Курс математики для технических высших учебных заведений. Часть 3. Дифференциальные уравнения. Уравнения математической физики. Теория оптимизации: Учебное пособие [Н. А. Берков] (pdf) читать онлайн
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Лауреат второго Всероссийского конкурса НМС по математике
Министерства образования и науки РФ «Лучшее учебное издание по математике
в номинации «Математика в технических вузах»
Н. А. БЕРКОВ, В. Г. ЗУБКОВ,
В. Б. МИНОСЦЕВ, Е. А. ПУШКАРЬ
КУРС МАТЕМАТИКИ
ДЛЯ ТЕХНИЧЕСКИХ
ВЫСШИХ УЧЕБНЫХ
ЗАВЕДЕНИЙ
Часть 3
Дифференциальные уравнения.
Уравнения математической физики.
Теория оптимизации
Под редакцией
В. Б. Миносцева, Е. А. Пушкаря
Издание второе, исправленное
ДОПУЩЕНО
НМС по математике Министерства образования и науки РФ
в качестве учебного пособия для студентов вузов, обучающихся
по инженерно&техническим специальностям
•САНКТ/ПЕТЕРБУРГ•МОСКВА•КРАСНОДАР•
•2013•
ББК 22.1я73
К 93
Берков Н. А., Зубков В. Г., Миносцев В. Б.,
Пушкарь Е. А.
К 93
Курс математики для технических высших учебных
заведений. Часть 3. Дифференциальные уравнения.
Уравнения математической физики. Теория оптимизации:
Учебное пособие / Под ред. В. Б. Миносцева,
Е. А. Пушкаря. — 2+е изд., испр. — СПб.: Издательство
«Лань», 2013. — 528 с.: ил. — (Учебники для вузов.
Специальная литература).
ISBN 9785811415601
Учебное пособие соответствует Государственному образовательному
стандарту. Пособие включает в себя лекции и практические занятия.
Третья часть пособия содержит 25 лекций и 25 практических занятий по
следующим разделам: «Обыкновенные дифференциальные уравнения»,
«Дифференциальные уравнения в частных производных», «Элементы
вариационного исчисления и теории оптимизации».
Пособие предназначено для студентов технических, физико+
математических и экономических направлений.
ББК 22.1я73
Рецензенты:
À. Â. ÑÅÒÓÕÀ äîêòîð ôèçèêî-ìàòåìàòè÷åñêèõ íàóê, ïðîôåññîð,
÷ëåí ÍÌÑ ïî ìàòåìàòèêå Ìèíèñòåðñòâà îáðàçîâàíèÿ è íàóêè ÐÔ;
À. À. ÏÓÍÒÓÑ ïðîôåññîð ôàêóëüòåòà ïðèêëàäíîé ìàòåìàòèêè
è ôèçèêè ÌÀÈ, ÷ëåí ÍÌÑ ïî ìàòåìàòèêå Ìèíèñòåðñòâà îáðàçîâàíèÿ
è íàóêè ÐÔ; À. Â. ÍÀÓÌΠäîêòîð ôèçèêî-ìàòåìàòè÷åñêèõ íàóê,
äîöåíò êàôåäðû òåîðèè âåðîÿòíîñòåé ÌÀÈ; À. Á. ÁÓÄÀÊ äîöåíò,
çàì. ïðåäñåäàòåëÿ îòäåëåíèÿ ó÷åáíèêîâ è ó÷åáíûõ ïîñîáèé ÍÌÑ
ïî ìàòåìàòèêå Ìèíèñòåðñòâà îáðàçîâàíèÿ è íàóêè ÐÔ;
Ó. Ã. ÏÈÐÓÌΠïðîôåññîð, çàâ. êàôåäðîé âû÷èñëèòåëüíîé
ìàòåìàòèêè è ïðîãðàììèðîâàíèÿ ÌÀÈ (Òåõíè÷åñêèé óíèâåðñèòåò),
÷ëåí-êîððåñïîíäåíò ÐÀÍ, çàñëóæåííûé äåÿòåëü íàóêè ÐÔ.
Обложка
Е. А. ВЛАСОВА
Охраняется законом РФ об авторском праве.
Воспроизведение всей книги или любой ее части запрещается без письменного
разрешения издателя.
Любые попытки нарушения закона
будут преследоваться в судебном порядке.
© Издательство «Лань», 2013
© Коллектив авторов, 2013
© Издательство «Лань»,
художественное оформление, 2013
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. z(x) C , ' ' /01203245 6
% , "
7 - , + (' '
y = z(x)e− a(x)dx .
8"
9 (: . z(x) :
% : 8" $" 9 '
. 8")
y = Ce−
−
y = z(x) e
a(x)dx
−
+ z(x)e
a(x)dx
.
a(x)dx
− a(x)dx =
;"
0 F2y = −λ
dt
/ " "" .
) ) / " 0)
y = 0 ! $ "
/
1 **#
"
ma =
F.
022 3
a
F
! F1 F2 " ! Oy # $ !
% &&' Oy " $
d2 y
a Oy 2 " ((
dt
$
m
m
d2 y
dy
= −ky − λ
dt2
dt
d2 y
dy
+ λ + ky = 0.
dt2
dt
&&)
" "
* &&)
! +(($ !
, " " ! !
- . " Oy
F (t) ( $ t
&&) "
d2 y
dy
m 2 + λ + ky = F (t)
&&&
dt
dt
!
/ % &&& m %
λ
k
F (t)
= 2b,
= ω2,
= f (t),
m
m
m
" ! 0 ! %
( 1
dy
d2 y
+ 2b + ω 2 y = f (t).
&&2
dt2
dt
* &&2
! ! (( $
!
" " ! +(($
/
! ! +
3 # " ! (b = 0)
(f (t) ≡ 0) + &&2 "
d2 y
+ ω 2 y = 0.
&&4
2
dt
k2 + ω2 = 0 k1,2 = ±ωi !
!
Y (t) = C1 cos ωt + C2 sin ωt.
"
# $ C1 C2
$ N > 0 ϕ % &
C1 C2 !
C1 = N sin ϕ,
C2 = N cos ϕ.
' N % ϕ () C1
C2 %*
N = C12 + C22 ,
+ (
C1
tg ϕ =
C1
·
C2
C2 " &
Y (t) = N sin ϕ cos ωt + N cos ϕ sin ωt = N sin(ωt + ϕ).
,& !
( $
Y (t) = N sin(ωt + ϕ).
- % & !
( & ) +
T = 2π
& N . & ϕ . $ %
ω
& ω
/ +$ $ (b = 0)&
( f (t) ≡ 0& ! ) # 0
1
d2 y
dy
+ 2b + ω 2 y = 0.
dt2
dt
k 2 + 2bk + ω 2 = 0
√
2
2
k1,2 = −b ± b − ω
23
4
& b < ω # 0
√ )
* k1,2 = −b + ω i& ω = ω2 − b2 5 !
23
Y (t) = e−bt (C1 cos ω
t + C2 sin ω
t) = N e−bt sin(
ωt + ϕ),
C1 = N sin ϕ, C2 = N cos ϕ
N e−bt → 0 t → +∞
y
0
t
b > ω
" #
Y (t) = C1 e(−b+
√
$$ %
b−ω 2 )t
+ C2 e(−b−
√
b−ω 2 )t
! "
.
& '
( )
(
Y (t) → 0 t → +∞ # ( (
b = ω
Y (t) = C1 e−bt + C2 te−bt .
* + ! " )
(b = 0)
f (t) = a sin μt ,
( $$-
d2 y
+ ω 2 y = a sin μt.
dt2
. '
ȳ(t)
Y (t) .
$$
! )
$$
.
$$/
y + ω 2 y = 0.
Y (t) = N sin(ωt + ϕ).
! μ " #
" ω $ " " " % μi
" & " "' k2 + ω2 = 0!
ȳ "
ȳ(t) = A sin μt + B cos μt.
(() ȳ *! ȳ (t) = −μ2(A sin μt + B cos μt)
" * ȳ(t) ȳ(t)
"%(() A B +
−μ2 (A sin μt + B cos μt) + ω 2 A sin μt + ω 2 B cos μt = a sin μt,
A = ω2 −a μ2 , B = 0 $ " ,!
"
ȳ(t) =
a
sin μt,
ω 2 − μ2
%'
-
, #
.,
/ ," " " *a ω!
, ω2 − μ2 ," " / " ω2 − μ2
," , 0 ω = μ , ( - ,
$ " " " % μi = ωi " & " "'
k2 + ω2 = 0!
"
a
sin μt + N sin(ωt + ϕ).
ω 2 − μ2
- ! μ
y(t) = ȳ(t) + Y (t) =
0
ȳ(t) = (A sin μt + B cos μt)t.
ȳ(t)
ȳ (t) = 2(μA cos μt − μB sin μt) + μt(−A sin μt − B cos μt)
! μ = ω!
"%(() A B +
2μA cos μt − 2μB sin +μ2 t(−A sin μt − B cos μt)+
+μ2 t(A sin μt + B cos μt) = a sin μt,
A = 0, B = −
ȳ = −
a
2ω
ȳ(t)
at
cos ωt,
2ω
y(t) = Y (t) + ȳ(t) = N sin(ωt + ϕ) −
at
cos ωt.
2ω
t
! " #
$ % &'
at
cos ωt ) & "
'( −
2ω
*" +
" !, $ $
$ ! $ !
y
0
t
- $! ''( +! %
" ! ! #
$ (
R L C
U = U (t) ! ! "#$!
L
b
c
R
C
U(t)
a
d
%
& U ' !
( )*+
, -
I = I(t)!
Uab + Ubc + Ucd = U (t).
. +
Uab = RI(t)
Ubc = L
&
/$
dI
1
, Ucd =
dt
C
t
I(t)dt.
0
dI(t)
1 t
+
I(t)dt = U (t).
dt
C 0
t
RI(t) + L
0++
2
R
dI
dI
1
+ L 2 + I = U (t)
dt
dt
C
d2 I R dI
I
+
= U (t).
+
dt2
L dt LC
I
! U " #
$% U = 0 &
$
I +
R
I
I +
= 0.
L
LC
'' (
2R = 1, 8 H = 2, 45
2r = 6
√
k 2gh
g = 10 2
! h
k " "#$
# % &'
) * + $ $ t ,
$ $ h(t) ,
" (-% ΔV = Sh = Sv Δt , . $!
Δt S , !
S = r2 πv(t) v = v(t) , √
! "v = 0, 6 2gh%
R
H
h(t)
r
v
Δt
ΔV = πR2 Δh ! " ΔV # #
$Δh < 0% #
# & & Δh Δt
πR2 Δh = −πr2 · 0, 6 2ghΔt.
π & "
# "" ' (
Δt → 0
R2 dh = −0, 6r2
2gh dt.
)
# *
" ' h = h(t)(
√
r2
2 h = −0, 6 2
R
t
2gt + C.
+ C # h = H
h " ' # C (
t = 0 +
√
C = 2 H,
" ' h
(
√
√
t
g r2
t
2 R2
$,, -%
& #
h = h(t)(
h−
h=
H = −0, 6
√
H − 0, 6
g r2
t
2 R2
2
.
. & tk #
& * * $,, -% t
# " h = 0 + #
R2
tk =
0, 6 r2
√
2 √
H− h
g
$,, /%
0 # $,, /%
# # ( R = 0, 9 H = 2, 45 r = 0, 03
g = 9, 8 122
# (
0, 92
tk =
0, 6(0, 03)2
2
9, 8
.
h=0
2, 45 ≈ 1006 ≈ 16, 7
.
! " # $ %
& ' ( # $ (
v = v(t)
dv
t v(0) = 2 m = F (t)
dt
F (t) !" m
# $ " % $$&
F (t) = −kv(t) k > 0 '((& $$&
# # $ $ )" ")*
!" " +, -
' , % $$ "% $, % $' ((*
& )"
m
dv
= −kv.
dt
. " ! " $% "
$%, " $ #
m
dv
= −kdt
v
=⇒
m ln |v| = −kt + ln |C|
-!
k
v = Ce− m t .
#
v(0) = 2 $' C = 2
k
v(t) = 2e− m t .
/
( # # 0 %
k
−4
k
: 1 = 2e m
v(4) = 1 ) $ #
m
k
ln 2
=
m
4
t
- V (t) = 21− 4 1" T # *
2 3 4 , "
T
T
0, 25 = 21− 4
2−2 = 21− 4 −2 = 1 −
t
s(t) =
t
0
!"
#
T = 12
t
8
1 − 2− 4 .
ln 2
t
0
21− 4 dt =
v(t) dt =
T
4
8
ln 2
11, 5
! !
" # $
# %! " &
'( ) g !
$2*
$$%
&
'
(
)
'
+
,
#
v(t)
Fdrag
t
*
*
'
'
-
(
Fdrag = kv 2 (t),
k
* .+
,
+
'
+
'
(
ma = mg − Fdrag ,
a
*
/
'
-
dv
dt
'
dv
= mg − kv 2 .
dt
k
= η )
m
1 dv
1
= g − v2.
η dt
η
m
a =
v
+
t
dv
1
= dt;
−
η v2 − 1 g
η
dv
1
·
t=−
η
v 2 − η1 g
! v(t)
v=
√
g 1 − e−2 ηgt
√
·
η 1 + e−2 ηgt
" # η$
%
&' ( $ ) t → +∞ v(t) !
&' ( $ "
t → +∞ * η
lim v = lim
t→+∞
50 =
t→+∞
g
,
η
√
g 1 − e−2 ηgt
√
=
η 1 + e−2 ηgt
η
1
=
,
g
2500
η=
g
,
η
g
·
2500
η v(t)
g
1 − e− 25 t
v = 50
g ,
1 + e− 25
v = 50
e0,2t − e−0,2t
1 − e−0,4t
≡
50
·
1 + e−0,4t
e0,2t + e−0,2t
+
,
%* *
ds
e0,2t − e−0,2t
= 50 0,2t
,
dt
e + e−0,2t
s = s(t)
t
s=
50
0
e0,2t − e−0,2t
dt =
e0,2t + e−0,2t
= 250 ln(0, 5(e0,2t + e−0,2t ))|t0 = 250 ln(0, 5(e0,2t + e−0,2t )).
0,2t
e
et + e−t ≈ et t
!
250 ln (0, 5(e0,2t + e−0,2t )) = 1000;
+e
= 2e4 ⇒ e0,2t ≈ 2e4 ⇒ 0, 2t ≈ 4 + ln 2;
t ≈ 20 + 5 · 0, 7 = 23, 5 .
−0,2t
""#$
%
#
& !
'
(
Ox# )
! #
AM ! * MO + # ,-#
y
K1
M(x,y)
A
K
N
0
B
x
y=f(x)
y = f (x) x
y = f (x) KK1 MN
! AM Ox ∠OKM =
= ∠AMK1 = ∠OMK = α " OKM # $
% O |OM| = |OK| $
OMB |OM| = x2 + y2 & '
|OK| !
Y − y = y (X − x).
Y = 0 & Ox'
X = x − yy
|X| = |OK| = −X = −x +
OM
y
,
y
x2 + y 2 = −x +
OK
y
·
y
(( )
y =
y
x+
x2 + y 2
,
%
* % (( ) %
y $
! y = f x +
y =
y
x+
x2
+
y2
≡
1+
y
y/x
.
≡f
2
x
1 + (y/x)
&,, -'
. ) $
&,, -' x +
x+
dx
=
x =
dy
x2 + y 2
x
= +
y
y
2
x
x
.
+1=f
y
y
&,, /'
0 &,, /' t = xy
x = ty x = ty +t
t y + t = t +
√
t2 + 1 =⇒ t y =
√
t2 + 1.
√
dt
dy
√
=⇒ ln |y| = ln |t + t2 + 1| + ln C
=
2
y
t +1
√
y = C(t + 1 + t2 )
x
! ! t = y "
⎛
⎞
2
x ⎠
x
y =C⎝ + 1+
.
y
y
#
2
y
=x+
C
x
x2 + y 2 .
4
2
y
2y x
+ x2 = x2 + y 2
−
C2
C
$# $
y2
y2
2x
−
2
C
C
C
= y 2 =⇒ y 2 = 2Cx + C 2 =⇒ y 2 = 2C x +
.
2
$" !
C " % − C2 , 0 " & #
' " " $
! ! " & $
(()
◦ ◦
◦ !
" #◦$ %& ' ( "
) "*
((( +, " ( '- ,.
/ 0 )
'-1 / 2"(
3 ( , . " 3 !
g = 10
R = 6
H = 10 !" # $ %
% "
& ' # (
#
%
√
) 0, 6 2gh " g = 10 2 *
%+ h * % ,
(" "
- %
) " ,
(
+ . " % ./
" " 0 % "
" h 1 " /
% R ! + ,
2 "
- " ,
% v +" % +
p % ,
+ ρ - % +
μ
!
- ,
% )
% ,
% % 3 +
) 3 v - ) % % r ) " %
+
3 ,
% ) ) -
#) %
3 ) )
4 ,
)
! 2a
3
% ) ) %
,
3 %
) - ,
%
!
" #
$
# ! #
% & !
" #
" ! #
' !
% ( #
# %
) % % * (
( # * %
+ k # s1, s2, . . . , sk &
$ n = s1 + s2 + . . . + sk
# # % (* #
! !
,-. n
⎧
dy1
⎪
⎪
⎪ dx = f1 (x, y1 , . . . , yn ),
⎪
⎪
⎪
⎨ dy2
= f2 (x, y1 , . . . , yn ),
/,-.0
dx
⎪
⎪
.....................
⎪
⎪
⎪
⎪
⎩ dyn = fn (x, y1 , . . . , yn ).
dx
1 n
/,-.0
y1 (x), y2(x), . . . , yn (x),
! "
# $ %
y1 (x), . . . , yn (x) $
y1 (x0 ) = y10 , . . . , yn (x0 ) = yn0 ,
x0 , y10 , . . . , yn0 ! $
"
yi = yi (x), i = 1, . . . , n,
#
$ $ (x, y1 , . . . , yn ) %
& & ! ' ( !
yi (x0 ) = y10 , i = 1, . . . , n, ' ( !
#
$ !
!% (x0 , y10 , y20 , . . . , yn0 ).
) ! %
% ( % %%
** + ( (
%, !% %
$ $ , %
' yi = yi (x, C1, C2, . . . , Cm ),
!
" ! ) C1, . . . , Cm,
" ) $ * ( ) " ! +
, $ m = n.
. ' Φi(x, y1, . . . , yn) = 0,
i = 1, . . . , n( & ( )
& +
' Φi(x, y1, . . . , yn, C1, . . . , Cm ) = 0( &
i = 1, . . . , n( ! & (
C1, . . . , Cm " ) $ * * & * *
#
i = 1, . . . , n(
G, (
m
⎧
d2 x
dx dy dz
⎪
⎪
,
m
t,
x,
y,
z,
,
,
=
F
⎪
x
⎪
⎪
dt2
dt dt dt
⎪
⎪
⎨ d2 y
dx dy dz
m 2 = Fy t, x, y, z, , ,
,
⎪
dt
dt dt dt
⎪
⎪
⎪
⎪
d2 z
dx dy dz
⎪
⎪
.
⎩ m 2 = Fz t, x, y, z, , ,
dt
dt dt dt
!"#
$ x(t), y(t), z(t) % v
a
dy
dz
dx
i + j + k,
dt
dt
dt
!&#
d2 x
d2 y
d2 z
i
+
j
+
k.
dt2
dt2
dt2
v=
a=
$
F = F x i + F y j + F z k
!
"
#
% "" &
'!&
#& "" &
#& & " )
*
m
(
x = x(t), y = y(t), z = z(t)
%
dx
dy
dz
, v(t) = , w(t) =
" u(t) =
dt
dt
dt
d2 y
dv d2 z
dw
,
=
=
dt2
dt dt2
dt
$ (
d2 x
du
,
=
2
dt
dt
% +
)
⎧
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩
du
1
= Fx (t, x, y, z, u, v, w),
dt
m
dv
1
= Fy (t, x, y, z, u, v, w),
dt
m
dw
1
= Fz (t, x, y, z, u, v, w),
dt
m
dx
= u(t),
dt
dy
= v(t),
dt
dz
= w(t).
dt
!
"
fi (x, y1, . . . , yn) i = 1, 2, . . . , n
∂f
, i, j = 1, 2, . . . , n
∂y
x, y1 , . . . , yn ! G ! ! "#
$ x0, y10, y20, . . . , yn0 % ! G % #
& y1 = y1 (x) y2 = y2 (x), . . . ,
yn = yn (x) $'% ( " $)
i
j
y1 (x0 ) = y10 , y2 (x0 ) = y20 , . . . , yn (x0 ) = yn0 .
#
$
%
$
&
' $
+
"
* $
,
"
"
$ , (
n
n (
+
)
$ *
( *
$ &
"
(
"
$ &
"
!!
⎧
dx
⎪
⎪
= y 2 + sin t,
⎨
dt
dy
x
⎪
⎪
⎩ = ·
dt
2y
d2 x
dy
= 2y + cos t.
" #$
dt2
dt
dy
= x, &! " #$
% ! 2y
dt
!' (
d2 x
− x = cos t.
dt2
" )$
* + ,
! ,&
! -+! , ,
+ ./ λ2 − 1 = 0 0 ,
λ1,2 = ±1 &! / !
X(t) = C1 et + C2 e−t 1 /!
!! +2 ,&
x̄ = A cos t + B sin t
& ' ' +
" )$ ! A = − 12 , B = 0 , +!
x̄ = − 21 cos t
3, ! ! / " )$ !!
x = C1 et + C2 e−t −
1
cos t.
2
% ! !
y2 =
dx
1
− sin t = C1 et − C2 e−t − sin t.
dt
2
3, ! ! !! !
x = C1 et + C2 e−t − 12 cos t,
y=±
C1 et − C2 e−t − 12 sin t.
⎧
dx
⎪
= z − y,
⎪
⎪
⎪
dt
⎪
⎨
dy
= z,
⎪
dt
⎪
⎪
⎪
⎪
⎩ dz = z − x.
dt
d2 x
dz dy
− ·
=
dt2
dt
dt
! "
d2 x
+ x = 0.
dt2
# $$ !
$ " $#
% $&
$ λ2 + 1 = 0 ' $ ! λ1,2 = ±i
(
x = C1 cos t + C2 sin t.
)* + "
dz
− z = −x
dt
dz
− z = −C1 cos t − C2 sin t.
dt
( (
dz
− z = 0 +
dt
z = C3 et .
, ( &
-" $#
z = A cos t + B sin t.
$#
A B !
B − A = −C1 ,
−B − A = −C2 .
C2 − C1
C1 + C2
, B=
.
2
2
1
1
z = (C1 + C2 ) cos t + (C2 − C1 ) sin t + C3 et .
2
2
A=
1
dx
1
= (C1 − C2 ) cos t + (C1 + C2 ) sin t + C3 et .
y=z−
dt
2
2
⎧
⎪
x = C1 cos t + C2 sin t,
⎪
⎪
⎪
⎪
⎨
1
1
y = (C1 − C2 ) cos t + (C1 + C2 ) sin t + C3 et ,
2
2
⎪
⎪
⎪
⎪
1
1
⎪
⎩z = (C1 + C2 ) cos t + (C2 − C1 ) sin t + C3 et .
2
2
! "
#$
# ! "
% &
% ' (
n #
% ) $
! #* %
+,%-
⎧
dx
y
⎪
⎨
=− ,
dt
t
⎪
⎩ dy = − x ,
dt
t
t > 0.
. ! $ $ "
d
1
(x + y) = − (x + y),
dt
t
#*
d(x + y)
dt
=− ,
x+y
t
x+y =
C1
·
t
d
1
(x − y) = (x − y),
dt
t
x − y = C2 t.
⎧
⎨
C1
,
t
⎩x − y = C t
2
#
x = x(t)
x+y =
y = y(t) !"
⎧
1 C1
⎪
⎪
x
=
+
C
t
,
2
⎨
2 t
1 C1
⎪
⎪
⎩ y=
− C2 t .
2 t
$% &
⎧
dx
⎪
⎨
= x2 y,
dt
⎪
⎩ dy = y − xy 2 .
dt
t
' ( # ) ! y
* x
y
dx
dy
xy
+x
=
dt
dt
t
d
xy
(xy) =
·
dt
t
+,
xy = C1 t.
-$% ./0
xy
dx
= C1 tx
C1 t
dt
t2
x = C2 eC1 2 C2 = 0
!"#$%
y=
C1 −C1 t2
C1 t
2 .
=
te
x
C2
&' ( ( )
⎧
2
⎨x = C2 eC1 t2 ,
t2
⎩y = C1 te−C1 2 .
C2
* x = 0
y = 0 x = C
y = Ct+
, - n .
. /- ' ../0
dyi
=
aij yi + fi (x),
dx
j=1
n
i = 1, . . . , n,
!"##%
aij = 12345 6 ' ' ../
.'/ fi (x) !"##%
fi(x) ≡ 0,
7
- ) yi (x) ≡ 0.
,
!"##% 0
⎧
dy1
⎪
⎪
= a11 y1 + a12 y2 + . . . + a1n yn ,
⎪
⎪
dx
⎪
⎪
⎪
⎪
⎨ dy2 = a21 y1 + a22 y2 + . . . + a2n yn ,
dx
⎪ .................................
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩ dyn = an1 y1 + an2 y2 + . . . + ann yn .
dx
!"#8%
A n
dy
y
!
dx
⎛
a11
⎜ a21
A=⎜
⎝ ...
an1
a12
a22
...
an2
#
⎞
. . . a1n
. . . a2n ⎟
⎟,
... ... ⎠
. . . ann
⎛
⎛
⎞
⎜
⎜
dy ⎜
=⎜
dx ⎜
⎜
⎝
y1
⎜ y2 ⎟
⎟
y = ⎜
⎝ """ ⎠ ,
yn
dy1
dx
dy2
dx
""
"
$%&"''(
+
⎟
⎟
⎟
⎟.
⎟
⎟
⎠
dyn
dx
!
dy
= Ay.
dx
*
⎞
$%&"')(
n
$%&"',(
y1 = α1 eλx , y2 = α2 eλx , . . . , yn = αn eλx ,
- αk -
λ "
αk λ
yk = λαk eλx , k = 1, . . . , n
$%&"'.(" /
α1 , α2 , . . . , αn !
yk
$%&"',(
-
⎧
(a11 − λ)α1 + a12 α2 + . . . + a1n αn = 0,
⎪
⎪
⎪
⎪
⎨ a12 α1 + (a22 − λ)α2 + . . . + a2n αn = 0,
⎪
....................................
⎪
⎪
⎪
⎩
an1 α1 + an2 α2 + . . . + (ann − λ)αn = 0.
$%&"'0(
1 $%&"'0( +
2 " "
α1 , α2 , . . . , αn
(a11 − λ)
a12
a21
(a22 − λ)
...
...
an1
an2
...
a1n
...
a2n
...
...
. . . (ann − λ)
= 0.
$%&"'%(
n λ
! " #
n λ ! $ %
& # n ' λ1 & λ2 & . . . & λn & (
) A *
! + '
|A − λE| = 0,
E , ) n & -!
.
(A − λE) · b = 0,
b = (α1, α2 , . . . , αn) , ) A
) A b& (/
( Ab = λb!
+ &
!& ) A&
& + (
& % #
0 & ( λk
(α1k , α2k , . . . , αnk )& k = 1, 2, . . . , n * # 11
) n .
'
2 . y1 (x)& (/ ( λ = λ1& n
1) '
y11 = α11 eλ1 x , y21 = α21 eλ1 x , . . . , yn1 = αn1 eλ1 x ;
. y2 (x)& (/ ( λ = λ2& n
1) '
y12 = α12 eλ2 x , y22 = α22 eλ2 x , . . . , yn2 = αn2 eλ2 x ;
................................................................................................
n2 . yn (x)& (/ ( λ = λn & n
1) '
y1n = α1n eλn x , y2n = α2n eλn x , . . . , ynn = αnn eλn x .
y1(x), y2 (x), . . . , yn(x)
y = C1 y1 (x) + C2 y2 (x) + . . . + Cn yn (x)
⎧
y = C1 y11 (x) + C2 y12 (x) + . . . + Cn y1n (x),
⎪
⎪
⎨ 1
y2 = C1 y21 (x) + C2 y22 (x) + . . . + Cn y2n (x),
.................................
⎪
⎪
⎩ y = C y (x) + C y (x) + . . . + C y (x),
n
1 n1
2 n2
n nn
!" #
$ % yij (x)
⎧
y = C1 α11 eλ1 x + C2 α12 eλ2 x + . . . + Cn α1n eλ1 x ,
⎪
⎪
⎨ 1
y2 = C1 α21 eλ1 x + C2 α22 eλ2 x + . . . + Cn α2n eλ2 x ,
..........................................
⎪
⎪
⎩ y = C α eλ1 x + C α eλn x + . . . + C α eλn x .
n
1 n1
2 n2
n nn
!"&#
' (
) !" #
# *
) ) * +
) ) n) + %
! % #!
!,
⎧
dy
⎪
⎪
= −y − 2z,
⎨
dx
dz
⎪
⎪
= 3y + 4z.
⎩
dx
- . / + % a11 = −1, a12 = −2,
% y1 = y,
0 (
!" #.
a21 = 3, a22 = 4*
(−1 − λ)
−2
3
(4 − λ)
y2 = z !
= 0 ⇒ λ2 − 3λ + 2 = 0.
1 λ1 = 1, λ2 = 3!
2 y = α1 eλx, z = α2eλx *
λ1 = 1! 3 !",#* α1 , α2*
(−1 − λ)α1 − 2α2 = 0,
3α1 + (4 − λ)α2 = 0,
λ = λ1 = 1
!
!
"
#
α1 + α2 = 0,
α1 , α2 $
α1 = 1 α2 = −1 $ %
λ = 1
&
y1 (x) = ex ,
z1 (x) = −ex .
' !
"
λ = λ2 = 2
α1 = 2, α2 = −3 "
(
%
)
&
y2 (x) = 2e2x ,
z2 (x) = −3e2x .
* + , &
&
y = C1 y1 (x) + C2 y2 (x) = C1 ex + 2C2 e2x ,
z = C1 z1 (x) + C2 z2 (x) = −C1 ex − 3C2 e2x .
' ! ( # # & # (
( "
. %,, .
# ) ( ( "
( # #(
⎧
dx
⎪
⎪
= y,
⎨
dt
dy
⎪
⎪
= −x.
⎩
dt
d2 x
dy
=
dt2
dt
dy
dt
d2 x
+ x = 0.
dt2
λ2 + 1 = 0 λ1,2 = ±i
! x = C1 sin t + C2 cos t"
!
# $%&"'(
y = C1 cos t − C2 sin t" )* $%&"'(
x = C1 sin t + C2 cos t,
y = C1 cos t − C2 sin t.
%&"+
⎧
dx
⎪
⎪
= y,
⎪
⎪
⎪
⎨ dt
dy
= x,
⎪
dt
⎪
⎪
⎪
⎪ dz
⎩
= x + y + z.
dt
,
dy
d2 x
=
dt2
dt
-
d2 x
− x = 0.
dt2
)
λ2 − 1 = 0 *
λ1,2 = ±1 !
x = C1 et + C2 e−t .
dx
= C1 et − C2 e−t .
dt
x(t) y(t)
z
y=
dz
− z = 2C1 tet .
dt
z = C3 et + 2C1 tet .
!"#
⎧
⎨ x = C1 et + C2 e−t ,
y = C1 et − C2 e−t ,
⎩
z = C3 et + 2C1 tet .
$%&'()
%&'*
⎧
dx
x
⎪
⎪
=
,
⎨
dt
2x + 3y
dy
y
⎪
⎪
=
·
⎩
dt
2x + 3y
+
,
, " , +
dx
x
dx
dy
= ⇒
=
⇒ ln |x| = ln |y| + ln |C1 | ⇒ x = C1 y.
dy
y
x
y
-
' , ,
" , #. +
2
dy
2x
3y
dx
dy
dx
+3 =
+
⇔2 +3
=1⇔
dt
dt
2x + 3y 2x + 3y
dt
dt
⇔ d(2x + 3y) = dt ⇒ 2x + 3y = t + C2 .
x = C1 y,
2x + 3y = t + C2 ,
⎧
C1 (t + C2 )
⎪
⎪
,
⎨ x=
2C1 + 3
⎪
⎪ y = t + C2 ·
⎩
2C1 + 3
⎧
dx
⎪
⎪
= y − z,
⎪
⎪
⎪
⎨ dt
dy
= x + y + t,
⎪
dt
⎪
⎪
⎪
⎪ dz
⎩
= x + z + t.
dt
! "#$ # $ %
#
d
(y − z) = y − z,
dt
& $
ln |y − z| = t + ln |C1 |,
& $ #
y − z = C1 et .
' #( ) $% & $
% $
x = C1 et + C2 .
* +
*# $ y!
x = x(t)
dy
= y + C2 + C1 et + t.
dt
y = (C1 t + C3 )et − t − 1 − C2 .
x y
z=y−
dx
= (C1 t + C3 − C1 )et − 1 − C2 .
dt
!"#
⎧
x = C1 et + C,
⎪
⎪
⎨
y = (C1 t + C3 )et − t − 1 − C2 ,
⎪
⎪
⎩
z = (C1 t + C3 − C1 )et − 1 − C2 .
$%&'
⎧
dy
⎪
⎨
= 5y − z,
dx
⎪
⎩ dz = y + 3z.
dx
( ) "#
* +$%&,,-
a11 = 5, a12 = −1, a21 = 1, a22 = 3, "
y1 = y, y2 = z & . +$%&,$
(5 − λ)
−1
1
(1 − λ)
= 0 ⇒ λ2 − 8λ + 16 = 0 ⇒ (λ − 4)2 = 0
/ λ1,2 = 4 0&
" /
&
* λ1,2 = 4
y1 ≡ y(x) = e4x (A1 x + A2 ),
y2 ≡ z(x) = e4x (B1 x + B2 ).
1 y(x)
dy
= A1 e4x + 4(A1 x + A2 )e4x ,
dx
z(x)
dz
= B1 e4x + 4(B1 t + B2 )e4x ,
dx
dy
dz
,
, y
z
dx dx
e4x = 0
A1 + 4(A1 x + A2 ) ≡ 5(A1 x + A2 ) − (B1 x + B2 ),
B1 + 4(B1 x + B2 ) ≡ A1 x + A2 + 3(B1 x + B2 ).
x
A 1 , A 2 , B1 , B2
4A1 = 5A1 − B1 ,
4B1 = A1 + 3B1 ,
A1 + 4A2 = 5A2 − B2 ,
B1 + 4B2 = A2 + 3B2 .
A1 = B1
A2 − B2 = A1 ! A1 = C1 A2 = C2 B1 = C1 , B2 =
= A2 − A1 = C2 − C1 "# $ %
⎧
⎨ y = e4x (C1 x + C2 ),
⎩ z = e4x (C1 x + C2 − C1 ).
&'&
⎧
dy
⎪
⎪
= 2y − z,
⎨
dx
dz
⎪
⎪
= y + 2z.
⎩
dx
(%
) $
*&'++,
a11 = 2, a12 = −1, a21 = 1, a22 = 2, $
y1 = y, y2 = z - *&'+&,
(2 − λ)
−1
1
(2 − λ)
= 0 ⇒ λ2 − 4λ + 5 = 0
λ1 = 2 + i, λ2 = 2 − i
. %
y = α1 e(2+i)x ,
$
z = α2 e(2+i)x ,
λ1 = 2 + i
α1 , α2
!
(2 − λ)α1 − α2 = 0,
α1 + (2 − λ)α2 = 0,
" #" ! λ = λ1 = 2 + i #$ %& $ & %
# "$$! $ "' "
−iα1 − α2 = 0,
# " %$ α1 , α2 ! # $! ($
α1 = 1 #$% α2 = −i ( % λ = λ1 =
= 2 + i % )
y(x) = e(2+i)x ≡ e2x (cos x + i sin x),
z(x) = −ie(2+i)x ≡ e2x (sin x − i cos x).
* " "$'' % #$% +
) % # +" + ) + $
' $ ,, '$'' )
#$% " $ + + % + ) '
y1 (x) = e2x cos x,
z1 (x) = e2x sin x,
y2 (x) = e2x sin x,
z2 (x) = −e2x cos x.
* ) " $ ' ' #$%
+ + % + ) # $!
y = e2x (C1 cos x + C2 sin x),
z = e2x (C1 sin x − C2 cos x).
⎧
dx
⎪
⎪
= x + y,
⎨
dt
dy
⎪
⎪
= 3y − 2x.
⎩
dt
⎧
dx
⎪
⎪
= y + 1,
⎨
dt
dy
⎪
⎪
= 2et − x.
⎩
dt
⎧
dx
⎪
⎪
= y + z,
⎪
⎪
⎪
⎨ dt
dy
= x + z,
⎪
dt
⎪
⎪
⎪
⎪ dz
⎩
= x + y.
dt
⎧
dx
⎪
⎨
=
dt
⎪
⎩ dy =
dt
! "#
$
x
,
x+y
y
.
x+y
$ %! &
⎧
dx
⎪
⎪
= y − z,
⎪
⎪
⎪
⎨ dt
dy
= x2 + y,
⎪
dt
⎪
⎪
⎪
⎪
⎩ dz = x2 + z.
dt
! "# &
$ %! ' & & $
(! ' x $
⎧
dx
⎪
⎪
= 7x + 3y,
⎨
dt
dy
⎪
⎪
= 6x + 4y.
⎩
dt
⎧
dx
⎪
⎪
= x − y,
⎨
dt
dy
⎪
⎪
= x + 3y.
⎩
dt
⎧
dx
⎪
⎨
= x − 2y,
dt
dy
⎪
⎩
= x − y.
dt
⎧
dx
⎪
⎨
= x − 3y,
dt
⎪
⎩ dy = 3x + y.
dt
⎧
dx
⎪
⎨
= 2x + y,
dt
⎪
⎩ dy = 4y − x.
dt
y − y = e2x cos x
! "
#$ % & '!! ( ) * ! ""&!
"* +)
, - , . ( /!
*) ( !!
. & *)
"& " &
0!
* )!&1 ) 2 )!&31 "
)!
4)!& !" *$ )!& "! 5
)! !1 * ! *& 1 !!
- "* $ &)! $ !&
+ & )!&
"* & ! 6 !" * !
! * $#! " 1
1 ' "
7 ! 8 ! ) ! " )
" )!& " * 9 &
" ! "& & ! " )
)!& 6 & & !! 1
!" #
y =
y−x
y+x
1
y
ln(y 2 + x2 ) + arctg = C.
2
x
$!" #%
$!" &%
' ( y(x)
(
$!" &%
)) ( $!" #% *
+
,
- $!" #%
)
$ &.% / (r, ϕ)
$!" &% r = Ceϕ
,
(
) (
,
0
1 +
(
/ ,
2 y(x)
un (x) n → +∞ , un (x) +
) ( 3
n ,
y(x) *
,
!!" # $
% #
!!" &
%
y(x) % '
Dh : {xn }
"
# ( % )
)
% * % $
+
% , '
%
(
% # % -
% %
% .'
%
'
% '
%
y = y − x,
y(0) = 1.
0 x 100,
y = 1 + x + Cex .
y(0) = 1 C = 0 !
y(100) = 101"
y(0) = 1,000001 # C = 10−6
y(100) = 2,7 · 1037 "" "
# $$ %
& !
# " '($$ % & (
&
) * % +,
$ *+,"+-.." / # !
$$ % &
* ' .
F (x, y, y , y , . . . , y (n−1) , y (n) ) = 0,
*01"-.
*01"2.
3 & & $$ %
(n−1)
y(x0 ) = y0 , y (x0 ) = y0 , y (x0 ) = y0 , . . . , y (n−1) (x0 ) = y0
(n−1)
F (x0 , y0 , y0 , y , . . . , y0
(n)
, y0 ) = 0,
n!
x = x0
.
*01"4.
5 $$ % *01"-.
x, y, y, y, . . . , y(n) (n + 1)!
y(n+1)
F1 (x, y, y , y , . . . , y (n) , y (n+1) ) = 0.
*01"0.
(n)
(n−1)
y0 = f (x0 , y0 , y0 , y0 , . . . , y0
).
(n)
y0
(n)
(n+1)
F1 (x0 , y0 , y0 , y0 , . . . , y0 , y0
) = 0,
! n + 1"
(n+1)
x = x0 # y (n+1) (x0 ) = y0
! $ % & ! "
& x = x0
)
'& y = y(x) (( %
! * #
y(x) = y(x0 )+y (x0 )
x − x0
(x − x0 )2
(x − x0 )n
+y (x0 )
+. . .+y (n) (x0 )
+. . .
1!
2!
n!
+
+ x0
& , "
,!- & (( %"
#
n
x − x0
(x − x0 )2
(n) (x − x0 )
+ y0
+ . . . + y0
+ ...,
1!
2!
n!
"
-
.
y(x) = y0 + y0
⎧
⎨ y = 2xy + 4y,
y(0) = 0,
⎩
y (0) = 1.
/
' & # 0 (( %
(( % "
1 $ !
x0 2
3 / y = 2xy + 4y ⇒ y0 = 2 · 0 · 1 + 4 · 0 = 0
4(( %
/
y = 2y + 2xy + 4y = 6y + 2xy ⇒ y0 = 6 · 1 + 2 · 0 · 0 = 6.
56
4(( %
56
IV
y = 6y + 2y + 2xy = 8y + 2xy ⇒ y0IV = 8 · 0 + 2 · 0 · 6 = 0.
55
V
y = 8y + 2y + 2xy
IV
= 10y + 2xy IV ⇒ y0V = 10 · 6 + 2 · 0 · 0 = 60.
y0V I = 0, y0V II = 840
! " " x0 = 0 #$
% " & '
y=
!
6
1
60
840 7
x + x3 + x5 +
x + ...
1!
3!
5!
7!
y =x+
x3 x5 x7
+
+
+...
1!
2!
3!
( )* * ! % " "
+) , C0, C1, C2, . . . , Cn, . . .'
+∞
Cn (x − x0 )n .
n=0
- +) , ) Cn(n = 0, 1, 2, . . .) .) ! $
) + " " * ! "$
"" , " ")* * .$
(x − x0) " " " * " "
/ * % " " $
+)* , " /0 "
" #)"
1
y + y + y = 0.
x
1
2 x = 0 " 3 . !
x = 0 0 "" "
4% . ! 0!/ )* "'
y = c0 + c1 x + c2 x2 + c3 x3 + . . . + cn xn + . . .
, /0 ")$
#" )*'
y = c1 + 2c2 x + 3c3 x2 + . . . + ncn xn−1 + . . .
y = 2c2 + 2 · 3c3 x + 3 · 4c4 x2 + . . . + (n − 1)ncn xn−2 + . . .
(2c2 + 2 · 3c3 x + 3 · 4c4 x2 + . . . + (n − 1)ncn xn−2 + . . .)+
1
+ (c1 + 2c2 x + 3c3 x2 + . . . + ncn xn−1 + . . .)+
x
+(c0 + c1 x + c2 x2 + c3 x3 + . . . + cn xn + . . .) = 0.
x! " #
$ c0, c1, c2 , . . .
%"& " '( x1 ) &
1
c1 ! c1 = 0! $ " "
*+
x
! " , '! c1 = y x=0= 0
- y - x = 0 - 3
. , "& x, x ! x5
2 · 3c3 + 3c3 + c1 = 0,
4 · 5c5 + 5c5 + c3 = 0,
6 · 7c7 + 5c7 + c5 = 0.
/ c1 = 0! "&
"
c3 = 0,
c5 = 0,
...,
c2k+1 = 0.
$ + "&
0 ( x0, x2, x4, . . .
2 · c2 + 2c2 + c0 = 0,
3 · 4c4 + 4c4 + c2 = 0,
5 · 6c6 + 4c6 + c4 = 0.
% , +
(n + 2)(n + 1)cn+2 + (n + 2)cn+2 + cn = 0
. -
c2 = −
c2
c0
c4
c0
= 2 2 ; c6 = − 2 = − 2 2 2 ;
2
4
24
6
246
c0
c0
n
n
= (−1) = 2 2
= (−1) n
·
2 4 . . . (2n)2
2 (n!)2
c0
;
22
c2n
(n + 2)2 cn+2 + cn = 0.
" - c0
c2 = −
x2
x4
x6
y(x) = c0 1 − 2 + 4
− 6
+...
2
2
2 (2!)
2 (3!)2
+∞
x2n
x2n
. . . + (−1)n 2n
+
.
.
.
=
c
(−1)n 2n
·
0
2
2 (n!)
2 (n!)2
n=0
! c0 = 1 " # $% y1(x) = J0(x)
& '
( "
) %* "+ &
y1 (0) = 1, y1 (0) = 0
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$
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+
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y := 1 Y := rkf ixed(y, 0, 1, 5, f )
f (x, y) :=
2
0 0.2
0.4
0.6
0.8
1
Y =
1 1.124 1.326 1.698 2.581 6.857
Y :=Rkadapt(y, 0, 1, 5, f )
0 0.2
0.4
0.6
0.8
1
Y =
1 1.124 1.326 1.698 2.582 7.097
5 %
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2
= 7, 09929356
% 012 64 y(1) =
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⎪
⎪
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n
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⎩ dyn = fn (x, y1 , y2 , · · · , yn ).
dx
y1 (x0 ) = y01 , y2 (x0 ) = y02 , · · · , y1 (x0 ) = y0n .
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x ∂x y ∂y
y
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º
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% ) u = arcsin(x + ay);
∂u
∂u
=a .
∂y
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2
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% * u = et sin x; ∂u
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% u = cos(xy) + y2; uxx + uy + y2u = y4.
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56,
m
n
m
h
(n+1,m-1)
(n+1,m)
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m
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m
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n
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m
n
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λ
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0
0
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1
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b
1+ a r
1
1
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b
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r ! " #
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(b/a)r > 1 τ > (a/b)h ()+
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i
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τ
h
a
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ha
3 τ /h) = r "& λ "#
|λ| =
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a2
(
r ! " "#
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(τ /h) (a/b) ' h → 0 "# (
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2
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1
+
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*,
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h
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.
b
b
1 − r + reiωh
a
a
5 ! (
|λ| =
b
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a
1
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b
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a
2
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(b/a)r 1 τ (a/b)h 6 8 . (
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a
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2π
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n
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a0
+
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2
n=1
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f (x) =
! 0#
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. ) ,
1
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2
1
cos x cos y = (cos(x − y) + cos(x + y)) ;
2
1
sin x cos y = (sin(x − y) + sin(x + y)) ;
2
1
1
sin2 x = (1 − cos 2x); cos2 x = (1 + cos 2x);
2
2
1
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2
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3 ! 2#. &*
4 n k 5 n = k. &
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π
π
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sin nx sin kx dx = 0,
−π
π
−π
π
π
−π
−π
−π
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sin nx dx = 0,
cos nx dx = 0,
π
n=k
π
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sin kx cos kx dx = 0.
−π
$
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π
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* #
−π
π
⎞
⎛
π
π
π
π
+∞
a0
⎝an cos nx dx + bn sin nx dx⎠ .
dx +
f (x) dx =
2
n=1
−π
−π
−π
+ $
a0 =
1 π
f (x) dx.
π −π
,
k∈N
!-
−π
π a0
dx = a0 π
−π 2
!- !
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cos kx
*
.
π
a0 π
f (x) cos kx dx =
cos kx dx+
2 −π
−π
π
+∞
π
+
an cos nx cos kx dx + bn sin nx cos kx dx .
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n=1
−π
/*
! !
! * *
*
*
ak
π
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n=k
% $ ! 1#
−π
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sin kx
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π
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[−π, π] '$$ ( ," - $ )
1 π
1 π
f (x) dx, an =
f (x) cos nx dx,
π −π
π −π
1 π
f (x) sin nx dx.
bn =
π −π
a0 =
.
, . ," !
" / # # "
! ," 0 !
$ &
1 f (x)
[a, b]
f (x)
[a, b] !
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* " $ 0 ! -+-
# $( f (x) ,"
( $ f (x) $
2π ) *
)+
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f (x) S(x) )
x0
x → x0
S(x0 ) =
1
lim f (x) + lim f (x) .
x→x0 +
2 x→x0−
x ∈ (−π, π]
! f (x) = x
"!
# $ # %
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&
*
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1 π
1 x2 π
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π
1 π
1 π
1
x sin nx −
an =
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xd sin nx =
π −π
πn −π
nπ
−π
π
π
1
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cos nx
=
πn2
−π
1
cos nπ − cos(−nπ) = 0;
=
πn2 π
π
1
1 π
1
bn =
x cos nx −
x sin nx dx = −
xd cos nx = −
π −π
πn
nπ
−π
−π
π
π
1
1
=
− cos nx dx = −
π cos nπ + π cos(−nπ) − sin nx
πn
n
−π
−π
n
2
2(−1)
= − cos nπ = −
.
n
n
a0 =
0 ' ,# '!
1 &
a0 = a1 = · · · = an = · · · = 0; bn =
2(−1)n−1
.
n
(−1)n−1
sin x sin 2x sin 3x
−
+
− ··· +
sin nx + . . .
f1 (x) = 2
1
2
3
n
.
f1 (x) = 2
+∞
(−1)n−1 sin nx
n
n=1
.
! " ϕ(x)
ϕ(−x) = ϕ(x)
π
ϕ(x) dx =
−π
π
0
π
π
π
ϕ(x) dx + ϕ(x) dx = ϕ(−x) dx + ϕ(x) dx =
−π
0
0
π
π
= ϕ(x) dx + ϕ(x) dx = 2 ϕ(x) dx.
0
0
0
#$%!&'(
ϕ(−x) = −ϕ(x)! +
) *
π
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0
π
π
π
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−π
0
0
π
π
= − ϕ(x) dx + ϕ(x) dx = 0.
0
0
0
0
#$%!&&(
, -.
f (x)! / *
a0 ak bk 0
! +
a0 = 0,
1*
an = 0,
2
bn =
π
π
f (x) sin nx dx.
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0
f (x) =
) *
+∞
n=1
bn sin nx.
#$%!&2(
2
a0 =
π
π
2
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π
f (x)dx,
0
π
f (x) cos nxdx,
bn = 0.
0
a0
+
an cos nx.
2
n=1
+∞
f (x) =
& $ "
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!
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$
($
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y1 = y + c, x1 = x
a0
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2
n=1
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* +
a0 =
#& & $,$
2
π
π
f (x)dx,
0
, ,
f
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π
z = x
l
2π
x ∈ [−π, π]3
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2
π
π
f (x) sin nxdx,
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0
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f (x)
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l
f ( z)
π
1
2l
2l
/
0
2
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1 a
0
z =
+
(an cos nz + bn sin nz) ,
π
2
n=1
#& , 3
1 π l
1 π l
z dz,
an =
f
f z cos nzdz,
π −π
π
π −π
π
1 π l
f z cos nzdz.
bn =
π −π
π
a0 =
-
an = 0.
. $$
"
bn =
%
%
x = πl z, z = x πl ,
π
dz = dx z = −π, x = −l z = π, z = l
l
f (x)
2l
nπ
1 l
1 l
x dx,
f x)dx,
an =
f (x) cos
l −l
l −l
l
l
1
nπ
x dx.
f (x) sin
bn =
l −l
l
!"#$%
+∞
a0
nπ
nπ
an cos
+
x + bn sin
.
f x =
2
l
l
n=1
!""'%
a0 =
&
( f (x)
a0 = an = 0;
bn =
2
l
l
f (x) sin
0
2l )
nπ
x dx.
l
!""*%
+
+∞
nπ
bn sin
.
f x =
l
n=1
!""#%
( , f (x)
a0 =
2
l
l
f (x)dx,
0
an =
2
l
l
f (x) cos
0
nπ
x dx,
l
2l
)
bn = 0.
!"""%
+
+∞
a0
nπ
an cos
+
x .
f x =
2
l
n=1
!""-%
l
!"#$% &
' ()$ x = 0
x = l * +
∂u
∂ 2u
= a2 2 ,
∂t
∂x
x ∈ [0, l],
t > 0,
u(x, t)|t=0 = ϕ(x).
, '
' .
- '%
u(x, t) = X(x)T (t).
* +
!"#$.
/"$
∂u
∂ 2u
= X(x) · T (t),
= X (x) · T (t) ⇒ X(x)T (t) = a2 X (x)T (t),
2
∂t
∂x
T (t)
X (x)
=
. 2
a T (t)
X(x)
* ' ' x%
0 t% x% t % ' % +
.
T (t)
X (x)
=
= C.
a2 T (t)
X(x)
1
/!$
.
T (t) − a2 CT (t) = 0,
X (x) − CX(x) = 0.
/($
T (t) = BeCa t
t → +∞ u(x, t)
C < 0 C = −λ2
!"# $
T (t) = Be−λ a t .
!!#
2
% & C = −λ !"#
X (x) + λ2 X(x) = 0 $
X(x) = C1 cos λx + C2 sin λx,
!'#
2
2
s + λ = 0
( s1,2 = ±λi
) ( * +
* ,-
2
2 2
2 a2 t
u(t, x) = Be−λ
(C1 cos λx + C2 sin λx).
( * .* * +
C1 C2 * ( * - *
!// 0 +
-
u(t, x)|x=0 = 0.
1 2 ( * " # & !/#
X(0)T (t) = 0,
X(l)T (t) = 0.
) 2 ( * ( ,- T (t) +
(
X(0) = X(l) = 0.
!3#
4 X(x) * ( +
* !3# !3# !'#$
X(0) = 0
⇒
X(l) = 0
C1 = 0
⇒
C1 cos λl + C2 sin λl = 0
⇒ sin λl = 0 ⇒ λl = kπ, k ∈ Z ⇒ λk =
C1 = 0
⇒
C2 sin λl = 0
kπ
kπx
⇒ Xk (x) = Ck sin
,
l
l
Ck 5 * & (
k = 1, 2, 3, . . .
1( λk = kπ/l ( * ,+
- sin(kπx/l) 5
Tk (t)
kπ
2 2
; Tk (t) = Bk e−λk a t .
l
!" Xk (x) Tk (t)
λk =
uk (x, t)
&#'$
#$ %
%
(
2 2
uk (x, t) = Bk Ck e−λk a t sin
b k = Bk C k
u(x, t)
)
kπx
kπx
2 2
= bk e−λk a t sin
,
l
l
&#'$
* + ,
! %
-$ . ! ,
(
u(x, t) =
+∞
uk (x, t) =
k=1
0
/
*
*
!1 .
uk (x, t)
3
(k = 1, 2, . . . )
bk e−(kπ/l)
$
!$ .
4! .$
/
&#' !"
u(x, t)
* %
bk
, ,
ϕ(x)
%
+∞
bk sin
-(
kπx
= ϕ(x).
l
bk !2 " 1 %
(0, l) 5 , (
*
bk =
,
kπx
.
l
. , !
k=1
sin
u(x, 0) =
!"
2 a2 t
k=1
$ ! ,!
2
$
+∞
,$
2
l
l
ϕ(x) sin
0
-
*
kπx
dx.
l
6
&#'
* !" +
%
/$ !2 " bk 6
, ,
!"
1
$
u|x=0 = h0 ,
(
u|x=l = h1 ,
#7
u(x, t) w(x, t)
u(x, t) = w(x, t) + γ1 x + γ0 ,
γ0 γ1
w(x, t)
!
"
γ0 γ1
u|x=0 = w|x=0 + γ0 ,
u|x=0 = h0 ;
w|x=0 = 0 ⇒ h0 = γ0 ;
u|x=l = w|x=l + γ1 l + γ0 , u|x=l = h1 ; w|x=l = 0 ⇒ h1 = γ1 l + γ0 .
#$
γ1 γ0
γ0 = h0 ;
γ1 =
h1 − h0
.
l
%
h1 − h0
x + h0 .
&'( ))*
l
∂u
∂ 2u
∂w
∂ 2w
=
+
,
=
w(x, t)
2
∂t
∂t
∂x
∂x2
"
&'- ).*
u(x, t) / &'0 1*
u(x, t) "
w(x, t)
u(x, t) = w(x, t) +
h1 − h0
x − h0 ,
l
&'( )2* 3
w(x, t)|t=0 = ϕ(x) −
w|x=0 = 0;
w|x=l = 0.
&'( )-*
&'( )0*
#$
&'( )-*
&'( )0* $ w(x, t)
#$ u(x, t)
&'( )2*
&'( ))*
'( ) -
#
4
5 "
&'0 6*
∂u(t, x)
∂u(t, x)
|x=0 =
|x=0 = 0.
∂x
∂x
! " # $ % &
'( ' ( C " & (
% () C = −λ2
* ' $ " +
,-
* ' $ "
. "/ 0 )
X (0)T (t) = 0,
X (l)T (t) = 0.
( ! 1(2 3%$ T (t)
1 )
X (0) = X (l) = 0.
4$ X(x) !1- * ( +
! ! * $ X (x) ( +
& * ( ! !)
X (x) = −λC1 sin λx + λC2 cos λx.
X (0) = 0 ⇒ λC2 = 0 ⇒ C2 = 0.
X (l) = 0 ⇒ λ(−C1 sin λl + C2 cos λl) = 0 ⇒ sin λl = 0 ⇒
kπ
kπx
⇒ Xk (x) = Ck cos
.
l
l
' $ ( Ck
⇒ λl = kπ, k ∈ Z ⇒ λk =
#( ! ' - 1
+
k = 0, 1, 2, 3, . . . ! !
k = 0 ( 3%! ! !1 ! (
cos 0 = 1 = 0 5 1- ( λk = kπ/l
! Tk (t) !1 ! 3 ")
2 2
Tk (t) = Bk e−λk a t .
!! 3% Xk Tk (t) 3 +
uk (x, t) ! !1- * ( !
"/)
2 2
uk (x, t) = Bk Ck e−λk a t cos
kπx
kπx
2 2
= ak e−λk a t cos
,
l
l
* ak = Bk Ck k = 1, 2, 3 . . .
a
a
, ' B0 C0 = 0 ⇒ u0 (x, t) = 0
2
2
6
uk (x, t)
! "# $ % "& ' %
(
u(x, t) =
+∞
kπx
a0
2 2
+
,
ak e−(kπ/l) a t cos
2
l
k=1
+∞
uk (x, t) =
k=0
)
a0/2 ' λ0 = 0 *
+ ', - ./
"0 1' ' + % %
"&(
a0
kπx
+
= ϕ(x).
ak cos
2
l
k=1
+∞
2 '3 ak -../ 3,
./ ϕ(x) 3! (0, l) 4% +
2
ak =
l
l
ϕ(x) cos
0
kπx
dx.
l
)0
5 % ++
% "& +3 / $
) -../ ak + + . )0
)
ϕ(x) = u0
0 x l ! " u(x, t)
( 3 +3 /
$ ) -../ + + .
)0(
l
2u0
kπx
2u0 + k = 0
dx =
ak =
cos
0
+ k > 0 .
l
l
0
1 . ) + u(x, t) = a0 /2 = u0
1 ! 3% + +! +3
/ , + % +
$
+ t > 0
! " #
$ #
∂u(t, x)
|x=0 = u(x, t)|x=0 = 0.
∂x
% & '( " #
) * +, & # - # .
/ 0
X (0) = 0,
1
X(l) = 0.
2 & X(t) # ,
# 1 #
" 3 # X (x) x = 0
X (0) = 0 ⇒ λC2 = 0 ⇒ C2 = 0 " 3
x = l X(l) = 0 ⇒ C1 cos λl + C2 sin λl = 0 ⇒
π
⇒ C1 cos λl = 0 ⇒ cos λl = 0 ⇒ λl = + kπ k ∈ Z ⇒
2
(2k + 1)π
(2k + 1)πx
⇒ Xk (x) = Ck cos
.
⇒ λk =
2l
2l
+# ,( ( Ck
k = 0, 1, 2, . . . 4 , +
λk = (2k + 1)π/(2l) & # Tk (t) # # 5
% ## 5 Xk (x) Tk (t) 5 &
# )3 # ,
# 0
(2k + 1)πx
(2k + 1)πx
2 2
= ak e−λk a t cos
,
2l
2l
).
ak = Bk Ck k = 0, 1, 2, 3, . . .
6 & uk (x, t) # )3 # ,
* ,/ ( # 7 +
( #0
2 2
uk (x, t) = Bk Ck e−λk a t cos
u(x, t) =
+∞
k=0
uk (x, t) =
+∞
k=0
ak e−((2k+1)π/2l)
2 a2 t
cos
(2k + 1)πx
.
2l
)
8 (&
+ (# $ 5 # # # # &
# )3 # , #
ak (k = 0, 1, 2, . . . )
+∞
ak cos
k=0
(2k + 1)πx
= ϕ(x).
2l
! " #
$ %& cos (2k +2l1)πx "
(0, l) ' #
l
(2k + 1)πx
(2n + 1)πx
l/2 k = n
cos
dx =
cos
k = n .
0
2l
2l
0
( $ )$$% *
+ cos (2k +2l1)πx "
(0, l) , $ )$$%
ak =
2
l
l
ϕ(x) cos
0
(2k + 1)πx
dx.
2l
-. /0
" // , / "
)$$% 1 $
l
t > 0
u|t=0 = l − x
- . 2 . 1 #
"
∂u
= u|x=l = 0.
0
∂x x=0
u|t=0 = l − x
-. , / " )$$% ak 1
$
ak =
2
l
l
0
(l − x) cos (2k+1)πx
dx =
2l
4
(l
(2k+1)π
l
− x) sin (2k+1)πx
+
2l
0
4
+ (2k+1)π
−
=
l
0
8l
(2k+1)2 π 2
8l
(2k+1)2 π 2
sin (2k+1)πx
dx =
2l
cos (2k+1)πx
2l
l
0
u(x, t) =
8l
π2
k=0
l
− x) sin (2k+1)πx
−
2l
0
8l
cos (2k+1)π
= − (2k+1)
−1 =
2 π2
2
4
(l
(2k+1)π
.
+∞
1
(2k + 1)πx
2
2 2
2
.
e−a (2k+1) π t/4l cos
(2k + 1)2
2l
(U) m
1
(U) m 0,8
(U) m 0,6
(U) m 0,4
(U) m 0,2
(U) m
0
0,2
0,4
xm
0,6
0,8
1
! " # #!
$% ! & '& # $
& t → +∞ ! # ( )( "*
# !
#+ , ( # ! +'
" )& # # *
# ! ) & ) -)# -#) %
& Ox .
)
u(x) ( t! -#
! " t
" xm, m = 0, 1, · · · M
x ! # $ %
% "% & # #
"# "# #
#% ' ! " # "
()*+,)- ()./0) 1
#! 2 "% "
# 2 ak
3
#! "
!# #
" #
"
35 u|x=0 = u|x=l = 0.
u(x, t) =
bk =
5
∂u
∂x
65
∂u
∂x
+∞
bk e−(kπ/l)
2 a2 t
k=1
l
2
l
∂u
∂ 2u
= a2 2
∂t
∂x
u|t=0 = ϕ(x)
ϕ(x) sin
0
sin
4
kπx
l
kπx
dx k = 1, 2, . . .
l
∂u
= 0.
∂x x=l
x=0
+∞
a0
kπx
2 2
u(x, t) =
+
ak e−(kπ/l) a t cos
2
l
k=1
l
2
kπx
dx k = 0, 1, 2, . . .
ak =
ϕ(x) cos
l
l
=
0
= ux=l = 0.
x=0
+∞
u(x, t) =
k=0
ak e−λk a t cos λk x
2 2
2
l
ak =
l
ϕ(x) cos λk xdx λk =
0
(2k + 1)π
.
2l
! " #$ f (x)
%
1 +∞
|f (x)|dx &
l −∞
( #$ f (x)
&
' f (x)
)
& &
+∞
f (x) =
(A(α) cos αx + B(α) sin αx)dα,
'*+ ,*(
0
& "
-
f (x − 0) + f (x + 0)
=
2
.)## $
A(α)
+∞
(A(α) cos αx + B(α) sin αx)dα.
−∞
B(α) # -
1 +∞
f (τ ) cos ατ dτ ;
π −∞
1 +∞
B(α) =
f (τ ) sin ατ dτ.
π −∞
A(α) =
0
& "
kπ
π 2π
,...,
,...
- ,
l l
l
'*+ ,/(
0 "
$ " -
α ∈ [0, +∞) .
1
) $
*, )## $
f (x)
" 2 #$
A(α) B(α)
A(α) =
2
π
+∞
f (τ ) cos ατ dτ,
B(α) = 0,
0
!
f (x) =
+∞
A(α) cos αxdx.
"#
0
$ % %& ' f (x) (
A(α) = 0,
b(α) =
2
π
+∞
f (τ ) sin ατ dτ,
0
+∞
f (x) =
")
B(α) sin αxdx.
"
0
* % ! + ,
- + . /
%
+∞
f (x) =
0
0
1
π
+∞
f (τ )(cos ατ cos αx + sin ατ sin αx)dτ dα.
−∞
+ %&
% '
cos α(τ − x) = cos ατ cos αx + sin ατ sin αx
+
f (x) =
1
π
+∞ +∞
dα
f (τ ) cos α(τ − x)dx.
0
""
−∞
1+ 2 %
! % 3 2 2+ ' /
! l → +∞
!
" # $ % & "'
( $ ) * ( +& !
"
, #( -(& ( #
.+ /
u(x, t) = X(x)T (t)
0 ( -( !
# $ % & "' 1!
λ& "' /
2 a2 t
uλ (x, t) = (A(λ) cos λx + B(λ) sin λx)e−λ
.
0 # -( !
$ (& !
+ & λ & !
"' " (( λk &
#& "' ( & ( !
2 + + + !
λ ( " ( #&
"' ( $ ) & (
"' λ& !
-(/
+∞
2 2
u(x, t) =
(A(λ) cos λx + B(λ) sin λx)e−λ a t dλ.
$3 3
0
(& 4 . # !
$ % - . A(λ) B(λ) & #
$3 " $ ) 0 $3 3
t = 0& /
u|t=0
+∞
= ϕ(x) ⇒
(A(λ) cos λx + B(λ) sin λx)dλ = ϕ(x).
0
5 ( . ϕ(x) -(& !
$3 6 4. A(λ) B(λ)
1 +∞
ϕ(τ ) cos λτ dτ ;
π −∞
1 +∞
B(λ) =
ϕ(τ ) sin λτ dτ.
π −∞
A(λ) =
! "
!# $ %
u(x, t) =
1
π
+∞ +∞
2 2
dλ
ϕ(τ ) cos λ(τ − x)e−λ a t dτ.
0
&
−∞
' ( #
)# (#! *+* %
+∞
1 π −(τ −x)2/(4a2 t)
2 2
e
cos λ(τ − x)e−λ a t dλ =
.
2a t
0
& ( !#
%
1
u(x, t) =
π
⎛ +∞
⎞
+∞
2
2
ϕ(τ ) ⎝ cos λ(τ − x)e−λ a t dλ⎠ dτ =
−∞
0
1
= √
2a πt
+∞
2
2
ϕ(τ )e−(τ −x) /(4a t) dτ.
−∞
$ #
,( )# - .
)( # ! &
! %
u(x, t) =
1
√
2a πt
+∞
2
2
ϕ(τ )e−(τ −x) /(4a t) dτ.
−∞
/
⎧
⎨ 0,
u0 ,
u(x, t)|t=0 =
⎩
0,
x < x1 ,
x1 x x2 ,
x > x2 .
t > 0
u0
u(x, t) = √
2a πt
x2
e−(τ −x)
2 /(4a2 t)
dτ.
x1
! " # #! $ % &
' $ ( )(z)
1
)(z) = √
2π
x
e−S
2 /2
dS.
0
* % '
( $ + ' #
, % &
' - - ! . . /$ . $
! % ' ! % ' ( 0 % ' ( ! )(−z) = −)(z)- &
!# $1 ! ! 2 - )(−∞) = −0,5,
τ −x
√
= S -
)(+∞) = 0,5. 3 # .
a 2t
√
dτ = a 2tdS
u0
u(x, t) = √
2a πt
⎛
u0 ⎜
=√ ⎜
2π ⎝
x2 −x
√
a 2t
e−S
0
e−S
√
a 2t dS =
x1 −x
√
a 2t
x2 −x
√
a 2t
2 /2
2 /2
0
dS +
x1 −x
√
a 2t
⎞
e−S
2 /2
⎟
dS ⎟
⎠=
⎛
⎜ 1
= u0 ⎜
⎝ √2π
x2 −x
√
a 2t
e−S
2 /2
0
u(x, t) = u0
1
dS − √
2π
x2 − x
√
a 2t
e−S
2 /2
⎟
dS ⎟
⎠⇒
0
⎞
x1 −x
√
a 2t
−
x1 − x
√
a 2t
.
xa2√−2tx xa1√−2tx x < x1 x > x2
x1 < x < x2
! (z) !"! #! $ % (+∞) = 0,5
t = 0
! # & " x < x1 x > x2 '( #! "&
x1 < x < x2 u0 "& #" ! " ! #!
)#" & ! a x1 x2 *
a = 1 x1 = 1 x2 = 3 " +
u(x, t) = u0
3−x
√
2t
−
1−x
√
2t
&
.
, " # " !#& ' "
"& "!( - & *
"# "! "' # # .! # "
u(x, t) ! /01 "& #" !
2 ". " ! & #" x = 0 , # *
# #" x = 0 . #! " ! *
#"
+ u(0, t) = 0 3"! - ! 4 % # -%
#"& #! " " "! ' # % '*
"# 3"! 4 ". (x) ( "&
"& "#& x < 0 $ ' 5 *
- " % % #! " 6 ! #"
#".7 ' % # $ $ # (x) +
u(x, t) =
1
√
2a πt
⎛
⎞
(τ − x)2
(τ + x)2
+∞
−
⎜ − 4a2 t
4a2 t ⎟
f (τ ) ⎝e
−e
⎠ dτ.
8
0
5 - # -% "! "' # # .! *
" % " " #" ! +
∂u
= 0 " #! " 6 #" "& #"
∂x
x=0
u(x, t) =
(x)
! "
1
√
2a πt
⎛
⎞
(τ − x)2
(τ + x)2
+∞
−
⎜ − 4a2 t
4a2 t ⎟
f (τ ) ⎝e
+e
⎠ dτ.
#$% %&'
0
( )
) $*
+!
)"
$% &
u
=u
x=0
$% *
∂u
∂x
x=0
∂u
∂x
x=0
= 0,
u
x=L
,u
= 0,
u
∂u
∂x
= 0,
=
t=0
100
(L − x),
L
u
x=L
$% %
⎧ x ∈ (−∞, +∞)
u
=
t=0
$% -
,
x − L/2, x ∈ (0, L/2),
0,
x∈
/ (0, L/2).
=
t=0
x=L
=
x ∈ [0, L].
100, x ∈ (0, L/2],
20, x ∈ (L/2, L].
=
t=0
⎨ 100(1 − x/10), x ∈ [0, 10],
100(1 + x/10), x ∈ [−10, 0),
⎩
0,
|x| > 10.
x ∈ (0, +∞)
u x=0 = 0,
∂u
∂x
= 0,
x=0
u
U0 , x ∈ [2, 10],
0, x ∈ (0, 2) ∪ (10, +∞).
=
t=0
u
=
t=0
U0 , 0 < x < L,
0, x L.
! " #
$% & $
' $ # $
( ) (( &
x * ) )
) )
+%+ +
∂u
∂ 2u
− a2 2 = 0.
∂t
∂x
, -./
0
& $
0 x l) 0 t T &
u(x, 0) = ϕ0 (x),
, -1/
$ $% 2
⎧
⎪
⎪
⎨ α0 u
∂u
= γ1 (t);
∂x x=0
x=0
∂u
⎪
⎪
+ β1
= γ2 (t).
⎩ β0 u
x=l
∂x x=l
+ α1
, -3/
"
(n, m) 4 # $ & 5
& + + + 6
& 5 , ! / # +
2
n
=
unm+1 − 2unm + unm−1
+ O(h2 ).
h2
∂ u
∂x2
m
(n+1,m)
(n+1,m-1)
h
τ
h
τ
h
(n,m)
(n,m+1)
(n,m)
n ! "
⎧ n+1
un − 2unm + unm−1
um − unm
⎪
⎪
− a2 m+1
= 0,
⎪
⎪
⎪
τ
h2
⎪
⎪
m
=
1,
2,
.
.
.
,
M
−
1,
n = 0, 1, . . . , N − 1,
⎪
⎪
⎨ 0
m = 0, 1, 2, . . . , M,
um = ϕ0 (xm),
n+1
n+1
u
−
u
⎪
1
0
n+1
⎪
⎪
αu
= γ1 (tn+1 ), n = 0, 1, . . . , N − 1,
+ α1
⎪
⎪ 0 0
h n+1
⎪
n+1
⎪
⎪
uM − uM−1
⎪
⎩ β0 un+1
= γ2 (tn+1 ), n = 1, 2, . . . , N − 1.
M + β1
h
#
(n+1,m+1)
(n+1,m)
h
(n,m-1)
$%
&
∂ 2u
∂x2
n
m
(N + 1) · (M + 1) % !$ & % '
r = τ a2 /h2 ( ) *
⎧ 0
um = ϕ0 (xm ),
m = 0, 1, . . . , M,
(a)
⎪
⎪
⎪ n+1
n
n
n
⎪
=
(1
−
2r)u
+
r(u
+
u
),
(b)
u
⎪
m
m
m−1
m+1
⎪
⎪
⎪
m = 1, . . . , M − 1,
n = 1, 2, . . . , N − 1,
⎨
γ1 (tn+1 )
−α1 /h n+1
u1 +
, n = 0, 1, ..., N − 1, (c)
un+1
=
⎪
0
⎪
⎪
α0 − α1 /h
α0 − α1 /h
⎪
⎪
n+1
⎪
β1 /h
γ2 (t )
⎪
⎪
un+1 +
, n = 0, 1, ..., N − 1. (d)
⎩ un+1
M =
β0 + β1 /h M−1 β0 + β1 /h
+
n = 0 !"
#
t = 0
$ !" %
u (m = 1, 2, . . . , M − 1) n + 1#
t = (n + 1)τ
& ' !"
!" % ( t = (n + 1)τ
) n = n + 1
* n < N " $
! + + ,
$ - #
(n + 1, m) ' n + 1# " ) "
n+1
n
un+1 − 2un+1
un+1
m + um−1
m − um
− a2 m+1
= 0,
2
τ
h
m = 1, 2, . . .
n = 1, 2, . . .
-+ ' ' . / "
'
0 1 u #( (n+1)#
('+ ' .2
1 3
(n + 1)# #
' ( , ' " .2
. " (+ , "
'
. (' '
%+ ' / #
' ( " '
' (n + 1)# + '
% u n 4 ' un+1
, un+1
, . . . , un+1
1
2
M−1
.2 . '
n+1
n+1
am · un+1
m−1 + bm · um + cm · um+1 = fm ,
m = 1, 2, . . . , M − 1,
5
am = cm = −r" bm = 1 + 2r" fm = unm
6( % u
(n+1)#
0 ( ym = un+1
m
⎧
⎨ am ym−1 + bm ym + cm ym+1 = fm ,
y0 = L0 y1 + K0 ;
⎩
M yM−1 + K
M ,
yM = L
−α1 /h
;
α0 − α1 /h
β1 /h
;
=
β0 + β1 /h
m = 1, 2, . . . , M − 1;
γ1 (tn+1 )
;
α0 − α1 /h
γ2 (tn+1 )
.
=
β0 + β1 /h
L0 =
K0 =
M
L
M
K
! "" #"" ""$
% # &' # " () '& " " $
* ++ ,'! *
""" #"& #"('
-
." '" +"' '& y0 ! "
$
* ") *
ym = Lm ym+1 + Km ,
."
#"')
/"
m = 1, 2, . . . , M − 1.
% # m = m − 1
ym =
−cm
fm − am Km−1
ym+1 +
,
am Lm−1 + bm
am Lm−1 + bm
0
m = 1, 2, . . . , M − 1.
am (Lm−1 ym + Km−1 ) + bm ym + cm ym+1 = fm ,
'
-
m = 1, 2, . . . , M − 1.
1'" '! "
Lm =
−cm
fm − am Km−1
; Km =
, m = 1, 2, . . . , M − 1.
am Lm−1 + bm
am Lm−1 + bm
2
3"/++, Km Lm ( 4 & #"" ") "/++, $
) '&4 & #" +"' 2
" #! ) '" & # x = l 1" $
*
- # m = M − 1
yM = LM yM−1 + KM ;
yM−1 = LM−1 yM + KM−1 .
& #"')
yM =
LM KM−1 + KM
.
1 − LM LM−1
LM
yM =
M
K
β1 KM−1 + hγ2 (tn+1 )
.
β0 h + β1 − β1 LM−1
! " ! ym
m = 0 #$ $ %&& ' L0 K0
&
( m = m + 1 #$ $ %&& ' Lm Km
& )
) * m < M (
& $ !! yM
#$ ym & ( m = m − 1
+ * m > 0
#$
, $ ! " ! $
- $ . (n+1)/
0 . $ /
1 !
|bm| > |am | + |cm | $ !/
!
2 " ! !
n = 0 & +3 " /
t = 0
( ! " ! ! /
! " ! (n + 1/
t = (n + 1)τ
) n = n + 1
* n < M (
-
, " ! ! 1
! ! ' %
O(τ )+O(h2 ) ,
! ! ! /
$ 4 1 5 ' 6
m = 0, 1, . . . , M
n
δun − 2δunm + δunm−1
δun+1
m − δum
− a2 m+1
= 0.
τ
h2
n+1
n
δun+1 − 2δun+1
δun+1
m + δum−1
m − δum
− a2 m+1
= 0.
2
τ
h
δunm = λn eiwh
λn eiwh
! "
#
λ−1
eiwh − 2 + e−iwh
− a2
= 0.
τ
h2
$ " ! eiwh + e−iwh = 2 cos wh 1 − cos wh =
= 2 sin2 wh/2! eiwh − 2 + e−iwh = 2(cos wh − 1) =
= −4 sin2 wh/2 ! % !
λ=1−4
wh
τ a2
.
sin2
h2
2
&
# ! '( ) ) τ /h2 λ 1!
* % λ < −1! +
( " % , %- " " "! "
sin2 wh/2 = 1! ) + " !
λ = 1 − 4ra2 −1,
'
1
.
2a2
.
/ ! ) ! (" (" (" !
) t x * ("% * *-
r
τ /h2
0 +
1
.
2a2
λ eiwh − 2 + e−iwh
λ−1
2
−a
= 0.
τ
h2
" ! ( "
- "
1
λ=
.
1 + 4ra2 sin2 wh/2
01
"
0
|λ| 1 r
!
" #
!
$ % " & "
! ! ' "! ()*
+$ ! ' , ! " ()*
"! ' -./01.2 -.345.
6 ! 78 -./01.2 '
9 ! ' : 78 ;
" u t=0 = f (x) ) 0,
= ϕ(x),
t=0
ut (x, t)
= ψ(x).
t=0
1 &
2
2
u(x, t) = X(x)T (t).
,"
/ 0 ) ," !"&
T (t)
T (t)
=
= C.
2
a T (t)
X(x)
/ $ t&
% x& 3 t x 4 &
5 6 5 ) C
X (x) − CX(x) = 0.
! "#
T (t) − Ca2 T (t) = 0;
u(0, t) = X(0)T (t) = 0,
u(l, t) = X(l)T (t) = 0.
$ # % & u(x, t)' ! '
' T (t) = 0' ' #
X(0) = 0, X(l) = 0.
( " # ) * X(x) +
% " , - )) * #- +
. & /' % -
0
s2 − C = 0
C = λ2 > 0 ⇒ s1,2 = ±λ 1 . # ' " '
,% &
X(x) = C1 eλx + C2 e−λx
2" - 0 .
C1 + C2 = 0,
C1 eλl + C2 e−λl = 0.
3 & C1 = C2 = 0'
X(x) = 0 ⇒ u(x, t) = 0 0 x' t
4", . .
C = 0 ⇒ s1,2 = 0 1 . # ' ' ,%
&
X(x) = C1 + C2 x.
2" - 0 .
C2 = 0,
C1 + C2 l = 0,
0 x' t
4", . . !
C = −λ2 ⇒ s1,2 = ±λi 1 +! ' ,%
&
C1 = C2 = 0 ⇒ X(x) = 0 ⇒ u(x, t) = 0
X(x) = C1 cos λx + C2 sin λx.
2" - 0 . C1 = 0' C2 sin λl = 0
C2 = 0
sin λl = 0 ⇒ λl = πk (k = ±1, ±2, . . . ) ⇒ λ =
!" #
k Xk (x)&
Xk (x) = Ck sin
kπ
.
l
$ %
'"
kπx
,
l
Ck (
) Ck * *
k = 1, 2, . . . + *
λk = kπ/l %
* , #
* - λk = kπ/l
$ . sin(kπx/l) ( ,
!" /"
0 , $ . , %
(0, l)
+ , T (t) 1* λk = kπ/l $ %
. Tk (t)
2"
Tk (t) +
3
s2 +
kπa
l
2
Tk (t) = 0.
kπa
l
2
&
=0
i
s1,2 = ± kπa
l
4# 2" &
Tk (t) = Ak cos
kπat
kπat
+ Bk sin
,
l
l
5"
Ak Bk (
) * '" 5" $ 6" %
5 2" #
5 /"&
kπx
kπat
kπat
· Ak cos
+ Bk sin
.
uk (x, t) = Ck sin
l
l
l
Ck Ak = ak Ck Bk = bk uk (x, t)
kπx
kπat
kπat
+ bk sin
· sin
.
uk (x, t) = ak cos
l
l
l
uk (x, t)
! "# " " $% " "&
'"( ) # $* + ,' "&
" " ! " ! "
" $% - # ' " " &
' # " "# , . '
' (
u(x, t) =
+∞
ak cos
k=1
kπat
kπat
+ bk sin
l
l
· sin
/
kπx
.
l
0 . 1 u(x, t) " '"
$2 # " - . 1! 3 &
" " # ak bk . 1 / &
" #
u(x, 0) =
+∞
ak sin
k=1
kπx
= ϕ(x).
l
+ "' #' ! "
ut (x, t) =
+∞
kπa
k=1
l
ut (x, 0) =
−ak sin
kπat
kπat
+ bk cos
l
l
+∞
kπa
k=1
l
bk sin
sin
kπx
,
l
kπx
= ψ(x).
l
bk ,..1
4 " ak kπa
l
" - . 1! ϕ(x) ψ(x) ( (0, l) " 5"#
*%*
2 l
kπx
ϕ(x) sin
dx;
l 0
l
2 l
kπx
dx.
bk =
ψ(x) sin
kπa 0
l
ak =
6
! ak bk " # $
%! !& # uk (x, t) '
& (
uk (x, t) = Fk sin
kπx
sin
l
kπa
+ ϕk ,
t
)*
tg ϕk = ak /bk
+ #! $ # ! " '
ωk = kπa
!
l
# ! ϕk , #
!"
x Fk sin kπx
l
%
uk (x, t) (k + 1) -
(0, l)(
Fk =
a2k + b2k ,
sin
l 2l
(k − 1)l
kπx
= 0 ⇒ x = 0, , , . . . ,
, l.
l
k k
k
. !" , -# - ! '
!& " # / 0
# # wk '
"1 !# #, ! # # '
ω1 / , 2 ! ! #"1
ωk (k 2) !" 3# !
# ! &'
# # # 2
-# # 2 !& 4! Fk ! #!'
" k ! 2 !&, !
" 2 # ## / #
)
⎧
⎪
⎪
⎨
u|t=0 =
⎪
⎪
⎩
∂ 2u
∂ 2u
= a2 2 .
2
∂t
∂x
4hx
3l
x ∈ 0; ;
3l
4
4h(l − x)
3l
;l .
x ∈
l
4
ut |t=0 = 0,
u|x=0 = u|x=l = 0.
l
(u|x=0 = u|x=l = 0)
OAB !
"
#
(ut|t=0 = 0)
$ %
U
A
h
0
3l
4
B
l
x
$% !&& '" ( )**
* !&& +"
⎛ 3l
⎞
4
l
4h(l − x)
kπx
kπx ⎟
32h
2 ⎜ 4hx
3kπ
sin
dx +
sin
dx⎠ =
.
ak = ⎝
sin
l
3l
l
l
l
3(kπ)2
4
0
3l
4
, ( #
- b k = 0 ψ(x) = 0.
$% !&& '"
kπx
32h sin(3kπ/4)
kπat
sin
=
cos
3π 2 k=1
k2
l
l
+∞
u(x, t) =
32h
3π 2
1
πat 1
2πx
2πat
πx
√ sin
cos
− sin
cos
+
l
l
4
l
l
2
3πat
1
5πat
3πx
5πx
1
cos
+ √ sin
cos
+ ··· .
+ √ sin
l
l
l
l
9 2
25 2
=
.
πx
πat
32h
√ sin
sin
l
l
3π 2 2
32h
F1 = 2√
3π 2
u1 (x, t) =
ω1 = πa/l / #
x = 2l
4
√
2
x = l/4
2πat
8h
2πx
sin
sin
3π 2
l
l
2
F2 = 8h/3π
u2 (x, t) = −
3πx
3πat
12h
√ sin
cos
l
l
27 2
u3 (x, t) =
"
#
'
%
&
x = l/6
$
'
!
( )* +
'% '
'
(
' -)* .
-)* #. /'
,
%
'
'
-)* .
0
1
u(x, t) = X(x)T (t).
2
3 + (
X(0) = X (l) = 0
''
' '
'
(
2 '' 3
' -)* #.
+ (
-
)# 4
% '
'
-)* .
'
.1
T (t)
X (t)
=
= −λ2 .
2
a T (t)
X(x)
/
2
'
X (x) + λ X(x) = 0
$5
1
T (t) + (λa)2 T (t) = 0.
'
1
X(x) = C1 cos λx + C2 sin λx,
-)) 44.
s2 + λ2 = 0
s1,2 = ±λi
X(0) = C1 = 0,
X (l) = C2 λ cos λl = 0.
cos λl = 0
(2k + 1)π
, k = 0, 1, 2, . . .
λk =
2l
! λk
" #
Xk (x) = Ck sin((2k + 1)πx/(2l))
$
(2k + 1)πx
" # sin
(0; l)
2l
l
(2n + 1)πx
(2k + 1)πx
l/2, k = n
sin
dx =
.
sin
0,
k = n
2l
2l
0
%&
'(())*
(2k + 1)πat
(2k + 1)πat
+ Bk sin
,
2l
2l
!
uk (x, t) = Xk (x)Tk (t)
+&
λk
(2k + 1)πx
(2k + 1)πat
(2k + 1)πat
+ bk sin
sin
.
uk (x, t) = ak cos
2l
2l
2l
Tk (t) = Ak cos
, ak = Ak Ck bk = Bk Ck
- u(x, t) +& .
&/
+∞
(2k + 1)πx
(2k + 1)πat
(2k + 1)πat
+ bk sin
sin
u(x, t) =
ak cos
2l
2l
2l
k=0
'(()0*
!""# ak bk
'(12*
ut |t=0
+∞
(2k + 1)πx
= ϕ(x),
2l
k=0
+∞
(2k + 1)πa
(2k + 1)πx
bk sin
= ψ(x).
=
2l
2l
k=0
u|t=0 =
ak sin
'(()2*
ak (2k +2l1)πx bk
ϕ(x) ψ(x)
sin (2k +2l1)πx ! !
" ! #!"
" " ak !
#$$ %&' ! sin (2k +2l1)πx
(0; l)
bk
2 l
(2k + 1)πx
dx;
ϕ(x) sin
l 0
2l
l
4
(2k + 1)πx
bk =
dx.
ψ(x) sin
(2k + 1)πa 0
2l
ak =
!
$!
"
'(
ak
bk
# $!
# !
# % &
)
ρ
l x = 0 P
t = 0
$
*
"
"
- -
ak
bk !
# %
# $
0 &
% $! '
/
l
x sin
#
'(
0
8P l(−1)k
(2k + 1)πx
dx = 2
,
2l
π ES(2k + 1)2
&
u(x, t) =
, - -
.
2P
ak =
lES
* "
+
)
8l
π 2 ES
+∞
k=0
(
u(x, t)|x=0 = 0!
sin (2k+1)πl
= (−1)k
2l
bk = 0.
$
k
(2k + 1)πx
(2k + 1)πat
(−1)
sin
.
cos
2
(2k + 1)
2l
2l
1
(
x = 0
x = l 2
!
,!
l
! " Θ(x, t)
# $%& u(x, t) = Θ(x, t)& !'(
# )% ( # *+%
, " Θ(x, t) = X(x)T (t) -
! Θ(x, t) u(x, t) # $%
& # **% . ( ( (
# *+% / X(0) = 0
( " /
Θtt |x=l = −b2 Θt |x=l , ( b2 = GJ0 /J1
! Θ(l, t) = X(l)T (t) -/
T (t)X(l) = −b2 T (t)X (l).
0 & ( # **%
T (t) = −a2 λ2 T (t),
( /
T (t)X(l) = −b2 T (t)X (l) ⇒ −a2 λ2 T (t)X(l) =
= −b2 T (t)X (l) ⇒ b2 X (l) − a2 λ2 X(l) = 0.
1 & ( X(x) ! /
X(0) = 0, b2 X (l) − a2 λ2 X(l) = 0.
# *2%
3 ' & ' " (
# **% /
X(x) = C1 cos λx + C2 sin λx.
. # *2% C1 C2/
X(0) = 0 ⇒ C1 = 0 ⇒ X(x) = C2 sin λx,
b2 X (l) − a2 λ2 X(l) = 0 ⇒ λ(b2 cos λl − a2 λ sin λl)C2 = 0 ⇒
⇒ b2 X (l) − a2 λ sin λl = 0 ⇒ ctg λl = λa2 /b2 .
a2
λ,
b2
!" # $ %&
' ( λk )
' ' $ * λk (
*( + $( ,
C2 ' $ $ ' $ λk
ctg λl =
y=ctg(λ l)
2
y= a2 λ
b
λ2
λ1 0
λ1
π
l
2π
l
λ
λ2
,' * λk "+
Xk (x) = Ck sin λk x.
-. # *.
λk Tk (t) = Ak cos λk at + Bk sin λk at /
Θk (x, t) *. λk
Θk (x, t) = Xk (x)Tk (t) = (ak cos λk at + bk sin λk at) sin λk x,
# ak = Ak Ck bk = Bk Ck
0 *. $ .
Θ(x, t) =
+∞
k=1
(ak cos λk at + bk sin λk at) sin λk x
ak
Θt |t=0 =
Θ|t=0 =
+∞
bk
λk bk sin λk x = 0 ⇒ bk = 0,
k=1
+∞
ak sin λk x = α
k=1
x
l
k = 1, 2, 3, . . .
!" # ak $ "
sin λk x % (0; l)
ak "
# &' () % " *
" cos λk x % "
l
cos λk x cos λn xdx =
0
+
k l)
+ sin(2λ
,k = n
4λk
.
0, k = n
l
2
x
+∞
α
.
l
ak λk cos λk x =
k=1
,
ak
% - cos λn x % (0; l) .
% # cos λk x
l
ak λk
0
α
cos (λk x)dx =
l
2
%
ak =
λ2k l
0
l
cos λk xdx,
0
α sin λk l
.
sin(2λk l)
l
+
2
4λk
/
" 1 2
Θ(x, t) =
+∞
ak cos λk t sin λk x,
34
k=1
%
ak ' /
, " # λk 5 - #
* * "
" 6'
!
"
" !
# $ !
$
% &' ( ) *&'+,
!
- )
- x ∈ [0, +∞) ( .
*&'/0, 1 *&'/&, !
% && (
!
1
2
= 0,
t=0
= x,
t=0
= ex , u
t=0
x=0
t=0
t > 0.
= x2 ,
u
t=0
u
= 0,
t=0
∂ 2u
∂ 2u
= a2 2 , u
= sin 2x,
2
∂t
∂x
t=0
= 0, x ∈ (0, +∞).
∂ 2u
∂ 2u
=
, u
= 0,
2
∂t
∂x2
t=0
∂u
= ch x,
= 0, x ∈ (0, +∞).
∂x x=0
&&4
∂u
∂t
2
∂ 2u
∂ 2u
= 4 2,
2
∂t
∂x
x ∈ (−∞, +∞).
&&3
∂u
∂t
2
∂ u
∂ u
=
,
∂t2
∂x2
x ∈ (−∞, +∞).
&&+
∂u
∂t
"$
&&/
∂u
∂t
x ∈ [0, L]
= 0.
u t=0 = 0, ∂u
= 100x, u
= 0, u
∂t t=0
x=0
x=L
= 2x, u
= 0,
u t=0 = 0, ∂u
x=0
∂t t=0
∂u
∂x
∂u
,
u t=0 = sin πx
= 0,
2L ∂t t=0
u
∂u
∂x
= 0.
x=L
= 0,
x=0
= 0.
x=L
∂u
,
u t=0 = sin πx
= 0, u
= 0,
L ∂t t=0
x=0
= 0.
u
x=L
! !
" #
$ ! $ $ %&' ( !
"
) $ & !
" " * !
+* ,,)
-
+*+ + ,,) +
⎧ n+1
un − 2unm + unm−1
um − 2unm + un−1
m
⎪
⎪
− a2 m+1
= 0,
⎪
2
⎪
τ
h2
⎪
⎪
⎪
n = 1, 2, . . . , N − 1; m = 1, 2, . . . , M − 1;
⎪
⎪
⎪
⎪
m = 0, 1, . . . M;
u0m = ϕ0 (mh),
⎪
⎨ 1
um − u0m
= ϕ1 (mh),
m = 0, 1, . . . , M;
⎪
τ
⎪
⎪
n+1
n+1
⎪
u
−
u
⎪
0
⎪
= γ1 (tn+1 ), n = 0, 1, . . . , N − 1,
+ α1 1
α0 un+1
⎪
0
⎪
h
⎪
⎪
n+1
⎪
un+1
⎪
M − uM−1
⎩
β0 un+1
= γ2 (tn+1 ), n = 0, 1, . . . , N − 1.
M + β1
h
!
" #
# $% &
⎧ n+1
un − 2unm + unm−1
um − 2unm + un−1
m
⎪
2 m+1
⎪
−
a
= 0,
⎪
⎪
τ2
h2
⎪
⎪
⎪
n
=
1,
2,
.
.
.
,
N
−
1;
m
=
1,
2,
. . . , M − 1;
⎪
⎪
⎪
⎪
u0m = ϕ0 (mh),
m = 0, 1, . . . , M;
⎪
⎨ 1
um − u0m
τ a2
= ϕ1 (mh) +
ϕ (mh), m = 0, 1, . . . , M;
⎪
τ
2 0
⎪
⎪
n+1
n+1
⎪
u1 − u0
⎪
n+1
⎪
= γ1 (tn+1 ), n = 0, 1, . . . , N − 1,
⎪ α0 u0 + α1
⎪
h n+1
⎪
⎪
n+1
⎪
⎪
⎩ β un+1 + β uM − uM−1 = γ (tn+1 ), n = 1, 2, . . . , N − 1.
0 M
1
2
h
'!
( )
* + ,-! .
! &&
τ h / τ
# h
τ 0 '! !
. 1 ∂u(0, x)/∂t
τ
# h τ
$ ∂u(0, x)/∂t '!
ϕ0 (x)
u1m = u0m +
u1m
1 ∂ 2 u(0, x) 2
∂u(0, x)
τ+
τ + O(τ 3 ).
∂t
2 ∂t2
2
2
2
∂ u
∂ u
d ϕ0 (x)
= a2 2 = a2
∂t2
∂x
dx2
∂u(0, x) u1m − u0m a2 d2 ϕ0 (x)
=
−
τ + O(τ 2 ),
∂t
τ
2 dx2
!"#$
τ
u1m − u0m
= ϕ1 (mh) + a2 ϕ0 (mh) + O(τ 2 ).
τ
2
%
u1m = u0m + τ ϕ1 (mh) +
τ 2 2
a ϕ0 (mh).
2
!"&$ '
( ) ) *
u0m
2
un+1
= 2unm −un−1
m
m +a
u1m
!"&$
%
τ2 n
(u
−2unm +unm−1 ), n = 1, 2, . . . , N −1. !"+$
h2 m+1
, -
)*
#$ . u0m = ϕ0 (mh), m = 0, 1, 2, · · · M ) -
t = 0
/$ n = 1 . u1m = u0m + τ ϕ0 (mh) 0
!"&$ -
t = τ
1$ . !"+$ u
) ) ) (m = 1, 2, . . . , M − 1) n + 1
t = (n + 1)τ
&$ .
!2+ 3$
!2+ 4$ 0 t = (n + 1)τ
+$ n = n + 1
2$ 5 n < N ) 1
!$ 6 '
! " δunm = λn eiwmh !
"
n
n−1
δun − 2δunm + δunm−1
δun+1
m − 2δum + δum
− a2 m+1
= 0,
2
τ
h2
#! τ > rh
wh
λ + 1 = 0.
λ2 − 2 1 − 2r2 a2 sin2
$ %&
2
' ' '
( ! ) λ1 λ2 = 1 * +
) , + ' # ) , *
- # )
+
- , .! $ %&
/
wh
wh
2 2
2 wh
± 2ra sin
− 1.
λ = 1 − 2r a sin
r2 a2 sin2
$ &
2
2
2
0 # +
# - $ & # , /
r2 a2 sin2 (wh/2) < 1+ r2 a2 sin2 (wh/2) = 1 |λ| = 1 1
! +
sin2 (wh/2) = 1+ +
r2 a2 1+
1
τ
.
$ &
h
a
2 + # ! ) '
# ! (tn+1 , xm )
x − at = C1 x + at = C2 + " * $
3%& 4# - ! !
)# 0' 56+ '+ # !
# + # + # # !
) # (tn+1 , xm ) !
# # ! 77, ' +
τ
a+ *
h
+
! 2 4
r=
n+1
n+1
n
n−1
σ(un+1
un+1
m+1 − 2um + um−1 )
m − 2um + um
2
−
a
−
τ2
h2 n−1
n−1
n
n
n
n−1
(1
−
2σ)(u
−
2u
+
u
)
+
σ(u
−
2u
m+1
m+1
m
m−1
m + um−1 )
−a2
= 0,
h2
! σ = 1/2 " "# # #
$ σ = 1/4 % " "# # #
$ & ' σ = 0
# # (' )
*! " ! #
τ = rh +
1 + 4σr2 a2 sin2 (wh/2) λ2 − 2 1 − 2(1 − 2σ)r2 a2 sin2 (wh/2) λ+
+1 + 4σ 2 r2 a2 sin2 (wh/2) = 0.
,.! σ = 1/2 σ = 1/4 " " |λ| = 1
/ #" ! ! # "
" "
#
$ !
0 " " # " 1
23, 1 !! #
!
4 !5 6 7 8 "
' ! "# ! "# 1
9 " " ! " 1
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! : CADEFAG
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% $ ( $
$
( =>?@A< BA?>: % $ 1
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3+:
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3
3
3
3,:
3,:
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3 =
3:
3:
3,,
3>,
3+
3 =
3+
3=
3,:
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3==
3 !
3 !
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3+
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3+
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3=!
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3, 3
3
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3>: 3>!
1
t=0
t=0,2
t=0,4
t=0,6
t=0,8
t=1,0
0,8
0,6
0,4
0,2
0
0,2
0,4
0,6
0,8
1
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6-4*2049 *4(956?44 0/? GH%BI JBHK 4L(491 -4
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ORIGIN := 0
T OL := 0.00001
T max := 0.5
r := 1
M := 100
h
T max
τ := r ·
N :=
a
τ
hP r
τ P r := 0.1 hP r := 0.1 MP r :=
h
a2 := r2
a1 := 2 · 1 − r2
m := 0..MhP r
L := 1
a := 1
L
h :=
M
MhP r :=
L
hP r
xm := m · hP r
(U) m 1
(U) m 0,8
(U) m 0,6
(U) m 0,4
(U) m 0,2
(U) m
0
0,2
0,4
xm
0,6
0,8
1
U := |f or
m ∈ 0..M
xm ← m · h
ynm1m ← sin(π · xm )
ym ← ynm1m + τ · 0.5 · cos(0.5 · π · xm )
k←0
ngr ← 1
f or m ∈ 0, MP r..M
Uk,0 ← ynm1m
k ←k+1
f or n ∈ 2..N
t←n·τ
f or m ∈ 1..M − 1
ynp1m ← a1 · ym + a2 · (ym−1 + ym+1 ) − ynm1m
ynp10 ← 0
ynp1M−1 + h · 0.1 · sin(2 · π · t)
ynp1M ←
1+h
ynm1 ← y
y ← ynp1
if ngr · τ P r − t < 0.1 · τ
k←0
f or m ∈ 0, MP r..M
Uk,ngr ← ym
k ←k+1
ngr ← ngr + 1
U
U (x, t)
&
*+,-. /,+0
xm := m · hP r
! "# $
(
m := 0..MhP r
%
&
' !
%)
%
)
1
)
&
%
2+3
-
! " #$%&' (')*+,)!
. /012 34 5.
6 7 !0
6 78 ! 6 7
6 7 6 7 !!9.8 ! ! 6 7
6 7 6 7
: ! ";
;
. /012 34 5.
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. /012 34" 5.
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67
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3 3 3 -33 13 3 /3 3 3 (
! ! " #
$% & ! '
%#
( ) * #
!
! % ' & '
% ! " !
+
,- ( ! % !
#
! % ! ). * /- 0
#
! 1 2 % ! -
3 ( '
!
4 #
2 '
- 5#
- /
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2
6 %
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' ) ! % % ! !* 7 #
! '
2 %
( ( %
!
- 8 - / " !
- 1
9 : ; / " -
0, y1 > 0
y = y(t) - " ' ' #
A B +
( * )
* # ( ) "
= 2gy
( y = y(x) * ( # ds
dt
0 t !0 *
# ) A(0, 0) ) B(x1, y1) :
1
t[y(x)] = √
2g
x1
0
1 + y2
dx −→ inf;
√
y
y(0) = 0,
y(x1 ) = y1 .
1
& % 2
# !( ) !(*
, # . * ) m / # * x0
, !# # f
# * * v0 +''
' ) # x(t) ,
3*' mẍ = f 4, # , 5
f ∈ [f1 , f2 ] 4 ' # #
!0
, , #!( * ) # T
T −→ inf,
x(0) = x0 ,
mẍ = f,
ẋ(0) = v0 ,
f ∈ [f1 , f2 ],
x(T ) = ẋ(T ) = 0.
6
!"#$%
&' &' t[y(x)]
' ' ' ' ' (
! X "#)*% +
( ', &'
' ' ' -'
"#).
/ &0 & 1 &0
2
ai 3 ' &' (4 i,
1 i m5
bj 3 ! ( (% j 4 4
1 j n5
cij 3 ' &' i,
j , 4 5
xij 3 & ' &' '
i, j , 4
m n
64 ' cij xij &
i=1 j=1
7 4 8 &02
!*% xnij ∈ R+ ! 4 & '%5
!)% xij ai ! 4 %5
!9%
j=1
m
xij = bj
!& ' '' (
%
1 4 & &0 2
i=1
m
n
i=1 j=1
cij xij −→ inf,
n
j=1
xij ai ,
m
i=1
xij = bj ,
xij ∈ R+ .
!"#"%
& :&
n &' !&' '& % ; &
4 ( bj j 4 : 8
aij j 4
i4 &' ci 3
i4 &'
xi i
n
ci xi −→ inf,
i=1
n
aij xi bj ,
xi 0.
i=1
!
" # $% #
$& $ #$ '(
y
D
C
F
G
E
H
A
B
x
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$
#$ ABCD /# #
0 $ #$ ABCD M
hx AB N
hy AD - 1 hx = Ma hy = Nb
M1 M2 N 1 N 2 .0 0 0 . 0
av = 0 bv = 0
N 1 = N 2 = M1 = M2 = 0
!
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av > 0 bv > 0 0 < N 1 < N 2 < N M1 < M2 < M
$ !
%& ' " #
(" C av > 0 bv > 0 )
*
" DC BC % "
0 < N 1 < N N 2 = N 0 < M1 < M M2 = M
+ !
%& " #
(" AD av > 0 bv > 0 )
" AD "#%
M1 = 0 ( CD ,
N2 = N
- ' " &
&#
#% "
- & '# & ' &
" . # & #% " /
" #% &# ABCD " #
EHGF 01 & 2 |AB| = a |AD| = b |EF | = av
|EH| = bv O(xv0 , yv0 ) 3
- '
&# "
# ABCD N " AD
M " AB &
N 1 N 2 M1 M2 %4
& # " a b av bv 3
AB AD EF EH 5 .
xv0 yv0
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/ & "# 6
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hx, hy, r " Ox , xm
" Oy , ym m1, m2, n1, n2
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3469: 349; 34 345:; 34?=; 8
34 0 |y| − |x| =
|y| − i |x| =
= C1 , |y| + |x| = C2 xy < 0
|y| + i |x| = C2 xy
=
0
= C1 ,
√
√
√
√
xy < 0 y = 13 x3 + C1 , y = − 13 x3 + C2
√
√
xy > 0
√y = i 13 x3 +C1, √y = −i 13 x3 +C2
∂ 2z
∂ 2z
x2
+ 1, z(x) = xº
−
=0.
2
∂x2 ∂y 2
2
2
2
∂ u ∂ u ∂ u
2 + 2 + 2 = f (x, y, z).
∂x
∂y
∂z
ϕ − y
= 1
1 + y ϕ
y1 = ϕ(x1 )
y(x) = −
y =
1
x5
+ (x2 − x3 )
120 24
π
4
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Министерства образования и науки РФ «Лучшее учебное издание по математике
в номинации «Математика в технических вузах»
Н. А. БЕРКОВ, В. Г. ЗУБКОВ,
В. Б. МИНОСЦЕВ, Е. А. ПУШКАРЬ
КУРС МАТЕМАТИКИ
ДЛЯ ТЕХНИЧЕСКИХ
ВЫСШИХ УЧЕБНЫХ
ЗАВЕДЕНИЙ
Часть 3
Дифференциальные уравнения.
Уравнения математической физики.
Теория оптимизации
Под редакцией
В. Б. Миносцева, Е. А. Пушкаря
Издание второе, исправленное
ДОПУЩЕНО
НМС по математике Министерства образования и науки РФ
в качестве учебного пособия для студентов вузов, обучающихся
по инженерно&техническим специальностям
•САНКТ/ПЕТЕРБУРГ•МОСКВА•КРАСНОДАР•
•2013•
ББК 22.1я73
К 93
Берков Н. А., Зубков В. Г., Миносцев В. Б.,
Пушкарь Е. А.
К 93
Курс математики для технических высших учебных
заведений. Часть 3. Дифференциальные уравнения.
Уравнения математической физики. Теория оптимизации:
Учебное пособие / Под ред. В. Б. Миносцева,
Е. А. Пушкаря. — 2+е изд., испр. — СПб.: Издательство
«Лань», 2013. — 528 с.: ил. — (Учебники для вузов.
Специальная литература).
ISBN 9785811415601
Учебное пособие соответствует Государственному образовательному
стандарту. Пособие включает в себя лекции и практические занятия.
Третья часть пособия содержит 25 лекций и 25 практических занятий по
следующим разделам: «Обыкновенные дифференциальные уравнения»,
«Дифференциальные уравнения в частных производных», «Элементы
вариационного исчисления и теории оптимизации».
Пособие предназначено для студентов технических, физико+
математических и экономических направлений.
ББК 22.1я73
Рецензенты:
À. Â. ÑÅÒÓÕÀ äîêòîð ôèçèêî-ìàòåìàòè÷åñêèõ íàóê, ïðîôåññîð,
÷ëåí ÍÌÑ ïî ìàòåìàòèêå Ìèíèñòåðñòâà îáðàçîâàíèÿ è íàóêè ÐÔ;
À. À. ÏÓÍÒÓÑ ïðîôåññîð ôàêóëüòåòà ïðèêëàäíîé ìàòåìàòèêè
è ôèçèêè ÌÀÈ, ÷ëåí ÍÌÑ ïî ìàòåìàòèêå Ìèíèñòåðñòâà îáðàçîâàíèÿ
è íàóêè ÐÔ; À. Â. ÍÀÓÌΠäîêòîð ôèçèêî-ìàòåìàòè÷åñêèõ íàóê,
äîöåíò êàôåäðû òåîðèè âåðîÿòíîñòåé ÌÀÈ; À. Á. ÁÓÄÀÊ äîöåíò,
çàì. ïðåäñåäàòåëÿ îòäåëåíèÿ ó÷åáíèêîâ è ó÷åáíûõ ïîñîáèé ÍÌÑ
ïî ìàòåìàòèêå Ìèíèñòåðñòâà îáðàçîâàíèÿ è íàóêè ÐÔ;
Ó. Ã. ÏÈÐÓÌΠïðîôåññîð, çàâ. êàôåäðîé âû÷èñëèòåëüíîé
ìàòåìàòèêè è ïðîãðàììèðîâàíèÿ ÌÀÈ (Òåõíè÷åñêèé óíèâåðñèòåò),
÷ëåí-êîððåñïîíäåíò ÐÀÍ, çàñëóæåííûé äåÿòåëü íàóêè ÐÔ.
Обложка
Е. А. ВЛАСОВА
Охраняется законом РФ об авторском праве.
Воспроизведение всей книги или любой ее части запрещается без письменного
разрешения издателя.
Любые попытки нарушения закона
будут преследоваться в судебном порядке.
© Издательство «Лань», 2013
© Коллектив авторов, 2013
© Издательство «Лань»,
художественное оформление, 2013
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* b(x) ',-01( %
"
!
2$
" "
" $ %
"
" $
" "
dy
+ a(x)y = 0.
dx
!"
# $" b(x) ≡ 0
% & !" '
' %()
dy
⇒
= − a(x)dx ⇒
y
ln |y| = − a(x)dx + ln |C|.
dy
= −a(x)dx
y
* ' +
'
+
"
, %, - ,
, - + ('
' (b(x) ≡ 0)
", '
C ..
. z(x) C , ' ' /01203245 6
% , "
7 - , + (' '
y = z(x)e− a(x)dx .
8"
9 (: . z(x) :
% : 8" $" 9 '
. 8")
y = Ce−
−
y = z(x) e
a(x)dx
−
+ z(x)e
a(x)dx
.
a(x)dx
− a(x)dx =
;"
0 F2y = −λ
dt
/ " "" .
) ) / " 0)
y = 0 ! $ "
/
1 **#
"
ma =
F.
022 3
a
F
! F1 F2 " ! Oy # $ !
% &&' Oy " $
d2 y
a Oy 2 " ((
dt
$
m
m
d2 y
dy
= −ky − λ
dt2
dt
d2 y
dy
+ λ + ky = 0.
dt2
dt
&&)
" "
* &&)
! +(($ !
, " " ! !
- . " Oy
F (t) ( $ t
&&) "
d2 y
dy
m 2 + λ + ky = F (t)
&&&
dt
dt
!
/ % &&& m %
λ
k
F (t)
= 2b,
= ω2,
= f (t),
m
m
m
" ! 0 ! %
( 1
dy
d2 y
+ 2b + ω 2 y = f (t).
&&2
dt2
dt
* &&2
! ! (( $
!
" " ! +(($
/
! ! +
3 # " ! (b = 0)
(f (t) ≡ 0) + &&2 "
d2 y
+ ω 2 y = 0.
&&4
2
dt
k2 + ω2 = 0 k1,2 = ±ωi !
!
Y (t) = C1 cos ωt + C2 sin ωt.
"
# $ C1 C2
$ N > 0 ϕ % &
C1 C2 !
C1 = N sin ϕ,
C2 = N cos ϕ.
' N % ϕ () C1
C2 %*
N = C12 + C22 ,
+ (
C1
tg ϕ =
C1
·
C2
C2 " &
Y (t) = N sin ϕ cos ωt + N cos ϕ sin ωt = N sin(ωt + ϕ).
,& !
( $
Y (t) = N sin(ωt + ϕ).
- % & !
( & ) +
T = 2π
& N . & ϕ . $ %
ω
& ω
/ +$ $ (b = 0)&
( f (t) ≡ 0& ! ) # 0
1
d2 y
dy
+ 2b + ω 2 y = 0.
dt2
dt
k 2 + 2bk + ω 2 = 0
√
2
2
k1,2 = −b ± b − ω
23
4
& b < ω # 0
√ )
* k1,2 = −b + ω i& ω = ω2 − b2 5 !
23
Y (t) = e−bt (C1 cos ω
t + C2 sin ω
t) = N e−bt sin(
ωt + ϕ),
C1 = N sin ϕ, C2 = N cos ϕ
N e−bt → 0 t → +∞
y
0
t
b > ω
" #
Y (t) = C1 e(−b+
√
$$ %
b−ω 2 )t
+ C2 e(−b−
√
b−ω 2 )t
! "
.
& '
( )
(
Y (t) → 0 t → +∞ # ( (
b = ω
Y (t) = C1 e−bt + C2 te−bt .
* + ! " )
(b = 0)
f (t) = a sin μt ,
( $$-
d2 y
+ ω 2 y = a sin μt.
dt2
. '
ȳ(t)
Y (t) .
$$
! )
$$
.
$$/
y + ω 2 y = 0.
Y (t) = N sin(ωt + ϕ).
! μ " #
" ω $ " " " % μi
" & " "' k2 + ω2 = 0!
ȳ "
ȳ(t) = A sin μt + B cos μt.
(() ȳ *! ȳ (t) = −μ2(A sin μt + B cos μt)
" * ȳ(t) ȳ(t)
"%(() A B +
−μ2 (A sin μt + B cos μt) + ω 2 A sin μt + ω 2 B cos μt = a sin μt,
A = ω2 −a μ2 , B = 0 $ " ,!
"
ȳ(t) =
a
sin μt,
ω 2 − μ2
%'
-
, #
.,
/ ," " " *a ω!
, ω2 − μ2 ," " / " ω2 − μ2
," , 0 ω = μ , ( - ,
$ " " " % μi = ωi " & " "'
k2 + ω2 = 0!
"
a
sin μt + N sin(ωt + ϕ).
ω 2 − μ2
- ! μ
y(t) = ȳ(t) + Y (t) =
0
ȳ(t) = (A sin μt + B cos μt)t.
ȳ(t)
ȳ (t) = 2(μA cos μt − μB sin μt) + μt(−A sin μt − B cos μt)
! μ = ω!
"%(() A B +
2μA cos μt − 2μB sin +μ2 t(−A sin μt − B cos μt)+
+μ2 t(A sin μt + B cos μt) = a sin μt,
A = 0, B = −
ȳ = −
a
2ω
ȳ(t)
at
cos ωt,
2ω
y(t) = Y (t) + ȳ(t) = N sin(ωt + ϕ) −
at
cos ωt.
2ω
t
! " #
$ % &'
at
cos ωt ) & "
'( −
2ω
*" +
" !, $ $
$ ! $ !
y
0
t
- $! ''( +! %
" ! ! #
$ (
R L C
U = U (t) ! ! "#$!
L
b
c
R
C
U(t)
a
d
%
& U ' !
( )*+
, -
I = I(t)!
Uab + Ubc + Ucd = U (t).
. +
Uab = RI(t)
Ubc = L
&
/$
dI
1
, Ucd =
dt
C
t
I(t)dt.
0
dI(t)
1 t
+
I(t)dt = U (t).
dt
C 0
t
RI(t) + L
0++
2
R
dI
dI
1
+ L 2 + I = U (t)
dt
dt
C
d2 I R dI
I
+
= U (t).
+
dt2
L dt LC
I
! U " #
$% U = 0 &
$
I +
R
I
I +
= 0.
L
LC
'' (
2R = 1, 8 H = 2, 45
2r = 6
√
k 2gh
g = 10 2
! h
k " "#$
# % &'
) * + $ $ t ,
$ $ h(t) ,
" (-% ΔV = Sh = Sv Δt , . $!
Δt S , !
S = r2 πv(t) v = v(t) , √
! "v = 0, 6 2gh%
R
H
h(t)
r
v
Δt
ΔV = πR2 Δh ! " ΔV # #
$Δh < 0% #
# & & Δh Δt
πR2 Δh = −πr2 · 0, 6 2ghΔt.
π & "
# "" ' (
Δt → 0
R2 dh = −0, 6r2
2gh dt.
)
# *
" ' h = h(t)(
√
r2
2 h = −0, 6 2
R
t
2gt + C.
+ C # h = H
h " ' # C (
t = 0 +
√
C = 2 H,
" ' h
(
√
√
t
g r2
t
2 R2
$,, -%
& #
h = h(t)(
h−
h=
H = −0, 6
√
H − 0, 6
g r2
t
2 R2
2
.
. & tk #
& * * $,, -% t
# " h = 0 + #
R2
tk =
0, 6 r2
√
2 √
H− h
g
$,, /%
0 # $,, /%
# # ( R = 0, 9 H = 2, 45 r = 0, 03
g = 9, 8 122
# (
0, 92
tk =
0, 6(0, 03)2
2
9, 8
.
h=0
2, 45 ≈ 1006 ≈ 16, 7
.
! " # $ %
& ' ( # $ (
v = v(t)
dv
t v(0) = 2 m = F (t)
dt
F (t) !" m
# $ " % $$&
F (t) = −kv(t) k > 0 '((& $$&
# # $ $ )" ")*
!" " +, -
' , % $$ "% $, % $' ((*
& )"
m
dv
= −kv.
dt
. " ! " $% "
$%, " $ #
m
dv
= −kdt
v
=⇒
m ln |v| = −kt + ln |C|
-!
k
v = Ce− m t .
#
v(0) = 2 $' C = 2
k
v(t) = 2e− m t .
/
( # # 0 %
k
−4
k
: 1 = 2e m
v(4) = 1 ) $ #
m
k
ln 2
=
m
4
t
- V (t) = 21− 4 1" T # *
2 3 4 , "
T
T
0, 25 = 21− 4
2−2 = 21− 4 −2 = 1 −
t
s(t) =
t
0
!"
#
T = 12
t
8
1 − 2− 4 .
ln 2
t
0
21− 4 dt =
v(t) dt =
T
4
8
ln 2
11, 5
! !
" # $
# %! " &
'( ) g !
$2*
$$%
&
'
(
)
'
+
,
#
v(t)
Fdrag
t
*
*
'
'
-
(
Fdrag = kv 2 (t),
k
* .+
,
+
'
+
'
(
ma = mg − Fdrag ,
a
*
/
'
-
dv
dt
'
dv
= mg − kv 2 .
dt
k
= η )
m
1 dv
1
= g − v2.
η dt
η
m
a =
v
+
t
dv
1
= dt;
−
η v2 − 1 g
η
dv
1
·
t=−
η
v 2 − η1 g
! v(t)
v=
√
g 1 − e−2 ηgt
√
·
η 1 + e−2 ηgt
" # η$
%
&' ( $ ) t → +∞ v(t) !
&' ( $ "
t → +∞ * η
lim v = lim
t→+∞
50 =
t→+∞
g
,
η
√
g 1 − e−2 ηgt
√
=
η 1 + e−2 ηgt
η
1
=
,
g
2500
η=
g
,
η
g
·
2500
η v(t)
g
1 − e− 25 t
v = 50
g ,
1 + e− 25
v = 50
e0,2t − e−0,2t
1 − e−0,4t
≡
50
·
1 + e−0,4t
e0,2t + e−0,2t
+
,
%* *
ds
e0,2t − e−0,2t
= 50 0,2t
,
dt
e + e−0,2t
s = s(t)
t
s=
50
0
e0,2t − e−0,2t
dt =
e0,2t + e−0,2t
= 250 ln(0, 5(e0,2t + e−0,2t ))|t0 = 250 ln(0, 5(e0,2t + e−0,2t )).
0,2t
e
et + e−t ≈ et t
!
250 ln (0, 5(e0,2t + e−0,2t )) = 1000;
+e
= 2e4 ⇒ e0,2t ≈ 2e4 ⇒ 0, 2t ≈ 4 + ln 2;
t ≈ 20 + 5 · 0, 7 = 23, 5 .
−0,2t
""#$
%
#
& !
'
(
Ox# )
! #
AM ! * MO + # ,-#
y
K1
M(x,y)
A
K
N
0
B
x
y=f(x)
y = f (x) x
y = f (x) KK1 MN
! AM Ox ∠OKM =
= ∠AMK1 = ∠OMK = α " OKM # $
% O |OM| = |OK| $
OMB |OM| = x2 + y2 & '
|OK| !
Y − y = y (X − x).
Y = 0 & Ox'
X = x − yy
|X| = |OK| = −X = −x +
OM
y
,
y
x2 + y 2 = −x +
OK
y
·
y
(( )
y =
y
x+
x2 + y 2
,
%
* % (( ) %
y $
! y = f x +
y =
y
x+
x2
+
y2
≡
1+
y
y/x
.
≡f
2
x
1 + (y/x)
&,, -'
. ) $
&,, -' x +
x+
dx
=
x =
dy
x2 + y 2
x
= +
y
y
2
x
x
.
+1=f
y
y
&,, /'
0 &,, /' t = xy
x = ty x = ty +t
t y + t = t +
√
t2 + 1 =⇒ t y =
√
t2 + 1.
√
dt
dy
√
=⇒ ln |y| = ln |t + t2 + 1| + ln C
=
2
y
t +1
√
y = C(t + 1 + t2 )
x
! ! t = y "
⎛
⎞
2
x ⎠
x
y =C⎝ + 1+
.
y
y
#
2
y
=x+
C
x
x2 + y 2 .
4
2
y
2y x
+ x2 = x2 + y 2
−
C2
C
$# $
y2
y2
2x
−
2
C
C
C
= y 2 =⇒ y 2 = 2Cx + C 2 =⇒ y 2 = 2C x +
.
2
$" !
C " % − C2 , 0 " & #
' " " $
! ! " & $
(()
◦ ◦
◦ !
" #◦$ %& ' ( "
) "*
((( +, " ( '- ,.
/ 0 )
'-1 / 2"(
3 ( , . " 3 !
g = 10
R = 6
H = 10 !" # $ %
% "
& ' # (
#
%
√
) 0, 6 2gh " g = 10 2 *
%+ h * % ,
(" "
- %
) " ,
(
+ . " % ./
" " 0 % "
" h 1 " /
% R ! + ,
2 "
- " ,
% v +" % +
p % ,
+ ρ - % +
μ
!
- ,
% )
% ,
% % 3 +
) 3 v - ) % % r ) " %
+
3 ,
% ) ) -
#) %
3 ) )
4 ,
)
! 2a
3
% ) ) %
,
3 %
) - ,
%
!
" #
$
# ! #
% & !
" #
" ! #
' !
% ( #
# %
) % % * (
( # * %
+ k # s1, s2, . . . , sk &
$ n = s1 + s2 + . . . + sk
# # % (* #
! !
,-. n
⎧
dy1
⎪
⎪
⎪ dx = f1 (x, y1 , . . . , yn ),
⎪
⎪
⎪
⎨ dy2
= f2 (x, y1 , . . . , yn ),
/,-.0
dx
⎪
⎪
.....................
⎪
⎪
⎪
⎪
⎩ dyn = fn (x, y1 , . . . , yn ).
dx
1 n
/,-.0
y1 (x), y2(x), . . . , yn (x),
! "
# $ %
y1 (x), . . . , yn (x) $
y1 (x0 ) = y10 , . . . , yn (x0 ) = yn0 ,
x0 , y10 , . . . , yn0 ! $
"
yi = yi (x), i = 1, . . . , n,
#
$ $ (x, y1 , . . . , yn ) %
& & ! ' ( !
yi (x0 ) = y10 , i = 1, . . . , n, ' ( !
#
$ !
!% (x0 , y10 , y20 , . . . , yn0 ).
) ! %
% ( % %%
** + ( (
%, !% %
$ $ , %
' yi = yi (x, C1, C2, . . . , Cm ),
!
" ! ) C1, . . . , Cm,
" ) $ * ( ) " ! +
, $ m = n.
. ' Φi(x, y1, . . . , yn) = 0,
i = 1, . . . , n( & ( )
& +
' Φi(x, y1, . . . , yn, C1, . . . , Cm ) = 0( &
i = 1, . . . , n( ! & (
C1, . . . , Cm " ) $ * * & * *
#
i = 1, . . . , n(
G, (
m
⎧
d2 x
dx dy dz
⎪
⎪
,
m
t,
x,
y,
z,
,
,
=
F
⎪
x
⎪
⎪
dt2
dt dt dt
⎪
⎪
⎨ d2 y
dx dy dz
m 2 = Fy t, x, y, z, , ,
,
⎪
dt
dt dt dt
⎪
⎪
⎪
⎪
d2 z
dx dy dz
⎪
⎪
.
⎩ m 2 = Fz t, x, y, z, , ,
dt
dt dt dt
!"#
$ x(t), y(t), z(t) % v
a
dy
dz
dx
i + j + k,
dt
dt
dt
!&#
d2 x
d2 y
d2 z
i
+
j
+
k.
dt2
dt2
dt2
v=
a=
$
F = F x i + F y j + F z k
!
"
#
% "" &
'!&
#& "" &
#& & " )
*
m
(
x = x(t), y = y(t), z = z(t)
%
dx
dy
dz
, v(t) = , w(t) =
" u(t) =
dt
dt
dt
d2 y
dv d2 z
dw
,
=
=
dt2
dt dt2
dt
$ (
d2 x
du
,
=
2
dt
dt
% +
)
⎧
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩
du
1
= Fx (t, x, y, z, u, v, w),
dt
m
dv
1
= Fy (t, x, y, z, u, v, w),
dt
m
dw
1
= Fz (t, x, y, z, u, v, w),
dt
m
dx
= u(t),
dt
dy
= v(t),
dt
dz
= w(t).
dt
!
"
fi (x, y1, . . . , yn) i = 1, 2, . . . , n
∂f
, i, j = 1, 2, . . . , n
∂y
x, y1 , . . . , yn ! G ! ! "#
$ x0, y10, y20, . . . , yn0 % ! G % #
& y1 = y1 (x) y2 = y2 (x), . . . ,
yn = yn (x) $'% ( " $)
i
j
y1 (x0 ) = y10 , y2 (x0 ) = y20 , . . . , yn (x0 ) = yn0 .
#
$
%
$
&
' $
+
"
* $
,
"
"
$ , (
n
n (
+
)
$ *
( *
$ &
"
(
"
$ &
"
!!
⎧
dx
⎪
⎪
= y 2 + sin t,
⎨
dt
dy
x
⎪
⎪
⎩ = ·
dt
2y
d2 x
dy
= 2y + cos t.
" #$
dt2
dt
dy
= x, &! " #$
% ! 2y
dt
!' (
d2 x
− x = cos t.
dt2
" )$
* + ,
! ,&
! -+! , ,
+ ./ λ2 − 1 = 0 0 ,
λ1,2 = ±1 &! / !
X(t) = C1 et + C2 e−t 1 /!
!! +2 ,&
x̄ = A cos t + B sin t
& ' ' +
" )$ ! A = − 12 , B = 0 , +!
x̄ = − 21 cos t
3, ! ! / " )$ !!
x = C1 et + C2 e−t −
1
cos t.
2
% ! !
y2 =
dx
1
− sin t = C1 et − C2 e−t − sin t.
dt
2
3, ! ! !! !
x = C1 et + C2 e−t − 12 cos t,
y=±
C1 et − C2 e−t − 12 sin t.
⎧
dx
⎪
= z − y,
⎪
⎪
⎪
dt
⎪
⎨
dy
= z,
⎪
dt
⎪
⎪
⎪
⎪
⎩ dz = z − x.
dt
d2 x
dz dy
− ·
=
dt2
dt
dt
! "
d2 x
+ x = 0.
dt2
# $$ !
$ " $#
% $&
$ λ2 + 1 = 0 ' $ ! λ1,2 = ±i
(
x = C1 cos t + C2 sin t.
)* + "
dz
− z = −x
dt
dz
− z = −C1 cos t − C2 sin t.
dt
( (
dz
− z = 0 +
dt
z = C3 et .
, ( &
-" $#
z = A cos t + B sin t.
$#
A B !
B − A = −C1 ,
−B − A = −C2 .
C2 − C1
C1 + C2
, B=
.
2
2
1
1
z = (C1 + C2 ) cos t + (C2 − C1 ) sin t + C3 et .
2
2
A=
1
dx
1
= (C1 − C2 ) cos t + (C1 + C2 ) sin t + C3 et .
y=z−
dt
2
2
⎧
⎪
x = C1 cos t + C2 sin t,
⎪
⎪
⎪
⎪
⎨
1
1
y = (C1 − C2 ) cos t + (C1 + C2 ) sin t + C3 et ,
2
2
⎪
⎪
⎪
⎪
1
1
⎪
⎩z = (C1 + C2 ) cos t + (C2 − C1 ) sin t + C3 et .
2
2
! "
#$
# ! "
% &
% ' (
n #
% ) $
! #* %
+,%-
⎧
dx
y
⎪
⎨
=− ,
dt
t
⎪
⎩ dy = − x ,
dt
t
t > 0.
. ! $ $ "
d
1
(x + y) = − (x + y),
dt
t
#*
d(x + y)
dt
=− ,
x+y
t
x+y =
C1
·
t
d
1
(x − y) = (x − y),
dt
t
x − y = C2 t.
⎧
⎨
C1
,
t
⎩x − y = C t
2
#
x = x(t)
x+y =
y = y(t) !"
⎧
1 C1
⎪
⎪
x
=
+
C
t
,
2
⎨
2 t
1 C1
⎪
⎪
⎩ y=
− C2 t .
2 t
$% &
⎧
dx
⎪
⎨
= x2 y,
dt
⎪
⎩ dy = y − xy 2 .
dt
t
' ( # ) ! y
* x
y
dx
dy
xy
+x
=
dt
dt
t
d
xy
(xy) =
·
dt
t
+,
xy = C1 t.
-$% ./0
xy
dx
= C1 tx
C1 t
dt
t2
x = C2 eC1 2 C2 = 0
!"#$%
y=
C1 −C1 t2
C1 t
2 .
=
te
x
C2
&' ( ( )
⎧
2
⎨x = C2 eC1 t2 ,
t2
⎩y = C1 te−C1 2 .
C2
* x = 0
y = 0 x = C
y = Ct+
, - n .
. /- ' ../0
dyi
=
aij yi + fi (x),
dx
j=1
n
i = 1, . . . , n,
!"##%
aij = 12345 6 ' ' ../
.'/ fi (x) !"##%
fi(x) ≡ 0,
7
- ) yi (x) ≡ 0.
,
!"##% 0
⎧
dy1
⎪
⎪
= a11 y1 + a12 y2 + . . . + a1n yn ,
⎪
⎪
dx
⎪
⎪
⎪
⎪
⎨ dy2 = a21 y1 + a22 y2 + . . . + a2n yn ,
dx
⎪ .................................
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩ dyn = an1 y1 + an2 y2 + . . . + ann yn .
dx
!"#8%
A n
dy
y
!
dx
⎛
a11
⎜ a21
A=⎜
⎝ ...
an1
a12
a22
...
an2
#
⎞
. . . a1n
. . . a2n ⎟
⎟,
... ... ⎠
. . . ann
⎛
⎛
⎞
⎜
⎜
dy ⎜
=⎜
dx ⎜
⎜
⎝
y1
⎜ y2 ⎟
⎟
y = ⎜
⎝ """ ⎠ ,
yn
dy1
dx
dy2
dx
""
"
$%&"''(
+
⎟
⎟
⎟
⎟.
⎟
⎟
⎠
dyn
dx
!
dy
= Ay.
dx
*
⎞
$%&"')(
n
$%&"',(
y1 = α1 eλx , y2 = α2 eλx , . . . , yn = αn eλx ,
- αk -
λ "
αk λ
yk = λαk eλx , k = 1, . . . , n
$%&"'.(" /
α1 , α2 , . . . , αn !
yk
$%&"',(
-
⎧
(a11 − λ)α1 + a12 α2 + . . . + a1n αn = 0,
⎪
⎪
⎪
⎪
⎨ a12 α1 + (a22 − λ)α2 + . . . + a2n αn = 0,
⎪
....................................
⎪
⎪
⎪
⎩
an1 α1 + an2 α2 + . . . + (ann − λ)αn = 0.
$%&"'0(
1 $%&"'0( +
2 " "
α1 , α2 , . . . , αn
(a11 − λ)
a12
a21
(a22 − λ)
...
...
an1
an2
...
a1n
...
a2n
...
...
. . . (ann − λ)
= 0.
$%&"'%(
n λ
! " #
n λ ! $ %
& # n ' λ1 & λ2 & . . . & λn & (
) A *
! + '
|A − λE| = 0,
E , ) n & -!
.
(A − λE) · b = 0,
b = (α1, α2 , . . . , αn) , ) A
) A b& (/
( Ab = λb!
+ &
!& ) A&
& + (
& % #
0 & ( λk
(α1k , α2k , . . . , αnk )& k = 1, 2, . . . , n * # 11
) n .
'
2 . y1 (x)& (/ ( λ = λ1& n
1) '
y11 = α11 eλ1 x , y21 = α21 eλ1 x , . . . , yn1 = αn1 eλ1 x ;
. y2 (x)& (/ ( λ = λ2& n
1) '
y12 = α12 eλ2 x , y22 = α22 eλ2 x , . . . , yn2 = αn2 eλ2 x ;
................................................................................................
n2 . yn (x)& (/ ( λ = λn & n
1) '
y1n = α1n eλn x , y2n = α2n eλn x , . . . , ynn = αnn eλn x .
y1(x), y2 (x), . . . , yn(x)
y = C1 y1 (x) + C2 y2 (x) + . . . + Cn yn (x)
⎧
y = C1 y11 (x) + C2 y12 (x) + . . . + Cn y1n (x),
⎪
⎪
⎨ 1
y2 = C1 y21 (x) + C2 y22 (x) + . . . + Cn y2n (x),
.................................
⎪
⎪
⎩ y = C y (x) + C y (x) + . . . + C y (x),
n
1 n1
2 n2
n nn
!" #
$ % yij (x)
⎧
y = C1 α11 eλ1 x + C2 α12 eλ2 x + . . . + Cn α1n eλ1 x ,
⎪
⎪
⎨ 1
y2 = C1 α21 eλ1 x + C2 α22 eλ2 x + . . . + Cn α2n eλ2 x ,
..........................................
⎪
⎪
⎩ y = C α eλ1 x + C α eλn x + . . . + C α eλn x .
n
1 n1
2 n2
n nn
!"&#
' (
) !" #
# *
) ) * +
) ) n) + %
! % #!
!,
⎧
dy
⎪
⎪
= −y − 2z,
⎨
dx
dz
⎪
⎪
= 3y + 4z.
⎩
dx
- . / + % a11 = −1, a12 = −2,
% y1 = y,
0 (
!" #.
a21 = 3, a22 = 4*
(−1 − λ)
−2
3
(4 − λ)
y2 = z !
= 0 ⇒ λ2 − 3λ + 2 = 0.
1 λ1 = 1, λ2 = 3!
2 y = α1 eλx, z = α2eλx *
λ1 = 1! 3 !",#* α1 , α2*
(−1 − λ)α1 − 2α2 = 0,
3α1 + (4 − λ)α2 = 0,
λ = λ1 = 1
!
!
"
#
α1 + α2 = 0,
α1 , α2 $
α1 = 1 α2 = −1 $ %
λ = 1
&
y1 (x) = ex ,
z1 (x) = −ex .
' !
"
λ = λ2 = 2
α1 = 2, α2 = −3 "
(
%
)
&
y2 (x) = 2e2x ,
z2 (x) = −3e2x .
* + , &
&
y = C1 y1 (x) + C2 y2 (x) = C1 ex + 2C2 e2x ,
z = C1 z1 (x) + C2 z2 (x) = −C1 ex − 3C2 e2x .
' ! ( # # & # (
( "
. %,, .
# ) ( ( "
( # #(
⎧
dx
⎪
⎪
= y,
⎨
dt
dy
⎪
⎪
= −x.
⎩
dt
d2 x
dy
=
dt2
dt
dy
dt
d2 x
+ x = 0.
dt2
λ2 + 1 = 0 λ1,2 = ±i
! x = C1 sin t + C2 cos t"
!
# $%&"'(
y = C1 cos t − C2 sin t" )* $%&"'(
x = C1 sin t + C2 cos t,
y = C1 cos t − C2 sin t.
%&"+
⎧
dx
⎪
⎪
= y,
⎪
⎪
⎪
⎨ dt
dy
= x,
⎪
dt
⎪
⎪
⎪
⎪ dz
⎩
= x + y + z.
dt
,
dy
d2 x
=
dt2
dt
-
d2 x
− x = 0.
dt2
)
λ2 − 1 = 0 *
λ1,2 = ±1 !
x = C1 et + C2 e−t .
dx
= C1 et − C2 e−t .
dt
x(t) y(t)
z
y=
dz
− z = 2C1 tet .
dt
z = C3 et + 2C1 tet .
!"#
⎧
⎨ x = C1 et + C2 e−t ,
y = C1 et − C2 e−t ,
⎩
z = C3 et + 2C1 tet .
$%&'()
%&'*
⎧
dx
x
⎪
⎪
=
,
⎨
dt
2x + 3y
dy
y
⎪
⎪
=
·
⎩
dt
2x + 3y
+
,
, " , +
dx
x
dx
dy
= ⇒
=
⇒ ln |x| = ln |y| + ln |C1 | ⇒ x = C1 y.
dy
y
x
y
-
' , ,
" , #. +
2
dy
2x
3y
dx
dy
dx
+3 =
+
⇔2 +3
=1⇔
dt
dt
2x + 3y 2x + 3y
dt
dt
⇔ d(2x + 3y) = dt ⇒ 2x + 3y = t + C2 .
x = C1 y,
2x + 3y = t + C2 ,
⎧
C1 (t + C2 )
⎪
⎪
,
⎨ x=
2C1 + 3
⎪
⎪ y = t + C2 ·
⎩
2C1 + 3
⎧
dx
⎪
⎪
= y − z,
⎪
⎪
⎪
⎨ dt
dy
= x + y + t,
⎪
dt
⎪
⎪
⎪
⎪ dz
⎩
= x + z + t.
dt
! "#$ # $ %
#
d
(y − z) = y − z,
dt
& $
ln |y − z| = t + ln |C1 |,
& $ #
y − z = C1 et .
' #( ) $% & $
% $
x = C1 et + C2 .
* +
*# $ y!
x = x(t)
dy
= y + C2 + C1 et + t.
dt
y = (C1 t + C3 )et − t − 1 − C2 .
x y
z=y−
dx
= (C1 t + C3 − C1 )et − 1 − C2 .
dt
!"#
⎧
x = C1 et + C,
⎪
⎪
⎨
y = (C1 t + C3 )et − t − 1 − C2 ,
⎪
⎪
⎩
z = (C1 t + C3 − C1 )et − 1 − C2 .
$%&'
⎧
dy
⎪
⎨
= 5y − z,
dx
⎪
⎩ dz = y + 3z.
dx
( ) "#
* +$%&,,-
a11 = 5, a12 = −1, a21 = 1, a22 = 3, "
y1 = y, y2 = z & . +$%&,$
(5 − λ)
−1
1
(1 − λ)
= 0 ⇒ λ2 − 8λ + 16 = 0 ⇒ (λ − 4)2 = 0
/ λ1,2 = 4 0&
" /
&
* λ1,2 = 4
y1 ≡ y(x) = e4x (A1 x + A2 ),
y2 ≡ z(x) = e4x (B1 x + B2 ).
1 y(x)
dy
= A1 e4x + 4(A1 x + A2 )e4x ,
dx
z(x)
dz
= B1 e4x + 4(B1 t + B2 )e4x ,
dx
dy
dz
,
, y
z
dx dx
e4x = 0
A1 + 4(A1 x + A2 ) ≡ 5(A1 x + A2 ) − (B1 x + B2 ),
B1 + 4(B1 x + B2 ) ≡ A1 x + A2 + 3(B1 x + B2 ).
x
A 1 , A 2 , B1 , B2
4A1 = 5A1 − B1 ,
4B1 = A1 + 3B1 ,
A1 + 4A2 = 5A2 − B2 ,
B1 + 4B2 = A2 + 3B2 .
A1 = B1
A2 − B2 = A1 ! A1 = C1 A2 = C2 B1 = C1 , B2 =
= A2 − A1 = C2 − C1 "# $ %
⎧
⎨ y = e4x (C1 x + C2 ),
⎩ z = e4x (C1 x + C2 − C1 ).
&'&
⎧
dy
⎪
⎪
= 2y − z,
⎨
dx
dz
⎪
⎪
= y + 2z.
⎩
dx
(%
) $
*&'++,
a11 = 2, a12 = −1, a21 = 1, a22 = 2, $
y1 = y, y2 = z - *&'+&,
(2 − λ)
−1
1
(2 − λ)
= 0 ⇒ λ2 − 4λ + 5 = 0
λ1 = 2 + i, λ2 = 2 − i
. %
y = α1 e(2+i)x ,
$
z = α2 e(2+i)x ,
λ1 = 2 + i
α1 , α2
!
(2 − λ)α1 − α2 = 0,
α1 + (2 − λ)α2 = 0,
" #" ! λ = λ1 = 2 + i #$ %& $ & %
# "$$! $ "' "
−iα1 − α2 = 0,
# " %$ α1 , α2 ! # $! ($
α1 = 1 #$% α2 = −i ( % λ = λ1 =
= 2 + i % )
y(x) = e(2+i)x ≡ e2x (cos x + i sin x),
z(x) = −ie(2+i)x ≡ e2x (sin x − i cos x).
* " "$'' % #$% +
) % # +" + ) + $
' $ ,, '$'' )
#$% " $ + + % + ) '
y1 (x) = e2x cos x,
z1 (x) = e2x sin x,
y2 (x) = e2x sin x,
z2 (x) = −e2x cos x.
* ) " $ ' ' #$%
+ + % + ) # $!
y = e2x (C1 cos x + C2 sin x),
z = e2x (C1 sin x − C2 cos x).
⎧
dx
⎪
⎪
= x + y,
⎨
dt
dy
⎪
⎪
= 3y − 2x.
⎩
dt
⎧
dx
⎪
⎪
= y + 1,
⎨
dt
dy
⎪
⎪
= 2et − x.
⎩
dt
⎧
dx
⎪
⎪
= y + z,
⎪
⎪
⎪
⎨ dt
dy
= x + z,
⎪
dt
⎪
⎪
⎪
⎪ dz
⎩
= x + y.
dt
⎧
dx
⎪
⎨
=
dt
⎪
⎩ dy =
dt
! "#
$
x
,
x+y
y
.
x+y
$ %! &
⎧
dx
⎪
⎪
= y − z,
⎪
⎪
⎪
⎨ dt
dy
= x2 + y,
⎪
dt
⎪
⎪
⎪
⎪
⎩ dz = x2 + z.
dt
! "# &
$ %! ' & & $
(! ' x $
⎧
dx
⎪
⎪
= 7x + 3y,
⎨
dt
dy
⎪
⎪
= 6x + 4y.
⎩
dt
⎧
dx
⎪
⎪
= x − y,
⎨
dt
dy
⎪
⎪
= x + 3y.
⎩
dt
⎧
dx
⎪
⎨
= x − 2y,
dt
dy
⎪
⎩
= x − y.
dt
⎧
dx
⎪
⎨
= x − 3y,
dt
⎪
⎩ dy = 3x + y.
dt
⎧
dx
⎪
⎨
= 2x + y,
dt
⎪
⎩ dy = 4y − x.
dt
y − y = e2x cos x
! "
#$ % & '!! ( ) * ! ""&!
"* +)
, - , . ( /!
*) ( !!
. & *)
"& " &
0!
* )!&1 ) 2 )!&31 "
)!
4)!& !" *$ )!& "! 5
)! !1 * ! *& 1 !!
- "* $ &)! $ !&
+ & )!&
"* & ! 6 !" * !
! * $#! " 1
1 ' "
7 ! 8 ! ) ! " )
" )!& " * 9 &
" ! "& & ! " )
)!& 6 & & !! 1
!" #
y =
y−x
y+x
1
y
ln(y 2 + x2 ) + arctg = C.
2
x
$!" #%
$!" &%
' ( y(x)
(
$!" &%
)) ( $!" #% *
+
,
- $!" #%
)
$ &.% / (r, ϕ)
$!" &% r = Ceϕ
,
(
) (
,
0
1 +
(
/ ,
2 y(x)
un (x) n → +∞ , un (x) +
) ( 3
n ,
y(x) *
,
!!" # $
% #
!!" &
%
y(x) % '
Dh : {xn }
"
# ( % )
)
% * % $
+
% , '
%
(
% # % -
% %
% .'
%
'
% '
%
y = y − x,
y(0) = 1.
0 x 100,
y = 1 + x + Cex .
y(0) = 1 C = 0 !
y(100) = 101"
y(0) = 1,000001 # C = 10−6
y(100) = 2,7 · 1037 "" "
# $$ %
& !
# " '($$ % & (
&
) * % +,
$ *+,"+-.." / # !
$$ % &
* ' .
F (x, y, y , y , . . . , y (n−1) , y (n) ) = 0,
*01"-.
*01"2.
3 & & $$ %
(n−1)
y(x0 ) = y0 , y (x0 ) = y0 , y (x0 ) = y0 , . . . , y (n−1) (x0 ) = y0
(n−1)
F (x0 , y0 , y0 , y , . . . , y0
(n)
, y0 ) = 0,
n!
x = x0
.
*01"4.
5 $$ % *01"-.
x, y, y, y, . . . , y(n) (n + 1)!
y(n+1)
F1 (x, y, y , y , . . . , y (n) , y (n+1) ) = 0.
*01"0.
(n)
(n−1)
y0 = f (x0 , y0 , y0 , y0 , . . . , y0
).
(n)
y0
(n)
(n+1)
F1 (x0 , y0 , y0 , y0 , . . . , y0 , y0
) = 0,
! n + 1"
(n+1)
x = x0 # y (n+1) (x0 ) = y0
! $ % & ! "
& x = x0
)
'& y = y(x) (( %
! * #
y(x) = y(x0 )+y (x0 )
x − x0
(x − x0 )2
(x − x0 )n
+y (x0 )
+. . .+y (n) (x0 )
+. . .
1!
2!
n!
+
+ x0
& , "
,!- & (( %"
#
n
x − x0
(x − x0 )2
(n) (x − x0 )
+ y0
+ . . . + y0
+ ...,
1!
2!
n!
"
-
.
y(x) = y0 + y0
⎧
⎨ y = 2xy + 4y,
y(0) = 0,
⎩
y (0) = 1.
/
' & # 0 (( %
(( % "
1 $ !
x0 2
3 / y = 2xy + 4y ⇒ y0 = 2 · 0 · 1 + 4 · 0 = 0
4(( %
/
y = 2y + 2xy + 4y = 6y + 2xy ⇒ y0 = 6 · 1 + 2 · 0 · 0 = 6.
56
4(( %
56
IV
y = 6y + 2y + 2xy = 8y + 2xy ⇒ y0IV = 8 · 0 + 2 · 0 · 6 = 0.
55
V
y = 8y + 2y + 2xy
IV
= 10y + 2xy IV ⇒ y0V = 10 · 6 + 2 · 0 · 0 = 60.
y0V I = 0, y0V II = 840
! " " x0 = 0 #$
% " & '
y=
!
6
1
60
840 7
x + x3 + x5 +
x + ...
1!
3!
5!
7!
y =x+
x3 x5 x7
+
+
+...
1!
2!
3!
( )* * ! % " "
+) , C0, C1, C2, . . . , Cn, . . .'
+∞
Cn (x − x0 )n .
n=0
- +) , ) Cn(n = 0, 1, 2, . . .) .) ! $
) + " " * ! "$
"" , " ")* * .$
(x − x0) " " " * " "
/ * % " " $
+)* , " /0 "
" #)"
1
y + y + y = 0.
x
1
2 x = 0 " 3 . !
x = 0 0 "" "
4% . ! 0!/ )* "'
y = c0 + c1 x + c2 x2 + c3 x3 + . . . + cn xn + . . .
, /0 ")$
#" )*'
y = c1 + 2c2 x + 3c3 x2 + . . . + ncn xn−1 + . . .
y = 2c2 + 2 · 3c3 x + 3 · 4c4 x2 + . . . + (n − 1)ncn xn−2 + . . .
(2c2 + 2 · 3c3 x + 3 · 4c4 x2 + . . . + (n − 1)ncn xn−2 + . . .)+
1
+ (c1 + 2c2 x + 3c3 x2 + . . . + ncn xn−1 + . . .)+
x
+(c0 + c1 x + c2 x2 + c3 x3 + . . . + cn xn + . . .) = 0.
x! " #
$ c0, c1, c2 , . . .
%"& " '( x1 ) &
1
c1 ! c1 = 0! $ " "
*+
x
! " , '! c1 = y x=0= 0
- y - x = 0 - 3
. , "& x, x ! x5
2 · 3c3 + 3c3 + c1 = 0,
4 · 5c5 + 5c5 + c3 = 0,
6 · 7c7 + 5c7 + c5 = 0.
/ c1 = 0! "&
"
c3 = 0,
c5 = 0,
...,
c2k+1 = 0.
$ + "&
0 ( x0, x2, x4, . . .
2 · c2 + 2c2 + c0 = 0,
3 · 4c4 + 4c4 + c2 = 0,
5 · 6c6 + 4c6 + c4 = 0.
% , +
(n + 2)(n + 1)cn+2 + (n + 2)cn+2 + cn = 0
. -
c2 = −
c2
c0
c4
c0
= 2 2 ; c6 = − 2 = − 2 2 2 ;
2
4
24
6
246
c0
c0
n
n
= (−1) = 2 2
= (−1) n
·
2 4 . . . (2n)2
2 (n!)2
c0
;
22
c2n
(n + 2)2 cn+2 + cn = 0.
" - c0
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1
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π
π
π
π
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a0
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dx +
f (x) dx =
2
n=1
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−π
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π −π
,
k∈N
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.
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a0 π
f (x) cos kx dx =
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2 −π
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π
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n=1
−π
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n=k
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1 π
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π −π
1 π
f (x) sin nx dx.
bn =
π −π
a0 =
.
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! ," 0 !
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f (x)
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x → x0
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x→x0 +
2 x→x0−
x ∈ (−π, π]
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πn2 π
π
1
1 π
1
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x sin nx dx = −
xd cos nx = −
π −π
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nπ
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π
π
1
1
=
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π cos nπ + π cos(−nπ) − sin nx
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n
−π
−π
n
2
2(−1)
= − cos nπ = −
.
n
n
a0 =
0 ' ,# '!
1 &
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2(−1)n−1
.
n
(−1)n−1
sin x sin 2x sin 3x
−
+
− ··· +
sin nx + . . .
f1 (x) = 2
1
2
3
n
.
f1 (x) = 2
+∞
(−1)n−1 sin nx
n
n=1
.
! " ϕ(x)
ϕ(−x) = ϕ(x)
π
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0
π
π
π
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−π
0
0
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π
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0
0
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0
0
π
π
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0
0
0
0
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a0 = 0,
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2
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0
f (x) =
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n=1
bn sin nx.
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π
2
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a0
+
an cos nx.
2
n=1
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2
n=1
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#& & $,$
2
π
π
f (x)dx,
0
, ,
f
$",
π
z = x
l
2π
x ∈ [−π, π]3
$" ) $
" +
2
π
π
f (x) sin nxdx,
(! 1
0
"#,
f (x)
"#
$
l
f ( z)
π
1
2l
2l
/
0
2
+∞
1 a
0
z =
+
(an cos nz + bn sin nz) ,
π
2
n=1
#& , 3
1 π l
1 π l
z dz,
an =
f
f z cos nzdz,
π −π
π
π −π
π
1 π l
f z cos nzdz.
bn =
π −π
π
a0 =
-
an = 0.
. $$
"
bn =
%
%
x = πl z, z = x πl ,
π
dz = dx z = −π, x = −l z = π, z = l
l
f (x)
2l
nπ
1 l
1 l
x dx,
f x)dx,
an =
f (x) cos
l −l
l −l
l
l
1
nπ
x dx.
f (x) sin
bn =
l −l
l
!"#$%
+∞
a0
nπ
nπ
an cos
+
x + bn sin
.
f x =
2
l
l
n=1
!""'%
a0 =
&
( f (x)
a0 = an = 0;
bn =
2
l
l
f (x) sin
0
2l )
nπ
x dx.
l
!""*%
+
+∞
nπ
bn sin
.
f x =
l
n=1
!""#%
( , f (x)
a0 =
2
l
l
f (x)dx,
0
an =
2
l
l
f (x) cos
0
nπ
x dx,
l
2l
)
bn = 0.
!"""%
+
+∞
a0
nπ
an cos
+
x .
f x =
2
l
n=1
!""-%
l
!"#$% &
' ()$ x = 0
x = l * +
∂u
∂ 2u
= a2 2 ,
∂t
∂x
x ∈ [0, l],
t > 0,
u(x, t)|t=0 = ϕ(x).
, '
' .
- '%
u(x, t) = X(x)T (t).
* +
!"#$.
/"$
∂u
∂ 2u
= X(x) · T (t),
= X (x) · T (t) ⇒ X(x)T (t) = a2 X (x)T (t),
2
∂t
∂x
T (t)
X (x)
=
. 2
a T (t)
X(x)
* ' ' x%
0 t% x% t % ' % +
.
T (t)
X (x)
=
= C.
a2 T (t)
X(x)
1
/!$
.
T (t) − a2 CT (t) = 0,
X (x) − CX(x) = 0.
/($
T (t) = BeCa t
t → +∞ u(x, t)
C < 0 C = −λ2
!"# $
T (t) = Be−λ a t .
!!#
2
% & C = −λ !"#
X (x) + λ2 X(x) = 0 $
X(x) = C1 cos λx + C2 sin λx,
!'#
2
2
s + λ = 0
( s1,2 = ±λi
) ( * +
* ,-
2
2 2
2 a2 t
u(t, x) = Be−λ
(C1 cos λx + C2 sin λx).
( * .* * +
C1 C2 * ( * - *
!// 0 +
-
u(t, x)|x=0 = 0.
1 2 ( * " # & !/#
X(0)T (t) = 0,
X(l)T (t) = 0.
) 2 ( * ( ,- T (t) +
(
X(0) = X(l) = 0.
!3#
4 X(x) * ( +
* !3# !3# !'#$
X(0) = 0
⇒
X(l) = 0
C1 = 0
⇒
C1 cos λl + C2 sin λl = 0
⇒ sin λl = 0 ⇒ λl = kπ, k ∈ Z ⇒ λk =
C1 = 0
⇒
C2 sin λl = 0
kπ
kπx
⇒ Xk (x) = Ck sin
,
l
l
Ck 5 * & (
k = 1, 2, 3, . . .
1( λk = kπ/l ( * ,+
- sin(kπx/l) 5
Tk (t)
kπ
2 2
; Tk (t) = Bk e−λk a t .
l
!" Xk (x) Tk (t)
λk =
uk (x, t)
&#'$
#$ %
%
(
2 2
uk (x, t) = Bk Ck e−λk a t sin
b k = Bk C k
u(x, t)
)
kπx
kπx
2 2
= bk e−λk a t sin
,
l
l
&#'$
* + ,
! %
-$ . ! ,
(
u(x, t) =
+∞
uk (x, t) =
k=1
0
/
*
*
!1 .
uk (x, t)
3
(k = 1, 2, . . . )
bk e−(kπ/l)
$
!$ .
4! .$
/
&#' !"
u(x, t)
* %
bk
, ,
ϕ(x)
%
+∞
bk sin
-(
kπx
= ϕ(x).
l
bk !2 " 1 %
(0, l) 5 , (
*
bk =
,
kπx
.
l
. , !
k=1
sin
u(x, 0) =
!"
2 a2 t
k=1
$ ! ,!
2
$
+∞
,$
2
l
l
ϕ(x) sin
0
-
*
kπx
dx.
l
6
&#'
* !" +
%
/$ !2 " bk 6
, ,
!"
1
$
u|x=0 = h0 ,
(
u|x=l = h1 ,
#7
u(x, t) w(x, t)
u(x, t) = w(x, t) + γ1 x + γ0 ,
γ0 γ1
w(x, t)
!
"
γ0 γ1
u|x=0 = w|x=0 + γ0 ,
u|x=0 = h0 ;
w|x=0 = 0 ⇒ h0 = γ0 ;
u|x=l = w|x=l + γ1 l + γ0 , u|x=l = h1 ; w|x=l = 0 ⇒ h1 = γ1 l + γ0 .
#$
γ1 γ0
γ0 = h0 ;
γ1 =
h1 − h0
.
l
%
h1 − h0
x + h0 .
&'( ))*
l
∂u
∂ 2u
∂w
∂ 2w
=
+
,
=
w(x, t)
2
∂t
∂t
∂x
∂x2
"
&'- ).*
u(x, t) / &'0 1*
u(x, t) "
w(x, t)
u(x, t) = w(x, t) +
h1 − h0
x − h0 ,
l
&'( )2* 3
w(x, t)|t=0 = ϕ(x) −
w|x=0 = 0;
w|x=l = 0.
&'( )-*
&'( )0*
#$
&'( )-*
&'( )0* $ w(x, t)
#$ u(x, t)
&'( )2*
&'( ))*
'( ) -
#
4
5 "
&'0 6*
∂u(t, x)
∂u(t, x)
|x=0 =
|x=0 = 0.
∂x
∂x
! " # $ % &
'( ' ( C " & (
% () C = −λ2
* ' $ " +
,-
* ' $ "
. "/ 0 )
X (0)T (t) = 0,
X (l)T (t) = 0.
( ! 1(2 3%$ T (t)
1 )
X (0) = X (l) = 0.
4$ X(x) !1- * ( +
! ! * $ X (x) ( +
& * ( ! !)
X (x) = −λC1 sin λx + λC2 cos λx.
X (0) = 0 ⇒ λC2 = 0 ⇒ C2 = 0.
X (l) = 0 ⇒ λ(−C1 sin λl + C2 cos λl) = 0 ⇒ sin λl = 0 ⇒
kπ
kπx
⇒ Xk (x) = Ck cos
.
l
l
' $ ( Ck
⇒ λl = kπ, k ∈ Z ⇒ λk =
#( ! ' - 1
+
k = 0, 1, 2, 3, . . . ! !
k = 0 ( 3%! ! !1 ! (
cos 0 = 1 = 0 5 1- ( λk = kπ/l
! Tk (t) !1 ! 3 ")
2 2
Tk (t) = Bk e−λk a t .
!! 3% Xk Tk (t) 3 +
uk (x, t) ! !1- * ( !
"/)
2 2
uk (x, t) = Bk Ck e−λk a t cos
kπx
kπx
2 2
= ak e−λk a t cos
,
l
l
* ak = Bk Ck k = 1, 2, 3 . . .
a
a
, ' B0 C0 = 0 ⇒ u0 (x, t) = 0
2
2
6
uk (x, t)
! "# $ % "& ' %
(
u(x, t) =
+∞
kπx
a0
2 2
+
,
ak e−(kπ/l) a t cos
2
l
k=1
+∞
uk (x, t) =
k=0
)
a0/2 ' λ0 = 0 *
+ ', - ./
"0 1' ' + % %
"&(
a0
kπx
+
= ϕ(x).
ak cos
2
l
k=1
+∞
2 '3 ak -../ 3,
./ ϕ(x) 3! (0, l) 4% +
2
ak =
l
l
ϕ(x) cos
0
kπx
dx.
l
)0
5 % ++
% "& +3 / $
) -../ ak + + . )0
)
ϕ(x) = u0
0 x l ! " u(x, t)
( 3 +3 /
$ ) -../ + + .
)0(
l
2u0
kπx
2u0 + k = 0
dx =
ak =
cos
0
+ k > 0 .
l
l
0
1 . ) + u(x, t) = a0 /2 = u0
1 ! 3% + +! +3
/ , + % +
$
+ t > 0
! " #
$ #
∂u(t, x)
|x=0 = u(x, t)|x=0 = 0.
∂x
% & '( " #
) * +, & # - # .
/ 0
X (0) = 0,
1
X(l) = 0.
2 & X(t) # ,
# 1 #
" 3 # X (x) x = 0
X (0) = 0 ⇒ λC2 = 0 ⇒ C2 = 0 " 3
x = l X(l) = 0 ⇒ C1 cos λl + C2 sin λl = 0 ⇒
π
⇒ C1 cos λl = 0 ⇒ cos λl = 0 ⇒ λl = + kπ k ∈ Z ⇒
2
(2k + 1)π
(2k + 1)πx
⇒ Xk (x) = Ck cos
.
⇒ λk =
2l
2l
+# ,( ( Ck
k = 0, 1, 2, . . . 4 , +
λk = (2k + 1)π/(2l) & # Tk (t) # # 5
% ## 5 Xk (x) Tk (t) 5 &
# )3 # ,
# 0
(2k + 1)πx
(2k + 1)πx
2 2
= ak e−λk a t cos
,
2l
2l
).
ak = Bk Ck k = 0, 1, 2, 3, . . .
6 & uk (x, t) # )3 # ,
* ,/ ( # 7 +
( #0
2 2
uk (x, t) = Bk Ck e−λk a t cos
u(x, t) =
+∞
k=0
uk (x, t) =
+∞
k=0
ak e−((2k+1)π/2l)
2 a2 t
cos
(2k + 1)πx
.
2l
)
8 (&
+ (# $ 5 # # # # &
# )3 # , #
ak (k = 0, 1, 2, . . . )
+∞
ak cos
k=0
(2k + 1)πx
= ϕ(x).
2l
! " #
$ %& cos (2k +2l1)πx "
(0, l) ' #
l
(2k + 1)πx
(2n + 1)πx
l/2 k = n
cos
dx =
cos
k = n .
0
2l
2l
0
( $ )$$% *
+ cos (2k +2l1)πx "
(0, l) , $ )$$%
ak =
2
l
l
ϕ(x) cos
0
(2k + 1)πx
dx.
2l
-. /0
" // , / "
)$$% 1 $
l
t > 0
u|t=0 = l − x
- . 2 . 1 #
"
∂u
= u|x=l = 0.
0
∂x x=0
u|t=0 = l − x
-. , / " )$$% ak 1
$
ak =
2
l
l
0
(l − x) cos (2k+1)πx
dx =
2l
4
(l
(2k+1)π
l
− x) sin (2k+1)πx
+
2l
0
4
+ (2k+1)π
−
=
l
0
8l
(2k+1)2 π 2
8l
(2k+1)2 π 2
sin (2k+1)πx
dx =
2l
cos (2k+1)πx
2l
l
0
u(x, t) =
8l
π2
k=0
l
− x) sin (2k+1)πx
−
2l
0
8l
cos (2k+1)π
= − (2k+1)
−1 =
2 π2
2
4
(l
(2k+1)π
.
+∞
1
(2k + 1)πx
2
2 2
2
.
e−a (2k+1) π t/4l cos
(2k + 1)2
2l
(U) m
1
(U) m 0,8
(U) m 0,6
(U) m 0,4
(U) m 0,2
(U) m
0
0,2
0,4
xm
0,6
0,8
1
! " # #!
$% ! & '& # $
& t → +∞ ! # ( )( "*
# !
#+ , ( # ! +'
" )& # # *
# ! ) & ) -)# -#) %
& Ox .
)
u(x) ( t! -#
! " t
" xm, m = 0, 1, · · · M
x ! # $ %
% "% & # #
"# "# #
#% ' ! " # "
()*+,)- ()./0) 1
#! 2 "% "
# 2 ak
3
#! "
!# #
" #
"
35 u|x=0 = u|x=l = 0.
u(x, t) =
bk =
5
∂u
∂x
65
∂u
∂x
+∞
bk e−(kπ/l)
2 a2 t
k=1
l
2
l
∂u
∂ 2u
= a2 2
∂t
∂x
u|t=0 = ϕ(x)
ϕ(x) sin
0
sin
4
kπx
l
kπx
dx k = 1, 2, . . .
l
∂u
= 0.
∂x x=l
x=0
+∞
a0
kπx
2 2
u(x, t) =
+
ak e−(kπ/l) a t cos
2
l
k=1
l
2
kπx
dx k = 0, 1, 2, . . .
ak =
ϕ(x) cos
l
l
=
0
= ux=l = 0.
x=0
+∞
u(x, t) =
k=0
ak e−λk a t cos λk x
2 2
2
l
ak =
l
ϕ(x) cos λk xdx λk =
0
(2k + 1)π
.
2l
! " #$ f (x)
%
1 +∞
|f (x)|dx &
l −∞
( #$ f (x)
&
' f (x)
)
& &
+∞
f (x) =
(A(α) cos αx + B(α) sin αx)dα,
'*+ ,*(
0
& "
-
f (x − 0) + f (x + 0)
=
2
.)## $
A(α)
+∞
(A(α) cos αx + B(α) sin αx)dα.
−∞
B(α) # -
1 +∞
f (τ ) cos ατ dτ ;
π −∞
1 +∞
B(α) =
f (τ ) sin ατ dτ.
π −∞
A(α) =
0
& "
kπ
π 2π
,...,
,...
- ,
l l
l
'*+ ,/(
0 "
$ " -
α ∈ [0, +∞) .
1
) $
*, )## $
f (x)
" 2 #$
A(α) B(α)
A(α) =
2
π
+∞
f (τ ) cos ατ dτ,
B(α) = 0,
0
!
f (x) =
+∞
A(α) cos αxdx.
"#
0
$ % %& ' f (x) (
A(α) = 0,
b(α) =
2
π
+∞
f (τ ) sin ατ dτ,
0
+∞
f (x) =
")
B(α) sin αxdx.
"
0
* % ! + ,
- + . /
%
+∞
f (x) =
0
0
1
π
+∞
f (τ )(cos ατ cos αx + sin ατ sin αx)dτ dα.
−∞
+ %&
% '
cos α(τ − x) = cos ατ cos αx + sin ατ sin αx
+
f (x) =
1
π
+∞ +∞
dα
f (τ ) cos α(τ − x)dx.
0
""
−∞
1+ 2 %
! % 3 2 2+ ' /
! l → +∞
!
" # $ % & "'
( $ ) * ( +& !
"
, #( -(& ( #
.+ /
u(x, t) = X(x)T (t)
0 ( -( !
# $ % & "' 1!
λ& "' /
2 a2 t
uλ (x, t) = (A(λ) cos λx + B(λ) sin λx)e−λ
.
0 # -( !
$ (& !
+ & λ & !
"' " (( λk &
#& "' ( & ( !
2 + + + !
λ ( " ( #&
"' ( $ ) & (
"' λ& !
-(/
+∞
2 2
u(x, t) =
(A(λ) cos λx + B(λ) sin λx)e−λ a t dλ.
$3 3
0
(& 4 . # !
$ % - . A(λ) B(λ) & #
$3 " $ ) 0 $3 3
t = 0& /
u|t=0
+∞
= ϕ(x) ⇒
(A(λ) cos λx + B(λ) sin λx)dλ = ϕ(x).
0
5 ( . ϕ(x) -(& !
$3 6 4. A(λ) B(λ)
1 +∞
ϕ(τ ) cos λτ dτ ;
π −∞
1 +∞
B(λ) =
ϕ(τ ) sin λτ dτ.
π −∞
A(λ) =
! "
!# $ %
u(x, t) =
1
π
+∞ +∞
2 2
dλ
ϕ(τ ) cos λ(τ − x)e−λ a t dτ.
0
&
−∞
' ( #
)# (#! *+* %
+∞
1 π −(τ −x)2/(4a2 t)
2 2
e
cos λ(τ − x)e−λ a t dλ =
.
2a t
0
& ( !#
%
1
u(x, t) =
π
⎛ +∞
⎞
+∞
2
2
ϕ(τ ) ⎝ cos λ(τ − x)e−λ a t dλ⎠ dτ =
−∞
0
1
= √
2a πt
+∞
2
2
ϕ(τ )e−(τ −x) /(4a t) dτ.
−∞
$ #
,( )# - .
)( # ! &
! %
u(x, t) =
1
√
2a πt
+∞
2
2
ϕ(τ )e−(τ −x) /(4a t) dτ.
−∞
/
⎧
⎨ 0,
u0 ,
u(x, t)|t=0 =
⎩
0,
x < x1 ,
x1 x x2 ,
x > x2 .
t > 0
u0
u(x, t) = √
2a πt
x2
e−(τ −x)
2 /(4a2 t)
dτ.
x1
! " # #! $ % &
' $ ( )(z)
1
)(z) = √
2π
x
e−S
2 /2
dS.
0
* % '
( $ + ' #
, % &
' - - ! . . /$ . $
! % ' ! % ' ( 0 % ' ( ! )(−z) = −)(z)- &
!# $1 ! ! 2 - )(−∞) = −0,5,
τ −x
√
= S -
)(+∞) = 0,5. 3 # .
a 2t
√
dτ = a 2tdS
u0
u(x, t) = √
2a πt
⎛
u0 ⎜
=√ ⎜
2π ⎝
x2 −x
√
a 2t
e−S
0
e−S
√
a 2t dS =
x1 −x
√
a 2t
x2 −x
√
a 2t
2 /2
2 /2
0
dS +
x1 −x
√
a 2t
⎞
e−S
2 /2
⎟
dS ⎟
⎠=
⎛
⎜ 1
= u0 ⎜
⎝ √2π
x2 −x
√
a 2t
e−S
2 /2
0
u(x, t) = u0
1
dS − √
2π
x2 − x
√
a 2t
e−S
2 /2
⎟
dS ⎟
⎠⇒
0
⎞
x1 −x
√
a 2t
−
x1 − x
√
a 2t
.
xa2√−2tx xa1√−2tx x < x1 x > x2
x1 < x < x2
! (z) !"! #! $ % (+∞) = 0,5
t = 0
! # & " x < x1 x > x2 '( #! "&
x1 < x < x2 u0 "& #" ! " ! #!
)#" & ! a x1 x2 *
a = 1 x1 = 1 x2 = 3 " +
u(x, t) = u0
3−x
√
2t
−
1−x
√
2t
&
.
, " # " !#& ' "
"& "!( - & *
"# "! "' # # .! # "
u(x, t) ! /01 "& #" !
2 ". " ! & #" x = 0 , # *
# #" x = 0 . #! " ! *
#"
+ u(0, t) = 0 3"! - ! 4 % # -%
#"& #! " " "! ' # % '*
"# 3"! 4 ". (x) ( "&
"& "#& x < 0 $ ' 5 *
- " % % #! " 6 ! #"
#".7 ' % # $ $ # (x) +
u(x, t) =
1
√
2a πt
⎛
⎞
(τ − x)2
(τ + x)2
+∞
−
⎜ − 4a2 t
4a2 t ⎟
f (τ ) ⎝e
−e
⎠ dτ.
8
0
5 - # -% "! "' # # .! *
" % " " #" ! +
∂u
= 0 " #! " 6 #" "& #"
∂x
x=0
u(x, t) =
(x)
! "
1
√
2a πt
⎛
⎞
(τ − x)2
(τ + x)2
+∞
−
⎜ − 4a2 t
4a2 t ⎟
f (τ ) ⎝e
+e
⎠ dτ.
#$% %&'
0
( )
) $*
+!
)"
$% &
u
=u
x=0
$% *
∂u
∂x
x=0
∂u
∂x
x=0
= 0,
u
x=L
,u
= 0,
u
∂u
∂x
= 0,
=
t=0
100
(L − x),
L
u
x=L
$% %
⎧ x ∈ (−∞, +∞)
u
=
t=0
$% -
,
x − L/2, x ∈ (0, L/2),
0,
x∈
/ (0, L/2).
=
t=0
x=L
=
x ∈ [0, L].
100, x ∈ (0, L/2],
20, x ∈ (L/2, L].
=
t=0
⎨ 100(1 − x/10), x ∈ [0, 10],
100(1 + x/10), x ∈ [−10, 0),
⎩
0,
|x| > 10.
x ∈ (0, +∞)
u x=0 = 0,
∂u
∂x
= 0,
x=0
u
U0 , x ∈ [2, 10],
0, x ∈ (0, 2) ∪ (10, +∞).
=
t=0
u
=
t=0
U0 , 0 < x < L,
0, x L.
! " #
$% & $
' $ # $
( ) (( &
x * ) )
) )
+%+ +
∂u
∂ 2u
− a2 2 = 0.
∂t
∂x
, -./
0
& $
0 x l) 0 t T &
u(x, 0) = ϕ0 (x),
, -1/
$ $% 2
⎧
⎪
⎪
⎨ α0 u
∂u
= γ1 (t);
∂x x=0
x=0
∂u
⎪
⎪
+ β1
= γ2 (t).
⎩ β0 u
x=l
∂x x=l
+ α1
, -3/
"
(n, m) 4 # $ & 5
& + + + 6
& 5 , ! / # +
2
n
=
unm+1 − 2unm + unm−1
+ O(h2 ).
h2
∂ u
∂x2
m
(n+1,m)
(n+1,m-1)
h
τ
h
τ
h
(n,m)
(n,m+1)
(n,m)
n ! "
⎧ n+1
un − 2unm + unm−1
um − unm
⎪
⎪
− a2 m+1
= 0,
⎪
⎪
⎪
τ
h2
⎪
⎪
m
=
1,
2,
.
.
.
,
M
−
1,
n = 0, 1, . . . , N − 1,
⎪
⎪
⎨ 0
m = 0, 1, 2, . . . , M,
um = ϕ0 (xm),
n+1
n+1
u
−
u
⎪
1
0
n+1
⎪
⎪
αu
= γ1 (tn+1 ), n = 0, 1, . . . , N − 1,
+ α1
⎪
⎪ 0 0
h n+1
⎪
n+1
⎪
⎪
uM − uM−1
⎪
⎩ β0 un+1
= γ2 (tn+1 ), n = 1, 2, . . . , N − 1.
M + β1
h
#
(n+1,m+1)
(n+1,m)
h
(n,m-1)
$%
&
∂ 2u
∂x2
n
m
(N + 1) · (M + 1) % !$ & % '
r = τ a2 /h2 ( ) *
⎧ 0
um = ϕ0 (xm ),
m = 0, 1, . . . , M,
(a)
⎪
⎪
⎪ n+1
n
n
n
⎪
=
(1
−
2r)u
+
r(u
+
u
),
(b)
u
⎪
m
m
m−1
m+1
⎪
⎪
⎪
m = 1, . . . , M − 1,
n = 1, 2, . . . , N − 1,
⎨
γ1 (tn+1 )
−α1 /h n+1
u1 +
, n = 0, 1, ..., N − 1, (c)
un+1
=
⎪
0
⎪
⎪
α0 − α1 /h
α0 − α1 /h
⎪
⎪
n+1
⎪
β1 /h
γ2 (t )
⎪
⎪
un+1 +
, n = 0, 1, ..., N − 1. (d)
⎩ un+1
M =
β0 + β1 /h M−1 β0 + β1 /h
+
n = 0 !"
#
t = 0
$ !" %
u (m = 1, 2, . . . , M − 1) n + 1#
t = (n + 1)τ
& ' !"
!" % ( t = (n + 1)τ
) n = n + 1
* n < N " $
! + + ,
$ - #
(n + 1, m) ' n + 1# " ) "
n+1
n
un+1 − 2un+1
un+1
m + um−1
m − um
− a2 m+1
= 0,
2
τ
h
m = 1, 2, . . .
n = 1, 2, . . .
-+ ' ' . / "
'
0 1 u #( (n+1)#
('+ ' .2
1 3
(n + 1)# #
' ( , ' " .2
. " (+ , "
'
. (' '
%+ ' / #
' ( " '
' (n + 1)# + '
% u n 4 ' un+1
, un+1
, . . . , un+1
1
2
M−1
.2 . '
n+1
n+1
am · un+1
m−1 + bm · um + cm · um+1 = fm ,
m = 1, 2, . . . , M − 1,
5
am = cm = −r" bm = 1 + 2r" fm = unm
6( % u
(n+1)#
0 ( ym = un+1
m
⎧
⎨ am ym−1 + bm ym + cm ym+1 = fm ,
y0 = L0 y1 + K0 ;
⎩
M yM−1 + K
M ,
yM = L
−α1 /h
;
α0 − α1 /h
β1 /h
;
=
β0 + β1 /h
m = 1, 2, . . . , M − 1;
γ1 (tn+1 )
;
α0 − α1 /h
γ2 (tn+1 )
.
=
β0 + β1 /h
L0 =
K0 =
M
L
M
K
! "" #"" ""$
% # &' # " () '& " " $
* ++ ,'! *
""" #"& #"('
-
." '" +"' '& y0 ! "
$
* ") *
ym = Lm ym+1 + Km ,
."
#"')
/"
m = 1, 2, . . . , M − 1.
% # m = m − 1
ym =
−cm
fm − am Km−1
ym+1 +
,
am Lm−1 + bm
am Lm−1 + bm
0
m = 1, 2, . . . , M − 1.
am (Lm−1 ym + Km−1 ) + bm ym + cm ym+1 = fm ,
'
-
m = 1, 2, . . . , M − 1.
1'" '! "
Lm =
−cm
fm − am Km−1
; Km =
, m = 1, 2, . . . , M − 1.
am Lm−1 + bm
am Lm−1 + bm
2
3"/++, Km Lm ( 4 & #"" ") "/++, $
) '&4 & #" +"' 2
" #! ) '" & # x = l 1" $
*
- # m = M − 1
yM = LM yM−1 + KM ;
yM−1 = LM−1 yM + KM−1 .
& #"')
yM =
LM KM−1 + KM
.
1 − LM LM−1
LM
yM =
M
K
β1 KM−1 + hγ2 (tn+1 )
.
β0 h + β1 − β1 LM−1
! " ! ym
m = 0 #$ $ %&& ' L0 K0
&
( m = m + 1 #$ $ %&& ' Lm Km
& )
) * m < M (
& $ !! yM
#$ ym & ( m = m − 1
+ * m > 0
#$
, $ ! " ! $
- $ . (n+1)/
0 . $ /
1 !
|bm| > |am | + |cm | $ !/
!
2 " ! !
n = 0 & +3 " /
t = 0
( ! " ! ! /
! " ! (n + 1/
t = (n + 1)τ
) n = n + 1
* n < M (
-
, " ! ! 1
! ! ' %
O(τ )+O(h2 ) ,
! ! ! /
$ 4 1 5 ' 6
m = 0, 1, . . . , M
n
δun − 2δunm + δunm−1
δun+1
m − δum
− a2 m+1
= 0.
τ
h2
n+1
n
δun+1 − 2δun+1
δun+1
m + δum−1
m − δum
− a2 m+1
= 0.
2
τ
h
δunm = λn eiwh
λn eiwh
! "
#
λ−1
eiwh − 2 + e−iwh
− a2
= 0.
τ
h2
$ " ! eiwh + e−iwh = 2 cos wh 1 − cos wh =
= 2 sin2 wh/2! eiwh − 2 + e−iwh = 2(cos wh − 1) =
= −4 sin2 wh/2 ! % !
λ=1−4
wh
τ a2
.
sin2
h2
2
&
# ! '( ) ) τ /h2 λ 1!
* % λ < −1! +
( " % , %- " " "! "
sin2 wh/2 = 1! ) + " !
λ = 1 − 4ra2 −1,
'
1
.
2a2
.
/ ! ) ! (" (" (" !
) t x * ("% * *-
r
τ /h2
0 +
1
.
2a2
λ eiwh − 2 + e−iwh
λ−1
2
−a
= 0.
τ
h2
" ! ( "
- "
1
λ=
.
1 + 4ra2 sin2 wh/2
01
"
0
|λ| 1 r
!
" #
!
$ % " & "
! ! ' "! ()*
+$ ! ' , ! " ()*
"! ' -./01.2 -.345.
6 ! 78 -./01.2 '
9 ! ' : 78 ;
" u t=0 = f (x) ) 0,
= ϕ(x),
t=0
ut (x, t)
= ψ(x).
t=0
1 &
2
2
u(x, t) = X(x)T (t).
,"
/ 0 ) ," !"&
T (t)
T (t)
=
= C.
2
a T (t)
X(x)
/ $ t&
% x& 3 t x 4 &
5 6 5 ) C
X (x) − CX(x) = 0.
! "#
T (t) − Ca2 T (t) = 0;
u(0, t) = X(0)T (t) = 0,
u(l, t) = X(l)T (t) = 0.
$ # % & u(x, t)' ! '
' T (t) = 0' ' #
X(0) = 0, X(l) = 0.
( " # ) * X(x) +
% " , - )) * #- +
. & /' % -
0
s2 − C = 0
C = λ2 > 0 ⇒ s1,2 = ±λ 1 . # ' " '
,% &
X(x) = C1 eλx + C2 e−λx
2" - 0 .
C1 + C2 = 0,
C1 eλl + C2 e−λl = 0.
3 & C1 = C2 = 0'
X(x) = 0 ⇒ u(x, t) = 0 0 x' t
4", . .
C = 0 ⇒ s1,2 = 0 1 . # ' ' ,%
&
X(x) = C1 + C2 x.
2" - 0 .
C2 = 0,
C1 + C2 l = 0,
0 x' t
4", . . !
C = −λ2 ⇒ s1,2 = ±λi 1 +! ' ,%
&
C1 = C2 = 0 ⇒ X(x) = 0 ⇒ u(x, t) = 0
X(x) = C1 cos λx + C2 sin λx.
2" - 0 . C1 = 0' C2 sin λl = 0
C2 = 0
sin λl = 0 ⇒ λl = πk (k = ±1, ±2, . . . ) ⇒ λ =
!" #
k Xk (x)&
Xk (x) = Ck sin
kπ
.
l
$ %
'"
kπx
,
l
Ck (
) Ck * *
k = 1, 2, . . . + *
λk = kπ/l %
* , #
* - λk = kπ/l
$ . sin(kπx/l) ( ,
!" /"
0 , $ . , %
(0, l)
+ , T (t) 1* λk = kπ/l $ %
. Tk (t)
2"
Tk (t) +
3
s2 +
kπa
l
2
Tk (t) = 0.
kπa
l
2
&
=0
i
s1,2 = ± kπa
l
4# 2" &
Tk (t) = Ak cos
kπat
kπat
+ Bk sin
,
l
l
5"
Ak Bk (
) * '" 5" $ 6" %
5 2" #
5 /"&
kπx
kπat
kπat
· Ak cos
+ Bk sin
.
uk (x, t) = Ck sin
l
l
l
Ck Ak = ak Ck Bk = bk uk (x, t)
kπx
kπat
kπat
+ bk sin
· sin
.
uk (x, t) = ak cos
l
l
l
uk (x, t)
! "# " " $% " "&
'"( ) # $* + ,' "&
" " ! " ! "
" $% - # ' " " &
' # " "# , . '
' (
u(x, t) =
+∞
ak cos
k=1
kπat
kπat
+ bk sin
l
l
· sin
/
kπx
.
l
0 . 1 u(x, t) " '"
$2 # " - . 1! 3 &
" " # ak bk . 1 / &
" #
u(x, 0) =
+∞
ak sin
k=1
kπx
= ϕ(x).
l
+ "' #' ! "
ut (x, t) =
+∞
kπa
k=1
l
ut (x, 0) =
−ak sin
kπat
kπat
+ bk cos
l
l
+∞
kπa
k=1
l
bk sin
sin
kπx
,
l
kπx
= ψ(x).
l
bk ,..1
4 " ak kπa
l
" - . 1! ϕ(x) ψ(x) ( (0, l) " 5"#
*%*
2 l
kπx
ϕ(x) sin
dx;
l 0
l
2 l
kπx
dx.
bk =
ψ(x) sin
kπa 0
l
ak =
6
! ak bk " # $
%! !& # uk (x, t) '
& (
uk (x, t) = Fk sin
kπx
sin
l
kπa
+ ϕk ,
t
)*
tg ϕk = ak /bk
+ #! $ # ! " '
ωk = kπa
!
l
# ! ϕk , #
!"
x Fk sin kπx
l
%
uk (x, t) (k + 1) -
(0, l)(
Fk =
a2k + b2k ,
sin
l 2l
(k − 1)l
kπx
= 0 ⇒ x = 0, , , . . . ,
, l.
l
k k
k
. !" , -# - ! '
!& " # / 0
# # wk '
"1 !# #, ! # # '
ω1 / , 2 ! ! #"1
ωk (k 2) !" 3# !
# ! &'
# # # 2
-# # 2 !& 4! Fk ! #!'
" k ! 2 !&, !
" 2 # ## / #
)
⎧
⎪
⎪
⎨
u|t=0 =
⎪
⎪
⎩
∂ 2u
∂ 2u
= a2 2 .
2
∂t
∂x
4hx
3l
x ∈ 0; ;
3l
4
4h(l − x)
3l
;l .
x ∈
l
4
ut |t=0 = 0,
u|x=0 = u|x=l = 0.
l
(u|x=0 = u|x=l = 0)
OAB !
"
#
(ut|t=0 = 0)
$ %
U
A
h
0
3l
4
B
l
x
$% !&& '" ( )**
* !&& +"
⎛ 3l
⎞
4
l
4h(l − x)
kπx
kπx ⎟
32h
2 ⎜ 4hx
3kπ
sin
dx +
sin
dx⎠ =
.
ak = ⎝
sin
l
3l
l
l
l
3(kπ)2
4
0
3l
4
, ( #
- b k = 0 ψ(x) = 0.
$% !&& '"
kπx
32h sin(3kπ/4)
kπat
sin
=
cos
3π 2 k=1
k2
l
l
+∞
u(x, t) =
32h
3π 2
1
πat 1
2πx
2πat
πx
√ sin
cos
− sin
cos
+
l
l
4
l
l
2
3πat
1
5πat
3πx
5πx
1
cos
+ √ sin
cos
+ ··· .
+ √ sin
l
l
l
l
9 2
25 2
=
.
πx
πat
32h
√ sin
sin
l
l
3π 2 2
32h
F1 = 2√
3π 2
u1 (x, t) =
ω1 = πa/l / #
x = 2l
4
√
2
x = l/4
2πat
8h
2πx
sin
sin
3π 2
l
l
2
F2 = 8h/3π
u2 (x, t) = −
3πx
3πat
12h
√ sin
cos
l
l
27 2
u3 (x, t) =
"
#
'
%
&
x = l/6
$
'
!
( )* +
'% '
'
(
' -)* .
-)* #. /'
,
%
'
'
-)* .
0
1
u(x, t) = X(x)T (t).
2
3 + (
X(0) = X (l) = 0
''
' '
'
(
2 '' 3
' -)* #.
+ (
-
)# 4
% '
'
-)* .
'
.1
T (t)
X (t)
=
= −λ2 .
2
a T (t)
X(x)
/
2
'
X (x) + λ X(x) = 0
$5
1
T (t) + (λa)2 T (t) = 0.
'
1
X(x) = C1 cos λx + C2 sin λx,
-)) 44.
s2 + λ2 = 0
s1,2 = ±λi
X(0) = C1 = 0,
X (l) = C2 λ cos λl = 0.
cos λl = 0
(2k + 1)π
, k = 0, 1, 2, . . .
λk =
2l
! λk
" #
Xk (x) = Ck sin((2k + 1)πx/(2l))
$
(2k + 1)πx
" # sin
(0; l)
2l
l
(2n + 1)πx
(2k + 1)πx
l/2, k = n
sin
dx =
.
sin
0,
k = n
2l
2l
0
%&
'(())*
(2k + 1)πat
(2k + 1)πat
+ Bk sin
,
2l
2l
!
uk (x, t) = Xk (x)Tk (t)
+&
λk
(2k + 1)πx
(2k + 1)πat
(2k + 1)πat
+ bk sin
sin
.
uk (x, t) = ak cos
2l
2l
2l
Tk (t) = Ak cos
, ak = Ak Ck bk = Bk Ck
- u(x, t) +& .
&/
+∞
(2k + 1)πx
(2k + 1)πat
(2k + 1)πat
+ bk sin
sin
u(x, t) =
ak cos
2l
2l
2l
k=0
'(()0*
!""# ak bk
'(12*
ut |t=0
+∞
(2k + 1)πx
= ϕ(x),
2l
k=0
+∞
(2k + 1)πa
(2k + 1)πx
bk sin
= ψ(x).
=
2l
2l
k=0
u|t=0 =
ak sin
'(()2*
ak (2k +2l1)πx bk
ϕ(x) ψ(x)
sin (2k +2l1)πx ! !
" ! #!"
" " ak !
#$$ %&' ! sin (2k +2l1)πx
(0; l)
bk
2 l
(2k + 1)πx
dx;
ϕ(x) sin
l 0
2l
l
4
(2k + 1)πx
bk =
dx.
ψ(x) sin
(2k + 1)πa 0
2l
ak =
!
$!
"
'(
ak
bk
# $!
# !
# % &
)
ρ
l x = 0 P
t = 0
$
*
"
"
- -
ak
bk !
# %
# $
0 &
% $! '
/
l
x sin
#
'(
0
8P l(−1)k
(2k + 1)πx
dx = 2
,
2l
π ES(2k + 1)2
&
u(x, t) =
, - -
.
2P
ak =
lES
* "
+
)
8l
π 2 ES
+∞
k=0
(
u(x, t)|x=0 = 0!
sin (2k+1)πl
= (−1)k
2l
bk = 0.
$
k
(2k + 1)πx
(2k + 1)πat
(−1)
sin
.
cos
2
(2k + 1)
2l
2l
1
(
x = 0
x = l 2
!
,!
l
! " Θ(x, t)
# $%& u(x, t) = Θ(x, t)& !'(
# )% ( # *+%
, " Θ(x, t) = X(x)T (t) -
! Θ(x, t) u(x, t) # $%
& # **% . ( ( (
# *+% / X(0) = 0
( " /
Θtt |x=l = −b2 Θt |x=l , ( b2 = GJ0 /J1
! Θ(l, t) = X(l)T (t) -/
T (t)X(l) = −b2 T (t)X (l).
0 & ( # **%
T (t) = −a2 λ2 T (t),
( /
T (t)X(l) = −b2 T (t)X (l) ⇒ −a2 λ2 T (t)X(l) =
= −b2 T (t)X (l) ⇒ b2 X (l) − a2 λ2 X(l) = 0.
1 & ( X(x) ! /
X(0) = 0, b2 X (l) − a2 λ2 X(l) = 0.
# *2%
3 ' & ' " (
# **% /
X(x) = C1 cos λx + C2 sin λx.
. # *2% C1 C2/
X(0) = 0 ⇒ C1 = 0 ⇒ X(x) = C2 sin λx,
b2 X (l) − a2 λ2 X(l) = 0 ⇒ λ(b2 cos λl − a2 λ sin λl)C2 = 0 ⇒
⇒ b2 X (l) − a2 λ sin λl = 0 ⇒ ctg λl = λa2 /b2 .
a2
λ,
b2
!" # $ %&
' ( λk )
' ' $ * λk (
*( + $( ,
C2 ' $ $ ' $ λk
ctg λl =
y=ctg(λ l)
2
y= a2 λ
b
λ2
λ1 0
λ1
π
l
2π
l
λ
λ2
,' * λk "+
Xk (x) = Ck sin λk x.
-. # *.
λk Tk (t) = Ak cos λk at + Bk sin λk at /
Θk (x, t) *. λk
Θk (x, t) = Xk (x)Tk (t) = (ak cos λk at + bk sin λk at) sin λk x,
# ak = Ak Ck bk = Bk Ck
0 *. $ .
Θ(x, t) =
+∞
k=1
(ak cos λk at + bk sin λk at) sin λk x
ak
Θt |t=0 =
Θ|t=0 =
+∞
bk
λk bk sin λk x = 0 ⇒ bk = 0,
k=1
+∞
ak sin λk x = α
k=1
x
l
k = 1, 2, 3, . . .
!" # ak $ "
sin λk x % (0; l)
ak "
# &' () % " *
" cos λk x % "
l
cos λk x cos λn xdx =
0
+
k l)
+ sin(2λ
,k = n
4λk
.
0, k = n
l
2
x
+∞
α
.
l
ak λk cos λk x =
k=1
,
ak
% - cos λn x % (0; l) .
% # cos λk x
l
ak λk
0
α
cos (λk x)dx =
l
2
%
ak =
λ2k l
0
l
cos λk xdx,
0
α sin λk l
.
sin(2λk l)
l
+
2
4λk
/
" 1 2
Θ(x, t) =
+∞
ak cos λk t sin λk x,
34
k=1
%
ak ' /
, " # λk 5 - #
* * "
" 6'
!
"
" !
# $ !
$
% &' ( ) *&'+,
!
- )
- x ∈ [0, +∞) ( .
*&'/0, 1 *&'/&, !
% && (
!
1
2
= 0,
t=0
= x,
t=0
= ex , u
t=0
x=0
t=0
t > 0.
= x2 ,
u
t=0
u
= 0,
t=0
∂ 2u
∂ 2u
= a2 2 , u
= sin 2x,
2
∂t
∂x
t=0
= 0, x ∈ (0, +∞).
∂ 2u
∂ 2u
=
, u
= 0,
2
∂t
∂x2
t=0
∂u
= ch x,
= 0, x ∈ (0, +∞).
∂x x=0
&&4
∂u
∂t
2
∂ 2u
∂ 2u
= 4 2,
2
∂t
∂x
x ∈ (−∞, +∞).
&&3
∂u
∂t
2
∂ u
∂ u
=
,
∂t2
∂x2
x ∈ (−∞, +∞).
&&+
∂u
∂t
"$
&&/
∂u
∂t
x ∈ [0, L]
= 0.
u t=0 = 0, ∂u
= 100x, u
= 0, u
∂t t=0
x=0
x=L
= 2x, u
= 0,
u t=0 = 0, ∂u
x=0
∂t t=0
∂u
∂x
∂u
,
u t=0 = sin πx
= 0,
2L ∂t t=0
u
∂u
∂x
= 0.
x=L
= 0,
x=0
= 0.
x=L
∂u
,
u t=0 = sin πx
= 0, u
= 0,
L ∂t t=0
x=0
= 0.
u
x=L
! !
" #
$ ! $ $ %&' ( !
"
) $ & !
" " * !
+* ,,)
-
+*+ + ,,) +
⎧ n+1
un − 2unm + unm−1
um − 2unm + un−1
m
⎪
⎪
− a2 m+1
= 0,
⎪
2
⎪
τ
h2
⎪
⎪
⎪
n = 1, 2, . . . , N − 1; m = 1, 2, . . . , M − 1;
⎪
⎪
⎪
⎪
m = 0, 1, . . . M;
u0m = ϕ0 (mh),
⎪
⎨ 1
um − u0m
= ϕ1 (mh),
m = 0, 1, . . . , M;
⎪
τ
⎪
⎪
n+1
n+1
⎪
u
−
u
⎪
0
⎪
= γ1 (tn+1 ), n = 0, 1, . . . , N − 1,
+ α1 1
α0 un+1
⎪
0
⎪
h
⎪
⎪
n+1
⎪
un+1
⎪
M − uM−1
⎩
β0 un+1
= γ2 (tn+1 ), n = 0, 1, . . . , N − 1.
M + β1
h
!
" #
# $% &
⎧ n+1
un − 2unm + unm−1
um − 2unm + un−1
m
⎪
2 m+1
⎪
−
a
= 0,
⎪
⎪
τ2
h2
⎪
⎪
⎪
n
=
1,
2,
.
.
.
,
N
−
1;
m
=
1,
2,
. . . , M − 1;
⎪
⎪
⎪
⎪
u0m = ϕ0 (mh),
m = 0, 1, . . . , M;
⎪
⎨ 1
um − u0m
τ a2
= ϕ1 (mh) +
ϕ (mh), m = 0, 1, . . . , M;
⎪
τ
2 0
⎪
⎪
n+1
n+1
⎪
u1 − u0
⎪
n+1
⎪
= γ1 (tn+1 ), n = 0, 1, . . . , N − 1,
⎪ α0 u0 + α1
⎪
h n+1
⎪
⎪
n+1
⎪
⎪
⎩ β un+1 + β uM − uM−1 = γ (tn+1 ), n = 1, 2, . . . , N − 1.
0 M
1
2
h
'!
( )
* + ,-! .
! &&
τ h / τ
# h
τ 0 '! !
. 1 ∂u(0, x)/∂t
τ
# h τ
$ ∂u(0, x)/∂t '!
ϕ0 (x)
u1m = u0m +
u1m
1 ∂ 2 u(0, x) 2
∂u(0, x)
τ+
τ + O(τ 3 ).
∂t
2 ∂t2
2
2
2
∂ u
∂ u
d ϕ0 (x)
= a2 2 = a2
∂t2
∂x
dx2
∂u(0, x) u1m − u0m a2 d2 ϕ0 (x)
=
−
τ + O(τ 2 ),
∂t
τ
2 dx2
!"#$
τ
u1m − u0m
= ϕ1 (mh) + a2 ϕ0 (mh) + O(τ 2 ).
τ
2
%
u1m = u0m + τ ϕ1 (mh) +
τ 2 2
a ϕ0 (mh).
2
!"&$ '
( ) ) *
u0m
2
un+1
= 2unm −un−1
m
m +a
u1m
!"&$
%
τ2 n
(u
−2unm +unm−1 ), n = 1, 2, . . . , N −1. !"+$
h2 m+1
, -
)*
#$ . u0m = ϕ0 (mh), m = 0, 1, 2, · · · M ) -
t = 0
/$ n = 1 . u1m = u0m + τ ϕ0 (mh) 0
!"&$ -
t = τ
1$ . !"+$ u
) ) ) (m = 1, 2, . . . , M − 1) n + 1
t = (n + 1)τ
&$ .
!2+ 3$
!2+ 4$ 0 t = (n + 1)τ
+$ n = n + 1
2$ 5 n < N ) 1
!$ 6 '
! " δunm = λn eiwmh !
"
n
n−1
δun − 2δunm + δunm−1
δun+1
m − 2δum + δum
− a2 m+1
= 0,
2
τ
h2
#! τ > rh
wh
λ + 1 = 0.
λ2 − 2 1 − 2r2 a2 sin2
$ %&
2
' ' '
( ! ) λ1 λ2 = 1 * +
) , + ' # ) , *
- # )
+
- , .! $ %&
/
wh
wh
2 2
2 wh
± 2ra sin
− 1.
λ = 1 − 2r a sin
r2 a2 sin2
$ &
2
2
2
0 # +
# - $ & # , /
r2 a2 sin2 (wh/2) < 1+ r2 a2 sin2 (wh/2) = 1 |λ| = 1 1
! +
sin2 (wh/2) = 1+ +
r2 a2 1+
1
τ
.
$ &
h
a
2 + # ! ) '
# ! (tn+1 , xm )
x − at = C1 x + at = C2 + " * $
3%& 4# - ! !
)# 0' 56+ '+ # !
# + # + # # !
) # (tn+1 , xm ) !
# # ! 77, ' +
τ
a+ *
h
+
! 2 4
r=
n+1
n+1
n
n−1
σ(un+1
un+1
m+1 − 2um + um−1 )
m − 2um + um
2
−
a
−
τ2
h2 n−1
n−1
n
n
n
n−1
(1
−
2σ)(u
−
2u
+
u
)
+
σ(u
−
2u
m+1
m+1
m
m−1
m + um−1 )
−a2
= 0,
h2
! σ = 1/2 " "# # #
$ σ = 1/4 % " "# # #
$ & ' σ = 0
# # (' )
*! " ! #
τ = rh +
1 + 4σr2 a2 sin2 (wh/2) λ2 − 2 1 − 2(1 − 2σ)r2 a2 sin2 (wh/2) λ+
+1 + 4σ 2 r2 a2 sin2 (wh/2) = 0.
,.! σ = 1/2 σ = 1/4 " " |λ| = 1
/ #" ! ! # "
" "
#
$ !
0 " " # " 1
23, 1 !! #
!
4 !5 6 7 8 "
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3
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3+
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1
t=0
t=0,2
t=0,4
t=0,6
t=0,8
t=1,0
0,8
0,6
0,4
0,2
0
0,2
0,4
0,6
0,8
1
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ORIGIN := 0
T OL := 0.00001
T max := 0.5
r := 1
M := 100
h
T max
τ := r ·
N :=
a
τ
hP r
τ P r := 0.1 hP r := 0.1 MP r :=
h
a2 := r2
a1 := 2 · 1 − r2
m := 0..MhP r
L := 1
a := 1
L
h :=
M
MhP r :=
L
hP r
xm := m · hP r
(U) m 1
(U) m 0,8
(U) m 0,6
(U) m 0,4
(U) m 0,2
(U) m
0
0,2
0,4
xm
0,6
0,8
1
U := |f or
m ∈ 0..M
xm ← m · h
ynm1m ← sin(π · xm )
ym ← ynm1m + τ · 0.5 · cos(0.5 · π · xm )
k←0
ngr ← 1
f or m ∈ 0, MP r..M
Uk,0 ← ynm1m
k ←k+1
f or n ∈ 2..N
t←n·τ
f or m ∈ 1..M − 1
ynp1m ← a1 · ym + a2 · (ym−1 + ym+1 ) − ynm1m
ynp10 ← 0
ynp1M−1 + h · 0.1 · sin(2 · π · t)
ynp1M ←
1+h
ynm1 ← y
y ← ynp1
if ngr · τ P r − t < 0.1 · τ
k←0
f or m ∈ 0, MP r..M
Uk,ngr ← ym
k ←k+1
ngr ← ngr + 1
U
U (x, t)
&
*+,-. /,+0
xm := m · hP r
! "# $
(
m := 0..MhP r
%
&
' !
%)
%
)
1
)
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2+3
-
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6 7 6 7 !!9.8 ! ! 6 7
6 7 6 7
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!
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!
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2 '
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! '
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!
- 8 - / " !
- 1
9 : ; / " -
0, y1 > 0
y = y(t) - " ' ' #
A B +
( * )
* # ( ) "
= 2gy
( y = y(x) * ( # ds
dt
0 t !0 *
# ) A(0, 0) ) B(x1, y1) :
1
t[y(x)] = √
2g
x1
0
1 + y2
dx −→ inf;
√
y
y(0) = 0,
y(x1 ) = y1 .
1
& % 2
# !( ) !(*
, # . * ) m / # * x0
, !# # f
# * * v0 +''
' ) # x(t) ,
3*' mẍ = f 4, # , 5
f ∈ [f1 , f2 ] 4 ' # #
!0
, , #!( * ) # T
T −→ inf,
x(0) = x0 ,
mẍ = f,
ẋ(0) = v0 ,
f ∈ [f1 , f2 ],
x(T ) = ẋ(T ) = 0.
6
!"#$%
&' &' t[y(x)]
' ' ' ' ' (
! X "#)*% +
( ', &'
' ' ' -'
"#).
/ &0 & 1 &0
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1 j n5
cij 3 ' &' i,
j , 4 5
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i, j , 4
m n
64 ' cij xij &
i=1 j=1
7 4 8 &02
!*% xnij ∈ R+ ! 4 & '%5
!)% xij ai ! 4 %5
!9%
j=1
m
xij = bj
!& ' '' (
%
1 4 & &0 2
i=1
m
n
i=1 j=1
cij xij −→ inf,
n
j=1
xij ai ,
m
i=1
xij = bj ,
xij ∈ R+ .
!"#"%
& :&
n &' !&' '& % ; &
4 ( bj j 4 : 8
aij j 4
i4 &' ci 3
i4 &'
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n
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xi 0.
i=1
!
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F
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E
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0 $ #$ ABCD M
hx AB N
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N 1 = N 2 = M1 = M2 = 0
!
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*
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" AD "#%
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EHGF 01 & 2 |AB| = a |AD| = b |EF | = av
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M " AB &
N 1 N 2 M1 M2 %4
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|y| − i |x| =
= C1 , |y| + |x| = C2 xy < 0
|y| + i |x| = C2 xy
=
0
= C1 ,
√
√
√
√
xy < 0 y = 13 x3 + C1 , y = − 13 x3 + C2
√
√
xy > 0
√y = i 13 x3 +C1, √y = −i 13 x3 +C2
∂ 2z
∂ 2z
x2
+ 1, z(x) = xº
−
=0.
2
∂x2 ∂y 2
2
2
2
∂ u ∂ u ∂ u
2 + 2 + 2 = f (x, y, z).
∂x
∂y
∂z
ϕ − y
= 1
1 + y ϕ
y1 = ϕ(x1 )
y(x) = −
y =
1
x5
+ (x2 − x3 )
120 24
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