Добавил:
Upload Опубликованный материал нарушает ваши авторские права? Сообщите нам.
Вуз: Предмет: Файл:
Скачиваний:
37
Добавлен:
27.04.2015
Размер:
445.87 Кб
Скачать

11

+2%-" -0+1 #6) ?,"$ %$D +, (%"3 05+ 6"$%) 5 - $# +-$1,-1 , (%$-"+D"3 ,).

K -+ %- ) 5 %"+ +2%/" ,0 5+%$(%50, 5 . $$ ,0 $ +% 0$ " 3. 6) ("6"3 -"2$%/" 9 ? 608% $%) "-% D0$ (% $ 6:$ -: 05+ 6"$%) K , 8"- + , -+% -: ) + .#"6"$$ 3 $ 0+ $%:

05+ 6"$%) 46","$- ,% "-%,

05+ 6"$%) "-:J,

05+ 6"$%) 0 609 ,%,

05+ 6"$%) ?%.$" ,.

8#13 0 . $$13 0+ "$: 9+ $%/% "- 5+ (" 05+ 6"$%) 5+"#"6 2 5+"#"6"$$12 9+ $%(, %,""- J %$D +, (% $$0J , #"6: %-+0 -0+0 % . %, #"3 - 0"- #+09%,% 0+ $),%.

5+ 6"$%" ?%.$" ,

5+ 6"$%" 0 609 3

5+ 6"$%""-:J

5+ 6"$%""-" 1, 46","$- ,

"-" 1" 46","$-1

1

2

3

4

5

1- 05+ 6"$%" $D%90+ (%"3

2- 05+ 6"$%" 0 -+ $"$%", $"% 5+ $ -"3

3- 05+ 6"$%" /" - , 5"+"# /%

4- 05+ 6"$%" + /"- ,%

5- 05+ 6"$%" . I%- 3 %$D +, (%%

% . 3

12

+ "$: 05+ 6"$%) 46","$- ,% "-% 0I" - 6)"- $-+ 6: % $"5 +"# - "$$ " 05+ 6"$%" 46","$- ,% "-% % ) 6)"- ) % - /$% , %$D +, (%% - )$%% 46","$- "-% #6) 6"#0JI%2 0+ $"3.

+ "$: 05+ 6"$%) "-:J 0I" - 6)"- D0$ (%% 5 05+ 6"$%J"-:J %6% "" 0/ - , 6J/ JI%2 9" 9+ D%/" % + .$" "$$1" 46","$-1 "-%. K- - 0+ "$: . %, #"3 - 0"- 0+ $", 05+ 6"$%) 0 609 ,% 5 5+ , /" - , + . %-%) "-% % -.#.

+ "$: 05+ 6"$%) 0 609 ,% 0I" - 6)"- D0$ (%% 5. %, #"3 - %J #+09%,% 5"+ - + ,%, 5 - I% ,% 0 609 % 5 6:. -"6),%. K- - 0+ "$: 05+ 6"$%) ) 6)"- ) #,%$% -+ -% $1, %. %, #"3 - 0"- 0+ $", 05+ 6"$%) ?%.$" ,.

+ "$: 05+ 6"$%) ?%.$" , 0I" - 6)"- D0$ (%% ?I"3 #")-"6:$ -% 5"+ - + , ). $$1" 5+"# - 6"$%", 0 609 46" -+ ).%, % 15 6$)"- +#%$ (%J + ? - 5 ?%.$" 0. K- - 0+ "$: ) 6)"- ) 0+ $", 05+ 6"$%) 5"+ - + % . %, #"3 - 0"- 0+ $", 05+ 6"$%) 0 609 ,%.

8#13 5 6"#0JI%3 0+ "$: %,""- ? 6"" 1 0J -"5"$:? ?I"$%), /", 5+"#1#0I%3.

% -", 05+ 6"$%) 2 -1 "- " D0$ (% $ 6:$1" ?6 -% 05+ 6"$%) "-),%, ?" 5"/% JI%" 5 ##"+8 0 5"+ - + "9 #")-"6:$ -%, - 8" 05+ 6"$%" "-),% /+". 1/ 3$12 %-0 (%)2.

$ $1, D0$ (% $ 6:$1, ?6 -), 05+ 6"$%) "-),%-$ )- ):

05+ 6"$%" $D%90+ (%"3 (56 $%+ $%", D +,%+ $%" % + . %-%" 05+ 6)", 3 "-%, 0 - $ % # 4 560 - (%J $ 9 ? +0# $%), 0 - $ 6"$%" % %.,"$"$%" "#%$"$%3 ,"8#0 46","$- ,% "-%, 5+"# - 6"$%" "-" 12 +" 0+

5 6:. -"6), % -.#.);

05+ 6"$%" 0 -+ $"$%", $"% 5+ $ -"3 ( ?$ +08"$%", 6 6%. (%), +"9% -+ (%) % 0 -+ $"$%" $"% 5+ $ -"3 "-% % -.#.);

05+ 6"$%" /" - , 5"+"# /% ( ? +, ?+ ? - , +"9% -+ (%), 2+ $"$%" % - ?+ 8"$%" - -% -%/" %2 # $$12 D0$ (% $%+ $%% "-% % "" 46","$- , $ 6%. /" - "$$12 5 . -"6"3 % -.#.);

05+ 6"$%" + /"- ,% ( ? + % 0/"- 5+"# - 6)",12 0 609 ).%, $ /% 6"$%" 56 -1 . %2 % 5 6:. $%", 5 #9 - , + 16 %$-+ 6: 56 -1 /"- % -.#.);

05+ 6"$%" . I%- 3 %$D +, (%% ( ?" 5"/"$%"$D%#"$(% 6:$ -% % ("6 -$ -% 5"+"# ", 3 %$D +, (%%,1# / %9$ 6 -+" 9% 5+% $" $ (% $%+ $$ , # -05" %$D +, (%% % -.#.).

13

8# , 0+ $" 05+ 6"$%) D0$ (%% 5 05+ 6"$%J 15 6$)J- )5+"#"6"$$1,% +9 $%. (% $$ --"2$%/" %,% -+0 -0+ ,% ($ 5+%,"+,608? ,% 5+"#5+%)-%3) + .$ , D0$ (% $ 6:$ , ?@"," . % %, -%- #"+8 $%) +"> ",12 . # /.

&- & ( & 1

1%" ?"$$ -% 2 + -"+$1 #6) +","$$ 9 4- 5 + . %-%)-+ 6% ).% % . # /% #% (%56%$1 « "$"#8,"$- 5+"#5+%)-%3 46" -+ ).%».

2- - " -"2$%/" ) 4 560 - (%). %" ,"- #1 -"2$%/" 9? 608% $%) +" ,"$# $1 #6) 5+%,"$"$%) $ "-%.

3$ $1" 5+%.$ % * % -+ #%(% $$ " ," - "9 5+%,"$"$%) $

"-%.

40I$ -: % $ %? 6"" 5+"#5 /-%-"6:$ " ," - "9 5+%,"$"$%) $ "-%.

5?"$$ -% 5 + $"$%J * % % ," - "9$ $ 9 5+%,"$"$%) $ "-%.

6* $)-%): ?@" - -"2$%/" 3 4 560 - (%% % 46","$- "-%. ).: ,"8#0 $%,%.

7.% -", -"2$%/" 3 4 560 - (%% % % -", 05+ 6"$%). 2 , #%D% (%%, % 5 6:.0",1" $ "-%.

8?I ) 2", . %, ).% "-% 05+ 6"$%) 46" -+ ).:J % "-:J 46" -+ ).%. $ $1" 46","$-1 K. 2 $ .$ /"$%".

9+ $% 05+ 6"$%) K. 2 $ $1" D0$ (%%.

100$ (% $ 6:$1" ?6 -% 05+ 6"$%) K. 2 $ $1" 2 + -"+% -% %.

14

2 (- .!&! & ! '.$ (-6 ! '

2.1 (- &

,56" $1, 3 - ,, 2 + -"+%.0JI%, /" - D0$ (% $%+ $%) K, ) 6)"- ) %2 $ #"8$ -:.

! ' 1 4- 3 - ?@" - 2+ $)-: +","$% 0 - $ 6"$$12 5"+"#"6 2 .$ /"$%) "2 5 + ,"-+ , 2 + -"+%.0JI%25 ?$ -: 15 6$)-: -+"?0",1" D0$ (%% . # $$12 +"8%, 2 % 0 6 %)2 5+%,"$"$%), -"2$%/" 9 ? 608% $%), +", $- , 2+ $"$%) % -+ $ 5 +-%+ $%).

#"8$ -: ) 6)"- ) ,56" $1, 3 - ,, - + "

. % %, -% - $ .$ /"$%) ?@" - % 0 6 %3 "9 5+%,"$"$%) , 8"-6J/ -: ?". - .$ -:, # 69 "/$ -:, +", $- 5+%9 #$ -: %2+ $)", -: %6% 5+"#"6"$$1" /"- $%) 4-%2 3 - .

7 + !+ 1 4- 3 - ?@" - $"5+"+1 $ 2+ $)-: + ? - 5 ?$ " - )$%" -"/"$%" $" - + 9 +","$% %6% $ + ? - %.

" ( % 1 4- 3 - ?@" - 2+ $)-: + ? - 5 ?$ "

- )$%" # $ -056"$%) 5+"#"6:$ 9 - )$%) 5+% 0 - $ 6"$$ 3% -"," -"2$%/" 9 ? 608% $%) % +", $- .

* & " 1 4- 3 - ?@" - , . 6J/ JI"" ) 5+% 5 ?6"$$ -% 5 ##"+8 $%J % - $ 6"$%J + ? - 5 ?$ 9

- )$%) 50-", -"2$%/" 9 ? 608% $%) % +", $- .

.&! * 1 4- 3 - ?@" - 2+ $)-: . # $$12 5+"#"6 2 .$ /"$%) 5 + ,"-+ , 2 + -"+%.0JI%2 5 ?$ -% ?@" -15 6$)-: -+"?0",1" D0$ (%%, -"/"$%" % 5 6" 2+ $"$%) % (%6%) -+ $ 5 +-%+ $%)

!# # 4- - )$%" ?@" - , 5+% - + ,

.$ /"$%) "2 5 + ,"-+ , 2 + -"+%.0JI%2 5 ?$ -: 15 6$)-:

. # $$1" D0$ (%%, - "- - 0J- -+"? $%), $ +, -% $ 3 % (%6%)$ -+0 - + 3 (5+ " -$ 3) # 0,"$- (%%.

&!( 4- - )$%" ?@" - , 5+% - + , $

- "- - 0"- ", -+"? $%), $ +, -% $ 3 % (%6%) $ -+0 - + 3 (5+ " -$ 3) # 0,"$- (%%.

& 1 4- - )$%" ?@" - , 5+% - + , "9

# 6:$"3> ) 4 560 - (%) $"# 50 -%, %6% $"("6" ?+ .$ , 6%?

- $ 6"$%" "9 + ? - 5 ?$ 9 - )$%) $" ., 8$ %6% $"("6" ?+ .$ .

(& ' 4- ?1-%", . 6J/ JI"" ) $ +0>"$%% % 5+ $ 9 - )$%) ?@" - 5+% 2+ $"$%% + ? - 5 ?$ 9

- )$%).

15

!+ 4- ?1-%", . 6J/ JI"" ) $ +0>"$%% + ? - 5 ?$ 9 - )$%) ?@" - .

! ( 4- 5+ (" ?$ +08"$%) % 0 -+ $"$%) - . , -.". 5+ (" 5"+" # ?@" - + ? - 5 ?$ " - )$%" %. $"+ ? - 5 ?$ 9 .

- .1 60/ 3$1" ?1-%). .$% JI%" - .1 + .6%/ J- 5 + .$1, 5+%.$ ,:

5 2 + -"+0 %.,"$"$%):

-( +!- !+, 2 + -"+%.0JI%3 ) / ?+ .$1, %.,"$"$%", .$ /"$%3 #$ 9 %6% $" 6: %2 5 + ,"-+?@" - ;

-- !+, .$% JI%3 +".06:- -" 5 -"5"$$ 9 %.,"$"$%) .$ /"$%3 #$ 9 %6% $" 6: %2 5 + ,"-+ ?@" - ;

5 $">$",0 5+ ) 6"$%J:

-(- !+, ?$ +08% ",13 %.0 6:$ %6% >- -$1,% ,"- # ,% % +"# - ,% $-+ 6) % #% 9$ -%+ $%) 5+% 5 #9 - " ?@" - 5+%,"$"$%J %6% 5+ (" " "9 5+%,"$"$%) 5 $ .$ /"$%J;

-&- - !+, $" ?$ +08% ",13 %.0 6:$ %6% >- -$1,% ,"- # ,% % +"# - ,% $-+ 6) % #% 9$ -%+ $%), $1) 6)",13 5+% 5+ "#"$%% -"2$%/" 9 ? 608% $%) %6%5"(% 6:$1,% ,"- # ,% #% 9$ -% %;

. % %, -% - 5+%/%$1 .$% $ "$%):

-&$(- !+, .$% JI%3 5 5+%/%$", ). $$ 3 $" "+>"$ - , %6% $ +0>"$%", 0 - $ 6"$$12 5+ %6 % (%6%) $ +, 5+ " -%+ $%) % $ -+0%+ $%);

-& +( (- !+, .$% JI%3 5 5+%/%$", ). $$ 3$" "+>"$ - , %6% $ +0>"$%", 0 - $ 6"$$ 9 5+ ("

%.9 - 6"$%) %6% +", $- , 15 6$)", 9 $ +", $-$ , 5+"#5+%)-%%;

-/$! !- !+, .$% JI%3 5 5+%/%$",

). $$ 3 $ +0>"$%", 0 - $ 6"$$12 5+ %6 % (%6%) 0 6 %3 4 560 - (%%;

" K , 90- ?1-: 5 #+ .#"6"$1 ! ( ! !( (! *- ,

- +1" $" 5 #6"8 - - $ 6"$%J 5 6" .$% $ "$%) - . , % ( ! !( (! *- , - +1" , 90- ?1-: % 5 6:. $1 5 6"

- $ 6"$%).

2.2 !+! ! ' ( ! !( (! *-. #9 (

0$ (% $%+ $%" $" - $ 6% ",12 ?@" - 2 + -"+%.0"- ) 5+"%,0I" - "$$ 5 . -"6),% ?". - .$ -%. *0 -: ?@" -,

16

+ ? - 5 ?$13 $ / 6:$13 , ,"$- +","$% t=0, + ? - "- # - . -"/"$%" +","$% , - + " ) 6)"- ) 60/ 3$ 3 "6%/%$ 3.

". - .$ -: ?@" - 2 + -"+%.0"- ) 6"#0JI%,% 5 . -"6),%: 1 & 1 # + !+ &!#- p(t) "+ )-$ -: - 9 , /-

5+"#"6 2 . # $$ 3 $ + ? - % t - . $" 5+ %. 3#"-, -.".:

p(t) =1 q(t) = p{Z t}

+ ? - t 4- 5+ # 68%-"6:$ -: %6% ?@", + ? -1 ?@" - .

2 & 1 !+! q(t) "+ )-$ -: 5+ -% 5 6 8$ 9?1-%), /- - . 5+ %. 3#"- # , ,"$- t, -.".

q(t)=P{ <t}=F(t)

"+ )-$ -: q(t)=F(t) ) 6)"- ) D0$ (%"3 + 5+"#"6"$%) 60/ 3$ 3"6%/%$1 .

3 1 &! & !&!# !+! f(t) 5+"#"6)"- )

:

f(t) = dF(t)

= dq(t)

 

=

d p(t)

, - 0#

dt

 

dt

 

 

dt

t

q(t) = f (x) dx

0

4 ( 1 !+ ( (t) 0 6 $ ) 56 -$ -: "+ )-$ -%.$% $ "$%) - . ?@" - , ,"$- t 5+% 0 6 %%, /- # 4- 9 , ,"$- - . $" 5+ %. >"6:

 

f(t)

 

dF(t)

dp

a(t) =

=

dt

=

dt

 

 

p(t)

 

p(t)

1 F(t)

17

%5%/$ ) . % %, -: %$-"$ % $ -% - . 5+% "#"$ $ +% 0$ "

4.

I

 

II

 

 

 

 

III

 

 

 

 

 

 

 

 

 

 

t

% .4

% %, -: %,""- -+% ?6 -%:

I – ?6 -: $ / 6:$12 - . , %,"JI%2 ? 6:>0J %$-"$ % $ -:,?0 6 6"$$0J 96 $1, ?+ . , +1-1,% #"D" - ,% 5+ %. # - "$$ 9 2 + -"+ . (5"+% # 5+%+ ? - % 55 + -0+1, -"2$ 6 9%/" 9 5+ 9 $ , 51-$ 3 4 560 - (%%).

II – ?6 -: /% - 60/ 3$12 - . , ?0 6 6"$$12 60/ 3$1,% 5+%/%$ ,%, - + 3 (t)=const (5"+% # $ +, 6:$ 3 4 560 - (%%).

III – ?6 -: %.$ 12 - . , ?0 6 6"$$12 96 $1, ?+ . ,- +"$%", 46","$- ?@" - .

% %, -: (t) , 8"- ?1-: 55+ %,%+ $ D +,06 3"3?066 , 5% 1 JI"3 $ %? 6"" + 5+ -+ $"$$0J , #"6: - . (4 5 $"$(% 6:$13 . $ $ #"8$ -%):

p(t) = exp( at d ) ,

.#" ::

> 0, t 0, > 0

18

c 0/"- , "" 5 60/%,: a(t) = dp(t)dt p(t)1 = adtd 1 , - 0# %#$ , /-

< 1 – - "- - 0"- I – ?6 -%, -.".

a(t) = ad t1 d

=1 – - "- - 0"- II – ?6 -%, -.".

(t)=

> 1 – c - "- - 0"- III – ?6 -%, -.".

a(t) = adtd 1

5 C& !&!# ! !+! T0 , -", -%/" " 8%# $%" $ + ? - % ?@" - # 5"+ 9 - . , -.".

 

 

 

T = M[Z] = p(t) dt = tf (t)dt

o

o

o

 

# 6:$"3>", " + /"-1 5 . -"6"3 $ #"8$ -% ?0#", 5+ %. #%-: #6) ?6 -%, - "- - 0JI"3 5"+% #0 $ +, 6:$ 3

4 560 - (%% (5+% =1). 9# 5 . -"6% ?". - .$ -% 5+"#"6)J- )6"#0JI%, ?+ . ,:

p(t) = e at

q(t) = 1 e at

19

f(t) = ae at

T° = e at dt = 1a

°

2.3 !+! ! ' ( ! !( (! *-. #9 (

- )$%" - $ 6% ",12 ?@" - 5% 1 "- ) 60/ 3$ 3 D0$ (%"3 H(t), 5+"# - 6"$$ 3 $ +% 0$ " 5.

H1

 

 

1

 

2

 

3

 

1

2

3

 

 

t H0

% . 5

0$ (% $%+ $%" - $ 6% ", 9 ?@" - +","$% 4-

5 6"# -"6:$ -: %$-"+ 6 i "9 + ? - 5 ?$ -% H0 ($ +, 6:$ 9 D0$ (% $%+ $%)), /"+"#0JI%2 ) %$-"+ 6 ,% i "9

$"+ ? - 5 ?$ -% H1 (5+ - )).

#"8$ -: - $ 6% ",12 ?@" - 2 + -"+%.0"- ) 5 . -"6),% ?". - .$ -%, +", $- 5+%9 #$ -% % ,56" $1,% ( ,?%$%+ $$1,%).

* . -"6% ?". - .$ -% 5+"#"6)J- ) - 8", % #6) $" - $ 6% ",12 ?@" - , . % 6J/"$%", 0 . 6)

- $ 6% ", 9 ?@" - 5+"#"6)"- )

& !&!# ! ! !+

-$ >"$%" 0,, +$ 3 $ + ? - % t

, -", -%/" ,0 8%# $%J

/% 6 "9 - . r(t) -"/"$%" 4- 3 $ + ? - %:

T° =

t

M[r(t)]

 

 

20

 

", $- 5+%9 #$ -:

2 + -"+%.0"- )

( & 13

( ! ( pB (t) -

"+ )-$ -:J - 9 , /- +",) - $ 6"$%)

+ ? - 5 ?$ 9 - )$%) ?@" - $" 5+" 1 %- . # $$ " .$ /"$%" , -.".

pB (t)=p{ < t}

%? 6"" + 5+ -+ $"$ 5+% % 5 6:. $%% #6) + /"- 4 5 $"$(% 6:$13 . $ - $ 6"$%):

pB (t) = 1 e µt,

9#": - ( 1 ( ! ( - 0 6 $ ) 56 -$ -:"+ )-$ -% - $ 6"$%) + ? - 5 ?$ 9 - )$%) ?@" - ,5+"#"6"$$ ) #6) + , -+% ", 9 , ,"$- +","$% t 5+% 0 6 %%, /-

# 4- 9 , ,"$- - $ 6"$%" $" ?16 . "+>"$ .

5+"#"6)"- ) - 8" & (& * ( ! (

, -", -%/" " 8%# $%" +","$% - $ 6"$%) + ? - 5 ?$ 9

- )$%) ?@" - 5 6" - . :

TB = [l] = µ1

,56" $13 5 . -"6: $ #"8$ -% 2 + -"+%.0"- ) /"+".

/22 " ( M "+ )-$ -: - 9 , /- ?@" - 8"- )

+ ? - 5 ?$ , - )$%% 5+ %. 6:$13 , ,"$- +","$%, + ," 56 $%+0",12 5"+% # , -"/"$%" - +12 % 5 6:. $%" ?@" - 5

$ .$ /"$%J $" 5+"#0 , -+% "- ).

 

 

 

 

5+"#"6)"- )

M

/"+".

5 . -"6% ?". - .$ -% %

+", $- 5+%9 #$ -%:

 

 

 

 

 

 

 

 

 

B =

 

0

=

µ

 

 

 

0

+

µ+ a