&*
!" ,"+'$(
5!" ,"+'$(
$4 *
6&--$#" 7" ,/ #$%1'(+ !" "8&* '$( *# 9"4$,": ;&
y = Y0 + ǫY1 + . . . .
5!'# +'1"# *4 ", #&9# ' & '$( '( $ <,"4"8=3* )!>?
2Y0′ − Y0 = 0,
2Y1′ − Y1 = Y0′′ ,
<@:A?
*(. 9$&(.*,/ )$(.' '$(#
Y0 (0) = 0,
Y0 (1) = 1,
Y1 (0) = 0,
Y1 (1) = 0,
" ): B$7 !"," '# * -,$9%"3C 5!" $,.", $4 !" "8&* '$(# <@:A? '# ,".&)".D '" 7" ($7
!*1" E,#
$,.", "8&* '$(# 4$,
!"
Yi :
F$(#"8&"( %/ 7!')! 9$&(.*,/ )$(.' '$(#
x = 1:
.$ 7" )!$$#"G 5!" "2*) #$%& '$( #&++"# # 7" )*( #* '#4/ !" )$(.' '$( *
H" &# )$( '(&" 7' ! !" 9$&(.*,/ )$(.' '$( *
x = 1:
6$%& '$( $4 E,# $,.", -,$9%"3
2Y0′ − Y0 = 0,
+'1"#
Y0 = e
F%"*,%/
!'# #$%& '$( '# ($
#* '#E".: 0!"(
x
Y0
x
.
#'()"
!" )$(.' '$( *
#*/ 7!","
1
x
Y0 ∼ e− 2 (1 + ) = O(1).
2
n > 0 '# $ 9" 4$&(.: 5!"
1*,'*9%"
'((", 1*,'*9%" *(. '#
!" -##)& &).-!# $4 !')I("##
"8&* '$( <@:>? '(
'#
O(1) '(
",3# $4 X
ǫ1−2n
L$,
n>0
x = 0
'# ($
#$%& '$( '# !" %"*.'(+ $,.", !"+)& ,!("+-!#: B$7 7!"(
'# #3*%% 7" !*1"
x = ǫn X
x−1
2
'# #3*%% !" #$%& '$( 4*'%# *(. 7" ("". $ "2*3'(" !'# ,"+'$(
'( 3$," ." *'%: 5!"
;&
1*%'. 4$, *%%
Y0 (1) = 1,
!" .$3'(*(
n
O(ǫ ):
X
'# )*%%".
1 − 2n = −n
M 8&')I )$(#'# "()/ )!")I #!$7# !*
!"
5!" .'J","( '*%
dy
d2 y
+ 2ǫ−n
− y = 0.
2
dX
dX
",3# *," !" E,#
x
<@:K?
7$ ",3# *(. !"#" 9*%*()" '4
=⇒ n = 1.
!'# '# $ID <$ !", )!$')"# 4$,
*," ($ ?: N( !" '((", ,"+'$( '4 7" -&
y = y0 (X) + ǫα y1 (X) + . . .
AO
n
"+
n = 1/2
!"# α > 0 $%& '()'"!"("* !%"+ ,-./0 , !"# n = 10 * 1%& "#$" "#* 2*$&!%3 +4&*4
54+)2*6 !'
d 2 y0
dy0
+2
= 0,
2
dX
dX
y0 (X = 0) = 0.
$%& +%* )+(%&$47 8+%&!"!+% !'
9#* +"#*4 8+%&!"!+% 6('" 8+6* :4+6 !"#$%&' !"# "#* +("*4 '+2("!+% "$;!%3
X 2$43*. <+2=!%3 7!*2&'
y0 (X) = A + Be−2X
$%& y0(0) = 0 !652!*' "#$" A = −B . 9#('
y0 (X) = A(1 − e−X ).
9+ +)"$!% "#* 8+%'"$%" A * 6$"8# "#* !%%*4 '+2("!+% >('" &*4!=*& !"# "#* +("*4
'+2("!+%.
y0 (X) ∼ A :+4 X >> 1,
$%&
Y0 (x) ∼ e−
:+4 x → 0.
9#!' 3!=*' A = e− $%&
1
2
1
2
1
y0 (X) = e− 2 (1 − e−2X ).
!""#$% &' (#$) ? "*46 !%%*4 $%& ? "*46 +("*4 *@5$%'!+%'.
'!*+$)
x = O(1),
y = Y0 (x) + ǫY1 (x) + . . . ,
1
x
Y0 (x) = e− 2 e 2 .
,--+$)
x = ǫX,
y = y0 (X) + ǫα y1 (X) + . . . ,
1
y0 (X) = e− 2 (1 − e−2X ).
9#*'* $4* "#* )$'!8' +: )+(%&$47 2$7*4 "#*+47 $%& 6$"8#*& $'765"+"!8 *@5$%A
'!+%'. 9#* '+2("!+% 8$% )* 8+%"!%(*& "+ #!3#*4 +4&*4. B+"!8* "#$" "#* +("*4
'+2("!+% *@5$%&*& :+4 '6$22 x 3!=*'
1
y ∼ e− 2 (1 +
C#*% 4!""*% !% "*46' +:
54+8**& $'
x
+ . . . ) + ǫY1 (x) + . . . .
2
x = ǫX
"#!' '(33*'"' "#$" "#* !%%*4 '+2("!+% '#+(2&
y = y0 + ǫy1 + . . . .
C* #$& $''(6*& "#$" "#* +("*4 *@5$%'!+% 54+8**&*& !% 5+ *4' +: ǫ )(" "#!'
&+*' %+" #$=* "+ )* "#* 8$'*. D%* %**&' "+ 54+8**& +% $ "*46 )7 "*46 )$'!'
6$"8#!%3 "#* !%%*4 $%& +("*4 '+2("!+%' '7'"*6$"!8$227 $%& "#!' !22 !%:+46 #+
E-
!" #$$ %&'#( ")*+ ,"!#-". /" 0%(( 1&' %'2" & !" '"3 &)$") 4&) ,& ! !" %''")
#'$ &2 ") +&(2 %&'+. 5&0 4&) !" &2 ") +&(2 %&'
y = Y0 + ǫY1 + . . . ,
#'$ !" 6)&,("* 4&)
Y1
%+
1 x−1
2Y1′ − Y1 = Y0′′ = e 2 ,
4
Y1 (1) = 0.
7&(-%'8 8%-"+
(x − 1) x−1
e 2 .
8
x = ǫX #'$
Y1 =
9&) !" %''") 6)&,("*: 0" !#-"
y = y0 (X) + ǫy1 (X) + . . . .
;!" 6)&,("* 4&)
y1
%+
1
dy1
d 2 y1
+2
= y0 = e− 2 (1 − e−2X ),
2
dX
dX
y1 (X = 0) = 0.
<=.>?
;!" +&(2 %&' &4 <=.>? 8%-"+
0!")" 0" !#-"
+ #'
1
1
y1 = A(1 − e−2X ) + X(1 + e−2X )e− 2 ,
2
%'1&)6&)# "$ !" ,&2'$#)@ 1&'$% %&' #'$ A
%+ #' #),% )#)@ 1&'A
& ," $" ")*%'"$ 4)&* *# 1!%'8 0% ! !" &2 ") +&(2 %&'. ;!" &2 ") +&(2 %&'
"36#'$"$ 4&) +*#((
x
8%-"+
x−1
1
+ ǫ (x − 1)e 2 + . . . ,
8
1 (x − 1)
1
x
x
(1 + + . . . )).
∼ e− 2 (1 + + . . . ) + ǫe− 2 (
2
8
2
youter = e
/)%
x−1
2
"' %' ")*+ &4 %''") -#)%#,("+ !%+ %+
1
1
youter ∼ e− 2 + ǫe− 2 (
X
1
− ) + ....
2
8
;!" 0& ")* %''") +&(2 %&' %+
#+
1
1
1
yinn = e− 2 (1 − e−2X ) + ǫ[A(1 − e−2X ) + X(e−2X + 1)e− 2 ] + . . . ,
2
1
1
1
∼ e− 2 + ǫ(A + Xe− 2 ) + . . . ,
2
X → ∞. ;!%+ !#+ & *# 1! 0% ! !" 0& ")* &2 ") +&(2 %&' 0)% "' %'
<=.B?
")*+
&4 %''") -#)%#,("+: %.".:
1
1
youter ∼ e− 2 + ǫe− 2 (
C *# 1! %+ &'(@ 6&++%,(" %4
1
A = − 81 e− 2 .
X
1
− ) + ....
2
8
;!2+
1 1
X 1
y1 = − e− 2 (1 − e−2X ) + e− 2 (1 + e−2X ).
8
2
DE
<=.=?
!" #$%&'() *++(',%)*-%'$.
!"#$%&' ())&%*#'(+#%" +% +,- .%/!+#%" 0(/#1 #" +,- 2,%/- &-3#%" #. 1-4"-1 56
yunif = Youter + yinn − ymatch
2,-&- ymatch #. +,- ())&%*#'(+#%" +% y(x) #" +,- '(+7,#"3 &-3#%"8
9%& +,- (5%0- )&%5/-' 2- ,(1
1
Youter = e
− 12
x
e− 2
e +ǫ
(x − 1)e− 2 + O(ǫ2 ).
8
x
2
1
2x
1 −1
e− 2 x
− 2x
2
ǫ
yinn = e (1 − e ) + ǫ[− e (1 − e ) +
(1 + e− ǫ )] + . . . .
8
2 ǫ
'(+7,#"3 &-3#%" #. X(= x/ǫ) >> 1 ("1 x << 1; #-;
− 2x
ǫ
− 12
:,-
ǫ << x << 1.
:,!. ( %"-<+-&' !"#$%&' ())&%*#'(+#%" #.
1
1
x
2x
1
yunif = e− 2 e 2 + e− 2 (1 − e− ǫ ) − e− 2
#-
1
2x
x
yunif = e− 2 [e 2 − e− ǫ ].
+2% +-&' !"#$%&' ())&%*#'(+#%" #.
1
yunif = e
− 12
x
e− 2
e +ǫ
(x − 1)e− 2
8
x
2
1
+e
− 21
(1 − e
#-
− 2x
ǫ
2x
2x
1 1
e− 2 x
) + ǫ[− e− 2 (1 − e− ǫ ) +
(1 + e− ǫ )]
8
2 ǫ
1
1
x
1
−[e− 2 + ǫ(− + )e− 2 ].
8 2ǫ
1
yunif = e
− 21
x
2
(e − e
− 2x
ǫ
1
x
x 2x
e− 2
e− 2 − 2x
(x − 1)e− 2 +
e ǫ ).
+ e− ǫ ) + ǫ(
2
8
8
!/ 0'(1 '$ )*-23%$4 *$5 %$-1()15%*-1 6*(%*781.
=" +,- )&-0#%!. -*(')/- 2- 7%".+&!7+-1 (" %!+-& .%/!+#%" 2#+, x 4*-1 ("1 ǫ
+-"1#"3 +% >-&%; ("1 (" #""-& -*)(".#%" 2#+, X = x/ǫ 4*-1 ("1 ǫ 3%#"3 +%
>-&%8 ?&()#7(//6 +,- )&%7-.. '(6 5- &-)&-.-"+-1 (. #" 438 @ 2#+, +,- &-3#%"
&-)&-.-"+#"3 +,- %!+-& .%/!+#%" ("1 &-3#%" A +,- #""-& .%/!+#%"8 :,- 43!&(/.% .,%2. (" %0-&/() &-3#%" 2,-&- +,- +2% .%/!+#%". (3&--8 B%2-0-& 7/%.-&
-*('#"(+#%" %$ +,- 43!&- '#3,+ .!33-.+ +,(+ +,-&- #. ( )%..#5#/#+6 %$ ( &-3#%"
CD
ǫ
ǫ0
B
A
C
x
!"#$% &' (#)%$ *+,#)!+- $%.$%*%-)%/ 01 $%"!+*+,#)!+- 01 ! 2!)3 ǫ → 0 2!)3 X = x/ǫ 45%/8
2!)3 ǫ → 0 x 45%/6 7-/ !--%$
9 -+) 7::%**!0,% 01 )3% !--%$ +$ +#)%$ *+,#)!+-*8 ;- $%7,!)1 )3% 7:)#7, /+<7!-* +=
>7,!/!)1 += )3% )2+ *+,#)!+-* <71 0% ,7$"%$ )37- )3% 70+>% ,!<!)!-" .$+:%** 7,,+2*8
?3% /!@:#,)1 3%$% !* 7$!*%* =$+< )3% 271 )3% <7):3!-" !* /+-%8
A /!B%$%-) 271 )+ <7):3 )3% )2+ *+,#)!+-* !* )+ !-)$+/#:% 7- !-)%$<%/!7)%
>7$!70,%6 *71 x = ǫα ξ 2!)3 C!- )3% 70+>% %57<.,%D 0 < α < 18 E% 37>% X =
x/ǫ = ǫ−1+α ξ 7-/ *+ 7* ǫ → 0 2!)3 ξ 45%/ "!>%* X → ∞ 7-/ ǫ → 0 2!)3 ξ 45%/
7,*+ "!>%* x → 08 ?3#* ξ !* 7- !"#$%#& '"# >7$!70,% 7-/ !) !* !- )3!* >7$!70,%
)37) 2% 7))%<.) )+ <7):3 )3% !--%$ 7-/ +#)%$ *+,#)!+-*8 ?3% $%"!+- /%4-%/ 01
ξ = O(1) !* '! ()#$*'+ $#, (! =+$ )3% )2+ *+,#)!+-*6 7* *3+2- *:3%<7)!:7,,1 !- 4"8
FG8
E% 2!,, *3+2 3+2 )3!* 2+$H* 2!)3 7-+)3%$ %57<.,% !- 23!:3 )3% /!B%$%-)!7,
%I#7)!+- !* -+-,!-%7$8
"#$%&'( 9+-*!/%$
ǫy ′′ + y ′ + y 2 = 0,
y(0) = 0, y(1) = 1/2.
CJ8&D
K#..+*% 2% ,++H =+$ 7- +#)%$ *+,#)!+- += )3% =+$<
y = y0 + ǫy1 + . . . .
?3%- =$+< CJ8&D 2% +0)7!y0′ + y02 = 0,
y0′′ + y1′ + 2y0 y1 = 0.
LF
CJ8FGD
ǫ
ǫ0
x
!"#$% &'( )*%$+,- $%"!./ 012,3%34 5.$6!/" !/ 7%$81 .9 !/7%$8%3!,7% *,$!,:+%1
x = ǫα ξ ,/3 X = ǫ−1+α ξ 5!72 0 < α < 1; ,/3 ǫ → 0+<
=2% 1.+#7!./ .9 72% .#7%$ -$.:+%8 12.51 72,7
−
y0′
= 1,
y02
,/3 1.
y0 =
1
= x + k,
y0
1
.
x+k
=2% :.#/3,$> +,>%$ .??#$1 ,7 x = 0 052>@4 ,/3 1. 5% /%%3 7. #1% 72% ?./3!7!./
y0 (1) = 1/2 "!*!/" k = 1, ,/3 1.
y0 =
A7 /%B7 .$3%$
1
.
x+1
y1′ + 2y0 y1 + y0′′ = 0,
y1 (1) = 0.
C#:17!7#7!/" 9.$ y0 = 1/(x + 1) "!*%1
y1′ +
−2
2
y1 =
.
x+1
(1 + x)3
D%/?%
((1 + x)2 y1 )′ = −
2
,
1+x
(1 + x)2 y1 + k1 = −2 log(x + 1).
EF
!!"#$%& '() *+%,$'$+% y1(1) = 0 &$-). k1 = −2 log 2 /%, '(0.
2
2 log( 1+x
)
y1 =
.
(1 + x)2
1+2 '() $%%)2 .+"0'$+% 3) %)), '+ .))4 / .+"0'$+% $% ')25. +6 /% $%%)2 -/2$/7")
./# x = ǫnX /%, .07.'$'0'$+% $% 89:;< .(+3. '(/' n = 1 6+2 / ,$.'$%&0$.(), "$5$':
=() $%%)2 .+"0'$+% 5/# 7) )>!/%,), /.
y = Y0 (X) + ǫY1 (X) + . . . .
6')2 .07.'$'0'$+% $%'+ 89:;< /%, 0.$%& x = ǫX 3) +7'/$%
Y0′′ + Y0′ = 0,
Y1′′ + Y1′ + Y02 = 0.
=() 7+0%,/2# *+%,$'$+%. /2)
Y0 (0) = 0,
Y1 (0) = 0.
?+"-$%& 6+2 Y0 #$)",.
Y0 = A0 + B0 e−X ,
=(0.
/%,
A0 + B0 = 0.
Y0 = A0 (1 − e−X ).
=+ @%, A0 3) 5/'*( 3$'( $%')25),$/') -/2$/7"). /%, !0' x = ǫαξ, X = ǫ−1+αξ,
/%, 0 < α < 1 3$'( ξ = O(1): =() +%) ')25 +0')2 .+"0'$+% 32$'')% $% ')25. +6 ξ
$.
1
y = y0 (x) + · · · ∼
∼ 1 − ǫα ξ + . . . .
89:AA<
1 + ǫα ξ
?$5$"/2"# '() +0')2 .+"0'$+% $% ')25. +6 ξ $.
y = Y0 (X) + · · · ∼ A0 (1 − e−ǫ
−1+α ξ
) ∼ A0 .
=(0. 5/'*($%& 3$'( 89:AA< .(+3. '(/' A0 = 1 3$'( )22+2 O(ǫα):
B)6+2) 3) 5/'*( '+ .)*+%, +2,)2 3) %)), '+ @%, Y1 3($*( ./'$.@).
Y1′′ + Y1′ + Y02 = 0,
=(0.
Y1 (0) = 0.
Y1′′ + Y1′ = −(1 − e−X )2 .
?+"-$%& /%, /!!"#$%& '() *+%,$'$+% +% X = 0 &$-). 8*()*4<
1
Y1 (X) = A1 (1 − e−X ) + (1 − e−2X ) − X(1 + 2e−X ).
2
CD
!"# $! $%&#! #'! ()#!% *+, &++!% !"-*+.&(+. &+ #!%/. (0 #'! &+#!%/!,&*#!
1*%&*23!. *+, ,( #'! /*#4'&+56 7'! ()#!% !"-*+.&(+ $%&##!+ &+ #!%/. (0 ξ &.
yout =
=
1
2
1
2 log(
+ǫ
) + ...,
2
1+x
(1 + x)
1+x
1
1
2
+
ǫ
2
log(
) + ...,
1 + ǫα ξ
(1 + ǫα ξ)2
1 + ǫα ξ
∼ 1 − ǫα ξ + ǫ2α ξ 2 + · · · + 2 log 2(ǫ − 2ǫα+1 ξ + O(ǫ2α )) − 2ǫ(1 − 2ǫα ξ)(ǫα ξ − O(ǫ2α )).
896:;<
!"# #'! &++!% .(3)#&(+ $%&##!+ &+ #!%/. (0 ξ &.
1
yinn = (1 − e−X ) + ǫ(A1 (1 − e−X ) + (1 − e−2X ) − X(1 + 2e−X )) + . . . ,
2
1
−ǫα−1 ξ
−ǫα−1 ξ
−2ǫα−1 ξ
α−1
−ǫα−1 ξ
= (1−e
)+ǫ A1 (1 − e
) + (1 − e
) − ǫ ξ(1 + 2e
) +. . . ,
2
ǫ
896:=<
∼ 1 + ǫA1 + − ǫα ξ + . . .
2
>+ 896:;< &0 $! ?!!- #!%/. #( (%,!% ǫ *+, *..)/&+5 #'*# #'! #!%/. O(ǫ2α) *%!
./*33!% #'*+ #!%/. (0 O(ǫ) $! %!@)&%! 0 < α < 1/26 7'&. 5&1!.
896:A<
B(/-*%&+5 896:A< *+, 896:=< $! .!! #'*# #'! #!%/. (0 O(ǫα) /*#4' *)#(/*#&4*33C
*+, #( /*#4' #'! O(ǫ) #!%/. $! %!@)&%!
yout ∼ 1 − ǫα ξ + ǫ2 log 2 + O(ǫ2 , ǫ1+α , ǫ2α ).
ǫA1 +
5&1&+5
D#
ǫ
= 2ǫ log 2,
2
1
A1 = − + 2 log 2.
2
#'! +!"# (%,!% (0 /*#4'&+5 #'! #!%/. (0 O(ǫ2α)
7'! 4(/-(.&#! .(3)#&(+ #( O(ǫ2) &.
/*#4' *)#(/*#&4*33C6
ycomp = yout + yinn − ymatch .
>+ #'! *2(1! !"*/-3! $! E+,
ycomp =
2
2ǫ
1
− xǫ
log
+
+
(1
−
e
)+
x + 1 (x + 1)2
x+1
x
2x
x
1
x
1
ǫ (− + 2 log 2)(1 − e− ǫ ) + (1 − e− ǫ ) − (1 + 2e− ǫ )
2
2
ǫ
1 x
1
−(1 + ǫ(− + 2 log 2 + − )).
2
2
ǫ
896:F<
D 4(/-*%&.(+ (0 #'! +)/!%&4*3 .(3)#&(+ (0 896G< $&#' #'! 4(/-(.&#! .(3)#&(+
&. .'($+ &+ H&56 8::< *+, .'($. !"4!33!+# *5%!!/!+# 0(% ǫ ./*336
FA
0.6
0.5
0.4
u
0.3
0.2
0.1
x
0.1
0.2
0.3
0.4
O
0.5
1
!"
x
#"
!"#$% &&' ( )*+,-$!.*/ *0 12% /#+%$!)-3 .*3#1!*/ *0 45678 4.*3!9 3!/%.8 :!12
12% )*+,*.!1% .*3#1!*/ "!;%/ <= 456&>8 49-.2%9 3!/%8 1-?!/" 4-8 ǫ = 0.2@ -/9 4<8
ǫ = 0.1
!" #$%&'()' *+,&'A*/.!9%$
ǫy ′′ + a(x)y ′ + b(x)y = 0,
y(0) = A, y(1) = B.
456&B8
C2% *#1%$ ,$*<3%+ 4.%1 ǫ = 08 !. D#.1
a(x)y ′ + b(x)y = 0.
C-?% a(x) > 0 -/9 12%/
y′ = −
b(x)
y,
a(x)
y = Ce
−
Rx
b(s)
x0 a(s)
ds
.
("-!/ 12%$% -$% 1:* <*#/9-$= )*/9!1!*/. 1* .-1!.0= -/9 .* 12%$% +#.1 <% - <*#/9E
-$= 3-=%$@ <#1 :2%$% !. 12% <*#/9-$= 3*)-1%9F G#,,*.% 12-1 :% 2-;% - <*#/9-$=
3-=%$ -1 x = x̄ *0 12!)?/%.. γ(ǫ). H% :$!1%
:2%$%
x = x̄ + γ(ǫ)X,
X = O(1).
C2%/ .#<.1!1#1!/" !/1* 456&B8 :!12 y = Y "!;%.
a(x̄ + γX) dY
ǫ d2 Y
+
+ b(x̄ + γX)Y = 0.
2
2
γ dX
γ
dX
I*: %J,-/9 a, b -.
a(x̄ + γX) = a(x̄) + γXa′ (x̄) + . . . ,
1* "%1
b(x̄ + γX) = b(x̄) + γXb′ (x̄) + . . . ,
ǫ d2 Y
a(x̄) dY
+
+ b(x̄)Y + · · · = 0.
2
2
γ dX
γ dX
>>
456&K8
!" |γ| << 1 #$% &!'()*)# #%"'+ *"%
ǫ d2 Y
,
γ 2 dX 2
a dY
.
γ dX
!" * ,*-*).% /% "%01("%
1
ǫ
∼
2
γ
γ
=⇒ γ = O(ǫ).
2%).% +%# γ = ǫ (% x = x̄ + ǫX. "!' 345678 #$% "%&1.%& ())%" %01*#(!) (+
ǫ
−1
dY
d2 Y
+ a(x̄)
+ b(x̄)Y + · · · = 0.
dX 2
dX
345648
9$% -%*&(): !"&%" ())%" ;"!,-%' (+
d2 Y
dY
+ a(x̄)
= 0,
2
dX
dX
:(<():
Y = C0 + C1 e−a(x̄)X .
=!/ /% $*<% *++1'%& #$*# a(x̄) > 05 >? x̄ > 0 /% )%%& #! '*#.$ *+ /% :! !1# !?
#$% ,!1)&*"@ -*@%"A (% /% )%%& -('(#+ X → ±∞.
B+ X → ∞ %<%"@#$(): (+ !CA ,1# *+ X → −∞ (# +1::%+#+ #$*# C1 '1+# ,%
D%"! #! *<!(& %E;!)%)#(*- :"!/#$5
F1# C1 = 0 (';-(%+ )! ,!1)&*"@ -*@%"5 2%).% x̄ = 0 *)& #$% ,!1)&*"@ -*@%"
(+ *# x = 0 (? a(x) > 05
G('(-*"-@ (? a(x) < 0 #$%) /% $*<% * ,!1)&*@ -*@%" *# x = 1. >? a(x) = 0
()+(&% #$% "%:(!) /% $*<% *) ()#%")*- ,!1)&*"@ -*@%"5 9$% *,!<% *)*-@+(+ *-+!
,"%*C+ &!/)5
!" #$%&'(% )*+,-.(/0 12&(%13% .+4(%/
H!)+(&%"
ǫy ′′ + xy ′ − (ǫ2 x3 + 1)y = 0,
y(−1) = 1, y(1) = 2
*)& −1 ≤ x ≤ 1, 0 < ǫ << 1. 9$% *,!<% &(+.1++(!) +1::%+#+ *) ()#%"(!" -*@%"
*# x = 05
!" #$% !1#%" +!-1#(!) ;1#
y = y0 + ǫy1 + . . . ,
#! :%#
xy0′ − y0 = 0.
IJ
!"#
y0 = Ax.
$%&% '% !()% ( *%' +,-."/012 3!,.! 45"*+(&1 .5*+,0,5* +5 '% .!55#%6 3% .(*
#!5' 0!(0 0!%&% (&% *5 45"*+(&1 /(1%&# *%(& x = ±12 3% '&,0%
y = A± x
'!%&% 0!% + #0(*+# 75& x > 0 (*+ − 75& x < 02
8&59 0!% 45"*+(&1 .5*+,0,5*# ,0 #"::%#0# 0!(0
A+ = 2,
A− = −1.
3!%* x ,# #9(// 0!% ǫy′′ 0%&9 ,# *50 *%:/,:4/%; (*+ !%*.% '% /55< 75& (* ,*0%&,5&
/(1%& (0 x = 0 (*+ '&,0%
x = γ(ǫ)X,
γ(ǫ) << 1.
!,# :,)%# ',0! y = Y
ǫ d2 Y
γX dY
+
− Y + · · · = 0.
2
2
γ dX
γ dX
85& ( +59,*(*0 4(/(*.% 0!,# #"::%#0# 0!(0
ǫ
∼ O(1)
γ2
1
=⇒ γ = O(ǫ 2 ).
$%*.% #%0 x = ǫ X (*+ 7&59 0!% 5"0%& #5/"0,5* ,0 #"::%#0# 0!(0 '% %=>(*+ 0!%
,**%& #5/"0,5* (#
1
2
1
y = ǫ 2 Y0 + ǫY1 + . . . .
?"4#0,0"0,*: ,*05 0!% %@"(0,5* :,)%#
ǫǫ
−1
d 2 Y0
ǫ
+ ...
dX 2
1
2
1
2
+ ǫ Xǫ
− 21
dY0
ǫ
+ ...
dX
1
2
1
− ǫ 2 Y0 + · · · = 0.
$%*.% 0!% /%(+,*: 5&+%& >&54/%9 ,#
d 2 Y0
dY0
+X
− Y0 = 0.
2
dX
dX
AB2CDE
!% 45"*+(&1 .5*+,0,5*# #"::%#0 0!(0 '% 9"#0 9(0.! ',0! 0!% 5"0%& #5/"0,5* (#
!,# #"::%#0# 0!(0
Y0 ∼ A± X (# X → ±∞.
AB2FGE
!% %@"(0,5* AB2CDE .(* 4% #5/)%+ ,* 0%&9# 57 >(&(45/,. .1/,*+%& 7"*.0,5*#2 H7 '%
>"0
X → ±∞.
Y0 = e −
IJ
X2
4
W0
!"#
W0
$% &$'"$
() " !%
1
X2
W0′′ + ( − 2 −
)W0 = 0.
2
4
*) +&#"%,+- &#."/"#."# $)+0 &)#$ )1 !" "20% &)#
X2
1
)W = 0,
W +( +ν−
2
4
′′
%," !" /%,%3)+&4 4-+&#.", 10#4 &)#$
W = Dν (X)
%#.
D−ν−1 (iX).
Dν (x)
5# ),.", ) .) !" 6% 4!&#7 *" ,"20&," !" 3"!%8&)0,$ )1
:!" /,)/", &"$ )1
Dν (z)
1),
|x|
+%,7"9
%," $066%,&;". 3"+)* <$"" 1), "=%6/+" >3,%6)8& ; ?
@ "70# A
2
ν − z4
Dν (z) ∼ z e
∞
X
(−1)n an z −2n
n=0
2
ν − z4
Dν (z) ∼ z e
∞
X
n
(−1) an z
a0 = b0 = 1,
an =
−2n
n=0
z → ∞,
%$
F","
%$
z → ∞,
| arg(z)| <
3π
.
4
<B9CDE
1
∞
(2π) 2 iπν −ν−1 z2 X
4
e z
e
−
bn z −2n
Γ(−ν)
n=0
5π
π
< arg(z) <
.
4
4
<B9CCE
%#.
ν(ν − 1) . . . (ν − 2n + 1)
2n n!
bn =
(ν + 1)(ν + 2) . . . (ν + n)
.
2n n!
F"#4" *" 4%# *,& " !" $)+0 &)# )1 <B9DGE %$
Y0 = e −
X2
4
(CD−2 (X) + ED1 (iX)).
()* 0$&#7
D−2 (X) ∼ X −2 e−
X2
4
,
D1 (iX) ∼ (iX)e
%#.
2
D−2 (X) ∼ X
−2 − X4
e
X2
4
X→∞
1
(2π) 2 2πi X 2
−
e Xe 4 ,
Γ(2)
D1 (iX) ∼ (iX)e
%$
X2
4
%$
%$
X → −∞
X → −∞,
*" '#. !%
2
Y0 ∼ e
− X4
X2
C − X2
4 + E(iX)e 4
e
X2
X → ∞,
! "#$%&'()*+ %,- .! "! /*012, !"#$%%& %' (!)*+,!)-.!/ 01".)-%"
-.*" -#-+.-).$/
!
!"#$%& '($) #$%*+",
!
Y0 ∼ EiX
X → ∞.
"#
$!%&!
Ei = A+ .
' ( )"*)+
Y0 ∼ e −
X2
4
h
−C
i
p
X2
X2
(2π)Xe 4 + E(iX)e 4
X → −∞.
$!%&!
Y0 ∼ (−
, - %,
p
(2π)C + iE)X + O(1) X → −∞,
−
A±
.# %, /0! , -!% -")1!# 23*
p
(2π)C + iE = A− .
)!"4# /3
3
C=√ ,
2π
"%4 /0!
E = −2i,
%%!* #3)1/ 3% "#
Y0 =
X2
3
√ D−2 (X) − 2iD1 (iX) e− 4 .
2π
5 1% 23*( "66*37 ("/ 3% &"% 8! &")&1)"/!4 /3 , -!
yunif = ǫ
1
2
x2
3
ix
x
√ D−2 ( √ ) − 2iD1 ( √ ) e− 4ǫ .
ǫ
ǫ
2π
5 &3(6"* #3% 32 /0! 1% 23*( "66*37("/ 3%
yunif = ǫ
1
2
x2
ix
3
x
√ D−2 ( √ ) − 2iD1 ( √ ) e− 4ǫ .
ǫ
ǫ
2π
9 /0 " %1(!* &") #3)1/ 3% 32 /0! 4 :!*!%/ ") !;1"/ 3%
ǫy ′′ + xy ′ − (ǫ2 x3 + 1)y = 0,
"%4
−1 ≤ x ≤ 1,
0 < ǫ << 1,
23*
ǫ = 0.05
?@
y(−1) = 1, y(1) = 2
# #039%
% < ,= >=? 8!)39=
2.0
1.5
1.0
-1.0
0.5
-0.5
1.0
!"#$% &'( ) *+,-.$!/+0 +1 23% 4%5.*26 0#,%$!*.7 /+7#2!+0 2+ 23% 1#77 %8#.2!+0 ./
*+,-.$%9 :!23 23% #0!1+$, .--$+5!,.2!+0 49./3%9 7!0%6 1+$ ǫ = 0.05;
!"# $% &''()*+,&-+)./ 0123 4#-")5
<+#09.$= 7.=%$ 23%+$= 1.!7/ :3%0 :% 3.>% . $.-!9 >.$!.2!+0 !0 23% /+7#2!+0
23$+#"3+#2 23% $%"!+0 $.23%$ 23.0 7+*.77= .2 /+,% 7+*.2!+0;
!"#$%& ?+0/!9%$
ǫy ′′ + by = 0,
y(0) = 0,
y(1) = 1,
:3%$% b > .09 0 < ǫ << 1. @+2% 23.2 23% "%0%$.7 /+7#2!+0 !/
q
sin(x bǫ )
q .
y=
sin( bǫ )
A3% +#2%$ /+7#2!+0 !/ B#/2 y = 0; +$ 23% !00%$ /+7#2!+0C /#--+/% :% /%2
x = x̄ + γ(ǫ)X,
A3%0 23% %8#.2!+0/ "!>%/
γ << 1.
ǫ d2 y
+ by = 0.
γ 2 dX 2
) 9+,!0.02 D.7.0*% "!>%/ γ = ǫ .09 23% $%/#72!0" !00%$ -$+D7%, !/
1
2
d2 y
+ by = 0.
dX 2
A3% /+7#2!+0 "!>%/
√
√
y = A sin( bX) + B cos( bX).
E% *.0 *3++/% .0= x̄ D#2 0+2% 23.2 1+$ .0= *3+!*% +1 x̄ 23% /+7#2!+0 !/ 0+2 +1
D+#09.$= 7.=%$ 1+$, .09 *.00+2 D% ,.2*3%9 2+ 23% +#2%$ /+7#2!+0 ./ X → ±∞
D%*.#/% 23% !00%$ /+7#2!+0 +/*!77.2%/;
FG