- ,

. .

MATLAB

MAPLE

-

2010

............................................................................................. 3 1 ..................................................... 6 2 .......................................... 13

2.1 .............................................................................. 13 2.2 ........................................................................................... 15 2.3 ................................................................................... 17 2.4 .......................................................................... 20 2.5 ................................................................................ 22

2.5.1 , ................................ 22 2.5.2 , ............... 25

2.6 ........................................................................................................ 26 3 .................................. 28 4 ................................................... 30 .............................................................................................................. 32

5 ........................................................................ 33 5.1 .............................................................................. 35 5.2 .................................................................... 36 5.3 ................................................................... 38

6 ...................................................... 39 6.1 .................................................................................... 39 6.2 ............................................................. 41 6.3 .......................................................................................................................... 45 6.4 .................. 48

6.4.1 . ........................................................ 48 6.4.2. . .................................................................................... 48 6.4.3 . .............................................. 48

6.5 ............................................. 50 6.6 ............................................. 52 6.7 ...................................................... 53 6.8 ........................................................................................................ 54

7 .......................................................... 57 7.1 ................................................................. 57 7.2 ........................................ 59 7.3 ........................................................................................................ 61

8 ....................................................................... 62 .............................................................................................................. 64

9 ............................................................................ 66 9 ............................................................................ 67

2

.............................................................................................................. 70 10 ......................................................... 71 11 ....................................................... 73 .............................................................................................................. 75

12 .................................... 76 13 ........................................ 80

13.1 ......................................................................... 80 13.2 ........................................................... 82 13.3 ......................................................................... 84 13.4 ...................................................................................................... 88

14 ..................................... 91 14.1 ............................................ 93 14.2 ..................................................................................................... 95 14.3 .................................................................................................. 97

15 ................................................................ 100 15.1 ..................... 102 15.2 ............................................ 104 15.3 .................................................................................. 104 15.3 .................................................................................. 105 15.4 ..................................................................... 107 15.5 .................................................................................................... 109

16 ......................................... 111 16.1 () ........................................... 111 16.2 ............................................................................. 115

17 ................................................. 117

17.1 .................................................................... 118 17.2 ................................................................................ 121 17.3 , ..... 125 18.1 .................................................................................... 129 18.2 ........................................................... 133 18.2 ........................................................... 134 18.3 ...................................................................................... 137 18.4 ................................................................... 139

.................................................................................... 140 C ...................................................................... 141 ........................................................................................ 142

3

, , [1]. , , , . - , , .

, , , . , , , .

[2]. , , , .

, . , , -. , , , .

, .

4

, , , . . .

, , , S, . , , . , .

, , , - , : , , , , , , . .

, , , . . .

, , , . , . , , .

, , , S. .

5

, . , , , . . , . .

() S. S . , , - .

, .

MATLAB MAPLE [3,4].

6

1

. , , .

dx/dt = f(t) f = f(t), t, , , , , . , , [5]:

1.1. : 2 2dx x t

dt= +

x(t), '(t) = dx/dt. :

2

2 3 7 0d y dy ydx dx

+ + = -: ( ) ' ( ) ".

. 1. ,

. 2.

. 3. .

, - , 3 + 72 - 11 + 41 = 0. = (), '() = 2() . , .

1.2.

( ) 2xy x Ce= (1.1)

( ) ( )( )2 22 2 2x xdy C xe x Ce xydx = = =

7

, () (1.1) :

dy/dx=2xy (1.2) . , (1.1) , . [5] , (1.2) , (1.1).

. t, , , , .

1.3. , , : ( t) T(t) (. 1.1). ,

( )dT k T Adt

= , (1.3) k . , > , dT/dt < 0, t, . < , dT/dt > 0, .

, - (1.3). k , T(t), , , .

.1.1. , , .. (1.3), .

T

A

. 1.2. , .. (1.4), .

V y

8

1.4. ,

V , (. 1.2), :

ykdtdV = (1.4)

k . , V = , dV/dt = A(dy/dt). (1.4) :

yhdtdy =

h =k/A . 1.5.

P(t) . :

kPdtdP = (1.5)

k . 1.5 . ,

: P(t)=ekt (1.6) (1.5).

: P'(t) = Ckekt = k(Cekt) = kP(t)

t. (1.6) (1.5) , (1.5).

, k , dP/dt = k P(t)=ekt, "" , . , .

1.6. , P(t)=ekt t. t = 0 1000 . P(t) :

1000 = (0) = Ce0 = C, 2000 = (1) = Cek.

9

, = 1000 ek = 2, k = ln 2 0.693147. k (1.5) :

PPdtdP 693147.0)2(ln =

k = ln 2 = 1000 (1.6) P(t) = 1000(eln2)t = 10002t ( eln2 =2), . , . , (t = 1.5) :

(1.5) = 1000 23/2 2828. (0) = 1000 6 ,

, t = 0 " ". . 3 P(t) = ekt k = ln 2. dP/dt = k , . , P0 P (0). , , (0)=P0 , .

MATLAB k=log(2); t=-1:0.5:5; C=[ -0.12; -0.06; -0.03; -0.01; -0.005; 0.005; 0.01; 0.03; 0.06; 0.12]; P=C*exp(k*t); plot(t,P); title('P(t)=C*e^k^*^t'); xlabel('t'); ylabel('P(t)'); text(3.1,2.78, 'P(t)= 0.12*e^k^*^t\rightarrow'); text(3.45,1.78, 'P(t)= 0.06*e^k^*^t\rightarrow'); text(3.2,0.85, 'P(t)= 0.03*e^k^*^t\rightarrow'); text(3.1,-0.78, 'P(t)= -0.03*e^k^*^t\rightarrow'); text(3.45,-1.78, 'P(t)= -0.06*e^k^*^t\rightarrow'); text(3.1,-2.78, 'P(t)= -0.12*e^k^*^t\rightarrow'); grid on

1.5 1.6 (. 1.4), .

10

-1 0 1 2 3 4 5-4

-3

-2

-1

0

1

2

3

4P(t)=C*ek*t

t

P(t)

P(t)= 0.12*ek*t

P(t)= 0.06*ek*t

P(t)= 0.03*ek*t

P(t)= -0.03*ek*t

P(t)= -0.06*ek*t

P(t)= -0.12*ek*t

. 1.3. P(t) = ekt k = ln 2.

. 1.4.

-

-

11

1. . .

2. . 3.

, , .

, . (P t), , , (dP/dt = k, (0)=P0 ), .

( , t). , , .

, . , . , (1.5) P(t) = ekt , k P(t) . , , , . , 1.5, , . , (1.5) . , .

, . , , .

: , , , . ,

12

, . , , . , , . , . , , .

13

2 2.1

( , ), , [6]. ( ) :

x = f(t) .. , t. :

v(t)=f(t), .. v = dx/dt. a(t) a(t) = v'(t) = x"(t) ,

:

2

2

dtxd

dtdva ==

( ) , F(t) , , : ma(t) = F(t); .. F = ma, m .

F , x"(t) = F(t)/m . , x(t). , . 0= (0) v0 = v(0) .

. , F, = F/m, . :

dtdva = (a ).

, : ( ) 1Catadttv +==

, v = v0, t = 0, C1= v0.

( ) 0vatdtdxtv +== . (2.1)

:

( ) ( ) 2020 21 Ctvattdvatvdttx ++=+== ,

t = 0, x = x0 2 = x0. :

14

( ) 2 0 012x t at v t x= + + . (2.2) , (2.1),

, (2.2) t, , v0 x0.

15

2.2 . 2.1

w = 2. = , . , , :

= 2

2

0 1 axvvR . (2.3)

(2.3) , , vR = v0, vR = 0 .

, (, 0) vs . .2.1, ( ) vs vR. , :

S

R

vv=tan .

tan = dy/dx, (2.3) :

= 2

20 1

ax

vv

dxdy

S (2.4)

= () . 2.1. , 2 ,

v0 = 15 /. vs = 5 /, (2.4) :

y = 3(1 x2) = 3 3x2. :

( ) ( ) Cxxdxxxy +== 32 333

. 2.1. . x

y

vs

vR

(a , 0) (a , 0)

16

. (1) = 0 = 2. : (x) = 3x x3 + 2.

(1) = 3 1 + 2 = 4.

, 4 ,

2 .

17

2.3 , (, 0),

, (0,0). v0 , w. . 2.2, , . . 2.3 . :

++

==+

==

wyx

yvvdtdy

yx

xvvdtdx

22

00

22

00

sin

cos

, = f(x)

: ( )2200

1// yxwyv

xvdtdxdtdy

dxdy +== . (2.5)

k=w/v0, . k, , (2.5) :

1/ 22

1dy y ykdx x x

= + (2.6)

y = xv, ' = v + xv' :

=+ .1

11

2dx

xkdv

v

, , : ( )2ln 1 lnv v k x C+ + = + ,

. 2.3.

. 2.2.

18

v(a) = 0 C = klna. , :

( )2ln 1 ln ln ln ln kx xv v k x k a k a a + + = + = = 21kxv v

a

+ + =

k kx xa a

+ = ( )22 2 2 22

2 2 2

1 11 2 1 1 111 1 1

v v v v v vv vv v v v v v

+ + + + + + + ++ + + = =+ + + + + +

= ( )22 2

2 2

2 12 2 1 2 21 1

v v vv v v vv v v v

+ ++ + + = =+ + + + 12

k kx xva a

= + .

y=xv, :

( ) 1 12

k ka x xy xa a

+ = + (2.7)

. , . 2.4,

(2.7), k < 1 (. . w < v0), . w = v0 ( k = 1), (2.7) y(x)=(1/2)(1 2/2), (0, /2), (0,0). , w > v0 ( k > 1) , (2.7), +, x 0.

MATLAB clc clear syms a v0 u x k a = vpa('800'); % v0= vpa('800'); % , / w = vpa('40'); % , / k = w/v0; y=a/2*( (x/a)^(1-k)-(x/a)^(1+k)); ezplot(y); if eval(k) < 1 d_y=diff(y,x); % , % , . y(x) 0

19

0 1 2 3 4 5 6

0

5

10

15

20

25

30

35

x

400.0 (0.00125 x)0.5 - 400.0 (0.00125 x)1.5

. 2.4. 2.7.

20

. 2.5. . x

v

a

y

2.4 2.2.

450 (/). () 2.5 (/2) ( , , ). , " " (v = 0 )?

. x(t) (. 2.5). t = 0. v0 = 450 (/, , x(t) ), = +2. 5, v ( |v|). (2.1) (2.2) :

v(t) = 2.5t - 450 (2.8)

x(t) = 1.25 t2 450t + x0, (2.9) x0 t = 0, .

(2.8) , v = 0 ( ) , t =450/2.5=180 (. . 3 ); t = 180, = 0 (2.2) :

x0= 0 (1.25)(180)2 + 450(180)= 40500 . .. x0= 40.5 . ,

, 40.5 . 3 .

21

MATLAB clc clear x=0; % a=2.5; % v0=-450; % t=-v0/a % x0=x-1/2*a*t^2-v0*t %

m . , (F=ma) , v , m, :

Gdvm Fdt

= (2.10) FG = mg , g ( ).

2.3. , (y0 = 0) v0 = 49 (/). (2.10) g = 9.8 :

9.8dvdt

= , v(t) = (9.8)t + v0 = (9.8)t + 49. , ( )

y(t) : ( ) ( ) ( ) ( )2 209.8 49 4.9 49 4.9 49 .y t t dt t t y t t= + = + + = +

, v(t) = (9.8)t + 49 = 0. , t = 5 (). , :

ymax = y(5) = (4.9)(52) + (49) (5) = 122.5 (). , = (4.9)t(t 10) = 0, .. 10

. . FR,

m, (2.10):

G Rdvm F Fdt

= +

22

2.5 (Principia Mathematica)

, , FR : FR = kv2. , . , FR = kvp, 1 2 k , . , =1 =2 , 1 < < 2. " " " " , k.

, . , FR , =1 =2. , , .

2.5.1 , m

. FG FR , ( = 1). , , . , = 0 , :

FG = mg FR = kv (2.11) k v = dy/dt . , "" (2.11) FR ( ), (v ) FR ( ) (v ). , , :

F = FR + FG = FG = kv mg, F = m(dv/dt)

mgkvdtdvm = .

,

gvdtdv = , (2.12)

23

= k/m > 0. , , (2.12) dv/dt = v + g.

(2.12) , :

( ) gegvtv t

+= 0 . (2.13)

v0 = v(0) . , :

( ) gtvv

tT== lim .

, , , , :

kmggvT == . (2.14)

, . , .

(2.13) :

( ) ( )0 tT Tdyv t v v e vdt = = + . (2.15) :

y(t) = (1/)(v0 vT)et + vT t + C. 0 t y0 = y(0)

, , = y0 + (v0 vT)/ , : y(t) = y0 + vT t + (1/)(v0 vT)(1 et). (2.16)

(2.15) (2.16) v , . y0 , v0 . ( , , - aR = v.) vT, (2.14).

, , 1.5, |vT| 6.5 . -, , , , , 0.5, |vT| 20 .

2.4. , v0 = 49 / .

24

, = 0.04 (2.12). , , 2.3.

. y0 = 0, v0 = 49 vT = /g = 245 (2.13) :

v(t) = 294et/25 245, y(t) = 7350 245t 7350et/25. , ,

( v = 0). : v(t) = 294et/25 245 = 0

tmax = 25ln(294/245)4.558 (). ymax= v(tmax)108.280 ( 122,5 , ). , , :

y(t) = 7350 245t 7350et/25= 0. ,

t0 = 8 tn+1 = tn y(tn)/y'(tn) . MATLAB. clc clear a=7350; b=245; c=25; % t = 8:0.001:10; y = a-b*t-a*exp(-t/c); plot(t,y); grid on

8 8.2 8.4 8.6 8.8 9 9.2 9.4 9.6 9.8 10-30

-20

-10

0

10

20

30

40

50

60

25

, y(t) Ot t = 9.4. : f = @(t)a-b*t-a*exp(-t/c); z = fzero(f,9)

z = 9.4109. , tf 9.41 (

10 , ). |v(tf )| 43,227 / ( 49 /).

, , . , , .

2.5.2 , ,

: FR = kv2 (2.17)

k > 0. , . FR < 0 ( v > 0) FR > 0 ( v < 0). , FR v, (2.17) :

FR = kv|v| :

vkvmgFFdtdvm RG =+= ,

..

vvgdtdv = , (2.18)

= k/m > 0. . . ,

y0 v0 > 0. (2.18) v > 0 :

2 21dv g v g vdt g

= = + . , ( )

y0 v0 0. (2.18) v

26

2.6 1. Maserati

250 / . 100 / 10 . , 200 /?

2. , , v , dv/dt = kv. ) , t : v(t) = v0ekt x(t) = x0 + v0(1 ekt)/k. b) , , .

3. , 1.5 /, , 10 0.5 /. , 2, , , . ?

4. , , v, dv/dt = kv2. ,

v(t) = v0/(1+ v0kt) x(t) = x0 + ln(1+ v0kt)/k. , 2, x(t)

t . 5.

( 4), , 3 ?

6. , , v, , dv/dt= kv3/2. ,

( ) ( )20 0 24

+=

vkt

vtv

( )

++= 2212

000 vkt

vk

xtx .

, 3/2, .

7. , 3 /2,

27

0.05 /2 . ) () . b) , 90% , ? 8. 7, ,

- (0,001 )v2 /2, v /.

9. 14000 , 2500 . , 100 / v .

.100250014000 vdtdv =

, ?

10. 3000 , 20 , . , ? , - : pv /2 = 0.15 =1.5 . (. .)

11. , , 400 , , , . 160 / 8 . . (. , , 160 /. , 400 .)

12. , W = 260 2 , (v0 = 0). , , :

Rdvm W B Fdt

= + + , , ( ), FR , , 1.5 - . 22.5 , , , ? (. 1000 /3.)

28

3

, , , . , M m, :

2

GMmFr

= , (3.1) G . , , . 2006 G =(6.674280.00067)1011 --1, -.

, . r .

3.1. , 53 , , 1477 /. , , = 4 /2. , " " (v = 0 )?

. r(t) t (. 3.1). () () F/m = GM/r2 (3.1), , :

2

2 2

d r GMTdt r

= (3.2) = 7.351022 () , R = 1.74106 ( 1740 , ). , t, :

2

2, dr d r dv dv dr dvv vdt dt dt dr dt dr

= = = = :

2

dv GMv Tdr r

= r. r

29

212

GMv Tr Cr

= + + , (3.3) ( = 0), ( = 4).

. = 0 (3.3) :

21

12

GMv Cr

= + (3.4)

1 = v02 GM/r0,

01 147701477 1000

3600 36 v

= = r0 = (1.74106) + 53000 = 1.793106 ( -).

. = 4 v = 0, r = R ( ) (3.3) :

22

1 42

GMv r Cr

= + + (3.5) 2 = 4R GM/R v = 0, r = R .

(3.4) (3.5). , h , (3.3) (3.4). r = (C1 C2)/4 = 1.78187106 , , h = r R = 41870 (. . 41.87 ). , r (3.4) v = 450 / .

. 3.1. , .

R

r R

30

4 , 1865

, , , , . , v0 , . , v=dr/dt t > 0, . r(t) t (. 4.1).

:

22

2

rGM

dtrd

dtdv == (4.1)

(3.2), = 0 ( ) M = 5.9751024 () , R = 6.378106 (). dv/dt = v(dv/dr) :

2rGM

drdvv = .

r : C

rGMv +=2

21 .

v = v0 r = R, t = 0, C = (1/2)v2 GM/R, , , v2 :

+=Rr

GMvv 112202 .

(4.1) v r . :

RGMvv 220

2 >

v(t) r

m

r(t)

M

R

. 4.1. .

31

v , 202GMv

R> .

:

02GMv

R= . (4.2)

G R, v0 11180 (/).

. (4.2) () , . , R , , v0 2375 /. . , (. ).

32

1. 329320 , , 109 .

) ( ) , , . . c = 3108 / ?

b) () . 2. , , ,

, , r(t) ( )

( ) ( ) ( )2

022 2 ; 0 , 0 ,e mGM GMd r r R r v

dt r S r= + = =

Me Mm ; R S = 384400 . , , . " . v0, .

33

5 , . 5.1.

t = 0. y v = dy/dt t. , ( ) ( ) . - m m(t) .

,

: dP Fdt

= , (5.1) ( ), a F (, . .).

m , (5.1) F = m(dv/dt) = ma. m .

, m m + m, v v + v t t + t. :

P (m + m)( v + v) mv = mv + vm + mv. ,

, m, v c. , t :

P (m + m)( v + v) mv = (mv + vm + mv) + (m)(v c) = mv + cm + mv.

x

v

F

c

y

. 5.1. .

34

t t 0, dP/dt (5.1), :

dv dmm c Fdt dt

+ = . (5.2) F = FG + FR, FG = mg FR = kv

, , (5.2) :

dv dmm c mg kvdt dt

+ = . (5.3)

35

5.1 ,

[0, t1], m0 m1. , m(0) = m0, m(t1) = m1, m(t) = m0 t, dm/dt = t t1, (5.4) t = t1.

5.1. (5.4) (5.3), :

( ) ( )0 0dvm t kv c m t gdt + = . (5.5) , :

( ) ( ) ( )/ / /00 1 1k k kgmcv t v M M Mk k = + + , v0 = v(0) M = m(t)/m0 = (m0 t)/m0 t.

36

5.2 5.2.

k = 0 (5.5) . : v(t) = v0 gt + cln(m0/(m0 t)). (5.6)

m0 t1 = m1, , (t = t1) :

v1 =v(t1) = v0 gt1 + cln(m0/m1). (5.7) 5.3. (5.6) .

: y(t) = (v0 + ct)t (1/2)gt2 (c/)(m0 c)ln(m0/(m0 t)).

, y1 = y(t1) = (v0 + ct)t1 (1/2)gt12 (cm1/)ln(m0/m1).

5.4. V-2, , 12850 , 68.5 % . , 70 , 2 /. , 1.45 . V-2 , , . Maple > # V-2 (-2) > restart: > # () > M_start_common:=12850: > # > p:=0.685:#68.5 % > # () > M_start_fuel:=M_start_common*p: > m[0]:=M_start_common: > # (Weight of useful loading) > M_useful_loading:=M_start_common-M_start_fuel: > # (/) > beta:=M_start_fuel/70: > # > g:=9.81: > # (/) (Speed of emission of exhaust gases) > c:=2000: > #

37

> k:=1.45: > # > P:=t->k/(m[0]-beta*t): > Q:=t->beta*c/(m[0]-beta*t)-g: > my_sys_ode:=diff(v(t),t)+P(t)*v(t)=Q(t): > # > ro:=simplify(exp(int(P(t),t))): > # > Q(t):=simplify(Q(t)*ro): > # > ro_v:=simplify(int(Q(t),t) + 1): > # - > V:=ro_v/ro: > `1`:=C1: > rrr:=eval(V,t=0)=0: > C1:=solve(rrr,C1): > # > h:=int(V,t=0..70);

:= h 41436.56270

> plot(V,t = 0..70);

38

5.3 , ,

, , , g = k = 0. g = 0 (5.7) , m0 m1 v = v1 v0 = cln(m0/m1).

, v m0/m1, . , (v0 = 0) = 5 / m0/m1 = 20, v1 = 5ln20 15 /. , , , , () .

39

6 6.1

: dx kxdt

= (k ) (6.1) , . .

[7]. , P(t) (, ), ( ). t P(t)t P(t)t , P(t) :

P(t) ( )P(t)t :

0lim

t

dP P kPdt t

= = , (6.2) k = .

[8]. A(t) ( ) t ( ). , r. ( , 10%- r = 0.10) , t A = rA(t)t, :

0lim

t

dA A rAdt t

= = .

N(t) t. , ( ) . , , ( ). N(t), (6.2), N , k > 0 = 0. , :

dN kNdt

= .

40

k . ,

, , 14 . , 14 , , 14 12. , , , , , .

14 , , 14 , - 14N ( ) 14 . , .

, , , 14. 14 , 14 . , , . k. 14 : k 0.0001216, t .

( , . , . , , , 14 . , , .)

. , A(t) , . :

dA Adt

= > 0. .

41

6.2 dx/dt = k x(t) > 0

( ) k :

1dx kdxx

= ln x = kt + C. :

eln x = ekt + C; x = x(t) = eCekt = Aekt. = .

= (0) = 0, (6.1) (0)=0 :

x(t) = 0ekt. (

, ), :

dx kxdt

= , . . 6.1 .6.2 x(t) k > 0 k < 0.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 11

2

3

4

5

6

7

8x(t)=x0*ek*t

t

x(t)

. 6.1. (k>0)

42

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1x(t)=x0*ek*t

t

x(t)

MATLAB clc clear k=2; % -2 t=0:0.01:1; x0=1; P=x0*exp(k*t); plot(t,P); title('x(t)=x0*e^k^*^t'); xlabel('t'); ylabel('x(t)'); grid on

6.1. , www.census.gov, 6.7 2008 . 21 18 . , , :

() k? () XXI

? () ,

, .. 60 ? ( , ,

. 6.2. (k < 0 )

43

.) ) P(t) , . t = 0 2008 , P0 = 6.7. (21-18)606024 = 259200, 0.0002592 t=0. ,

'(0) = (0.0002592) (365.25) 0.09467 . ' = kP t = 0 :

( )( )

0 0.09467 0.014130 6.7

Pk

P= .

, 1.41% ( , 2008 ). k :

P(t) = 6.70.01413t () t = 51

(51) = 6.7e(0.01413)(51)= 6.7e0.72063 13.77 () 2050 ( 2008 , ). (c) 60 ,

60 = 6.70.01413t; . . t = (1/0.01413)ln(60/6.7) 155 . . 2163 .

. , , . , , P(t) , , , , , .

, , . , :

(t) t;

(t) t.

44

, [t, t + t], () :

(t)P(t)t, (t)P(t)t. , P

[t, t + t] t : P={ }{ }(t)P(t)t (t)P(t)t, :

( ) ( ) ( ).P t t P tt

t0, :

( )dP Pdt

= , (6.3) = (t), = (t), P = P(t). (6.3) . , (6.3) k = . , t. . P(t).

6.2. , 100 = 0 ( ). = 0.0005. , , (6.3) ( ):

( ) ( )20.0005 , 0 100dP P Pdt

= = (t ). :

( )21 10.0005 ; =0.0005 .dP dt t CP P= + t = 0, = 100 = 1/100, pe

: P(t) = 2000/(20 t). , , (10) = 2000/10= 200, 10

. P t 20, "" 20 .

45

6.3

. . , . , , , , = 0 1P, 0 1 . = 0 , (6.3) :

( )0 1 0 ;dP P Pdt =

2 ,dP aP bPdt

= (6.4) = 0 0 b = 1.

b , (6.4) . P(t) , :

( ) ,dP kP M Pdt

= k = b M = a/b .

6.3. , :

dP/dt = 0.0004P(150 P) = 0.06P 0.0004P2. . ,

. :

( ) 0.0004150dP dt

P P= 1 1 1 0.0004150 150 dP dtP P + =

ln ln 150 0.06P P t C = + 0.06 0.06 .150

C t tP e e BeP

= = ( B = eC)

t = 0 = 0 150 , =0/(150 0). ,

0.060

0

.150 150

tP ePP P

= ,

:

( ) ( )0 0.060 0150

150 tPP t

P P e= +

46

t, 0 = (0).

. 6.3 , 0 = 20 0 = 300. , = 150. , , ( )lim 150

tP t =

0 > 0. MATLAB

clc clear k=-0.06; t=0:0.2:150; length_t = length(t); P=t; P0=[ 20; 50; 80; 100; 120; 150; 200; 240; 270; 300]; length_P0 = length(P0) for i_P0=1:length_P0 for i_t=1:length_t P(i_t)=150*P0(i_P0)/(P0(i_P0)+(150-P0(i_P0))*exp(k*t(i_t))); end plot(t,P); hold on end title('P(t)=150*P0/(P0+(150-P0)*e^-^0^.^0^6^*^t');% \lambda xlabel('t'); ylabel('P0'); hText=text(10,30,'\leftarrow P0=20'); hText=text(12,210,'\leftarrow P0=300'); grid on

47

0 50 100 1500

50

100

150

200

250

300P(t)=150*P0/(P0+(150-P0)*e-0.06*t

t

P0

P0=20

P0=300

. 6.3. P=0.06P 0.0004P2.

48

6.4 ,

. 6.4.1 .

. , ( - ) M P, MP . = k(M P),

( ) ( )dP P kP M Pdt

= = .

. 6.4.2. .

, , = ,

( ) ( ).dP P P kP M Pdt

= =

, - . , , .

6.4.3 . P(t)

, . . P'(t) M P, dP/dt = k(M P). , . - .

6.4. , t = 0

=100 10 . 1 P(t) , , (1) = 20 . , P(t) , , 80% .

.

49

( ) ,dP kP M Pdt

= P(0)=P0.

( ) ( )00 0 .kMtMPP t

P M P e= +

MAPLE : > my_diff_yravn2:=diff(P(t),t)=k*P(t)*(M-P(t));

:= my_diff_yravn2 = ddt ( )P t k ( )P t ( ) M ( )P t

> ics:=P(0)=P[0]; := ics = ( )P 0 P0

> v12:=dsolve({my_diff_yravn2,ics});

:= v12 = ( )P t P0 M + P0 e

( )k M tM e

( )k M tP0

0 = 10 = 100 () , :

P(t)=100/(1+9e-100kt). :

P(1)=100/(1+9e-100k)=20. e-100k :

e-100k = 4/9 k = 0.01ln(9/4) 0.008109. P(t)=80 :

80=100/(1+9e-100kt). t:

e-100kt = 1/36. , 80% ,

t = (ln36)/(100k) = (ln36)/(ln 9/4) 4/42, .. 4 3 .

50

6.5 P(t)

, . , /2 /2, , 2. , k2 ( , k ). ( // ) = kP. , (1)

( )2dP kP P kP P Mdt

= = (6.5) ( = /k > 0), .

, (6.5) , (6.4). , , , 0 , .

6.5. P(t),

:

( )20.0004 0.06 0.0004 150 .dP P P P Pdt

= = (6.6) P(t) ) (0) = 200; b) (0) = 100.

. (6.6), . :

( ) 0.0004150dP dt

P P= 1 1 1 0.0004150 150 dP dtP P =

ln ln 150 0.06P P t C = + 0.06 0.06 .150

C t tP e e BeP

= = ( B= eC)

a) t = 0 = 200 , = 4. ,

0.064150

tP eP

= ( )0.06 0.06 0.064 150 4 600t t tP e P e P e = = 0.06

0.06

6004 1

t

t

ePe

= . (6.7)

51

, t, T = (ln4)/0.06 23.105, (6.7) 0. , P t T. .

b) t = 0 = 100 = 2. :

0.06

0.06 0.06

300 3002 1 2

t

t t

ePe e

= =+ + (6.8)

, , t (6.8) . , P 0 t . .

, 6.5 ,

, , = 150. , .

52

6.6 :

2dx ax bx hdt

= (6.9) , b h ) . , , h .

6.6. (6.9) :

( )dx kx M x hdt

= , (6.10) h = 0, .. . , h > 0, k2 + k h = 0 :

( ) ( )2 24 1, 4 /2 2kM kM hkH N M M h kk = = . , h

, , 4h < k2, , H N , 0 < < N < . (6.10) :

( )( )dx k N x x Hdt

= .

. 6.7.

6.6 , k=1 =4 x(t) , , t . , 400 . , h=3, 300 ( ). (6.10) dx/dt = (4 x) 3,

x2 + 4 3 = (3 )( 1) = 0 = 1 N = 3. , 100 , () 300 . , 100 , t - 300 . 100 , , .

53

6.7

. , . k , t = N =(1/2)N0 N(t) = N0e-kt, (1/2)N0 = N0e-k. , , = ln2/k.

, 14 (ln2)/(0.0001216), . . 5700 .

6.8. , ,

63% 14 . ?

. t = 0 , , N0 14, . N=(0,63) N0, (0,63)N0= N0e-kt k = 0.0001216. ,

t = - ln (0.63)/0.0001216 3800 (). , 3800 .

- , , , 1800 .

54

6.8 1. ( .) 25000 2008 , 2010 30 000 . , . 2015 ? 2. ( .) , 10 . , ? 3. 30 120 000 . S(t) , S(t)=120t/20 t . 12% , 6%. () t, , A(t) t . (b) A(25) 55 . 4.( .) , , 14, , . ? 5.( .) , , , , 4.61010 14 . , , 5.01010 14 . . ? 6. ( .) cy 300000 . , 8% , . . ? 7. ( .) , , 50 100 . 5%- , , , ? 8.( .) , (pentobarbital) . , 45 .

55

, , 5 . , 50- 1 ? 9. 5.27 . , 100 , -. ? ( .) 10. , , , - , 238U ( 4.51109 ), 238U. 238U 0.9, ? 11. . , ( 1.28109 ) . , , ? 12. , , t , dA/dt = kA (k > 0). 25% 1 , , ? 13. I dI/dx = (1.4)I. () I0 ( = 0)? (b) 10 ( I0 10 )? (c) 1% ? 14. 100 000 . 10000 . , . , , . ? 15. , 235U 238U. 235U 137.7 238U. 238U (4.51109 ) 235U ( 7.1108 ), .

56

16. , . 15 , 5 10 . , 1 . (a) , A(t) ( ) t . (b) 8 ? (c) = 1 , ? 17. 3300 . , 1.5 6 . ? 18. . . , - , . , 1.5 6000 . 150 . ? 19. , . 7 . 8 3 , ( 10 ), 3 . (a) t = 0 , ; , t. , (. . , ). , :

1dxkdt t

= k (b). ? 20. C 7 , 19. , 8 6 9 4 . ? , , k, .

57

7 7.1

, T(t) , , T:

( )dT k T Adt

= k . :

dx ax bdt

= + . (b = 0)

. 7.1. ,

20, 180 17:00. 30 , T(t) 55 . 85 ( , ).

. t , t = 0 17:00. ( ), T(t) . T(t) < = 180, (0) = 20 (30) = 55. :

( )180= TkdtdT

; dtkdTT =1801 ; -ln(180-T) = kt + C; 180-T = Be-kt.

T(0) = 20 = 160, T(t) = 180 160e-kt. , = 55 t = 30. :

k = -(1/30) ln(125/160) 0.00822. , :

85 = 180 -160e(-0.00822)t tf = -[ln(95/160)]/(0.00822) 64 (), , . 17:00, 18:04.

MAPLE > restart: > ins:=T(0)=20: > A:=180: > my_diff:=diff(T(t),t)=-k*(T(t)-A);

:= my_diff = ddt ( )T t k ( ) ( )T t 180

> dsolve({my_diff,ins}); = ( )T t 180 160 e( )k t

58

> T:= t -> A-160*exp(-k*t): > r:=T(30)=55;

:= r = 180 160 e( )30 k 55 > k:=evalf(solve(r,k));

:= k 0.008228669263

> t[f]:=evalf(solve(T(t)=85,t)); := tf 63.35130347

59

7.2 ,

, , , :

A(t) = 0 + 1cos t + b1sin t. = /12, 24 (

). , . 21 t = 4 (), 33 , t = 16 (4 ). :

A(t) = 27 6cos((t 4)) =27-3cos(t) 3(3)1/2sin(t).

u(t) t, A A(t) , :

( )( );du k u A tdt

=

( )0 1 1cos sindu ku k a a t b tdt + = + + (7.1) () P(t)k Q(t)kA(t) 0.2 0.5 ( k , 0.5 , , 0.2 ( ) ).

. , ( t0 = 0) , . , .

(7.1) u(t)=u0 ( ). . :

u(t) = 0 + c0ekt + c1cos t + d1sin t,

2 2 2

1 1 1 1 1 10 0 0 1 12 2 2 2 2 2, , ,

k a k b k a k b k a k bc u a c dk k k

= = =+ + + = /12.

60

0 = 27, a1 = 3, b1 = 3(3)1/2 = /12 k = 0.2 (), () : u(t)=27+et/5(u023.3877)3.61225cos(t/12)+0.46772sin(t/12). (7.2)

, (7.2) t, " " :

usp(t) = 27 3.61225cos(t/12) + 0.46772sin(t/12). , 24

27 , .

, , k . . ( ?)

61

7.3 1. , 25. 0. , 20 15. 5? 2. 100 22 . 30 60. 38? 3. 22 . 12 27, 13:00 - 24. , 36,6 . ? 4. , = 1 , r = 0. r = kt (k ), ( ). ( ):

( ) ( ), 0 0,d mv mg vdt

= = m , v = dy/dt , . , dv/dt = g/4. , , .

62

8 , ,

. . y(t) t, V(t) . , , :

2v gy= (8.1) .. , . , , . , , :

2v c gy= 0 1 ( 0.6 ). = 1 .

(8.1) :

2dV av a gydt

= = ,

dV k ydt

= , 2k a g= .

, . () , :

( )0

y

V A y dy= , dV/dy = (), :

( )dV dV dy dyA ydt dy dt dt

= = (8.2) (8.1) (8.2) :

( ) 2dyA y a gy k ydt

= = . (8.3) .

63

8.1. () 4 . t=0 - 0.1 . ?

. . 8.1 : () = r2 = [16 - (4 - )2] = (8 - 2).

(8.3) :

( )28 dyy k ydt = 1 32 28 ky y dy dt

= 3 52 216 2

3 5ky y t C = + .

, = 0, . . : tf = C/k.

(0) = 4. : 3 52 216 2 4484 4 29.8666

3 5 15C = = .

: tf 3.1429.8666/[0.1 (29.81)1/2] 2118.298.

, 35 20 . 36 , .

. 8.1.

64

1. , 3 . t = 0 () . 1 1 . , ? 2. , 35 0.9 0.025 . , , 3 ? 3. t = 0 ( ) 5 . 1 3 . ? 4. , , V0 , T . , , tT V=V0[1-(t/T)]2. 5. , = 4/3 y. 12 , 4 . 13:00 2 . ? 6. , 2 = by y. 1.5 12 , . 13:00 0.4 . () y(t) , t . (b) ? () , () 0.6 . ? 7. 1.5 0.9 , . , 0.025 . , ? 8. 1.5 , 0.025 . , ? 9. , 1 . 1 . 13:00, ?

65

10. . r . 13:00 , 13:30 0.6 . () :

20.6 2dV r gydt

= ( ), , . (b) ? 11. (, .) () 12- , , . 8.2 (r = 0.5 , h = 1.5 ). , , = f(x) . ? , 10 ? 12. . 8.3 , Q. , , ( Q) . 1696 6 ( 1697 ). , , 29 1697 .

x

y

h

r

y=f(x)

x=g(x)

. 8.2.

66

: . , , = () , . , , , : (sin )/v = const (i) ( ) , ctg = '() (?) v = (2gy)1/2 , ( = (1/2)mv2 = mgy = ).

(i) : 2dy a y

dx y= (ii)

a . (b) = 2 sin2t, dy = 4 sint cost dt (ii), :

x = a(2t sin2t), = a(l cos2t), (iii) t = = 0, = 0. , = 2 (iii) = ( sin), = a(l cos) , , x.

. 8.3. ,

67

9

, , , . , x(t) t, x(0) = x0 t = 0. , - ci ri , r0 .

x(t), x [t, t+t]. , t ricit . , :

( ) i i r c t

, . ,

, 0(t) t. , . 9.1, 0(t) = x(t)/V(t), V(t) ( ri = r0) t.

x = { } { } ricit r0c0t. t:

0 0i ix rc r ct

. , t 0.

x(t) , , :

0 0i idx rc r cdt

= ri, ci r0 , c0

0(t)=x(t)/V(t) t. , x(t) :

68

0i i

rdx rc xdt V

= . (9.1)

V0 = V(0), V(t) = V0 + (ri r0)t, (9.1) (t) t.

. (9.1) . [t, t+t], .

9.1. , 480 3

( ) ( ) 350 3 . , t = 0 (), , ( !), , . , . , ?

. : V = 480 (3), ri = r0 = r = 350 (3/), ci = c(

) 0 = (0) = 5V, : x(t)=2cV? (9.1) :

dx rrc xdt V

= , :

dx px qdt

+ =

: r0c0t

c0=x/V V(t) x(t)

: ricit

. 9.1. .

69

p=r/V, q = rc. =ept. MATLAB.

clc clear x=dsolve('Dx=r*(c-x/V)','t')

: x(t) =V*c C1/exp((r*t)/V).

: (0) = 5V = V*c C1 C1 = 4V. , , x(t) = 2cV,

: V*c + 4V/exp(rt/V) = 2Vc 1+4/exp(rt/V) = 2 4/exp(rt/V) = 1 4 = exp(rt/V) ln 4 = rt/V t = (V/r) ln 4 = (480/350) ln 4 1.901 ().

9.2. 450- 40 340 . , 480 , 15 . 11 . ?

. - , V(t) = 340+t . t t + t () :

(15)(0.48) t 11x/(340+t)t, :

11 7.2340

dx xdt t

+ =+ . :

( ) ( ) ( )11 1111ln 340340 340dt ttx e e t ++= = = + . :

Dt[(340+t)11x] = 7.2(340+t)11 (340+t)11x = (7.2/12)(340+t)12 +C= 0.6(340+t)12 + C.

: (0) = 40. C = 164(340)11. t :

x(t) = 0.6(340+t) 164(340)11/(340+t)11. 27.5 , t = 27.5, :

x(27.5) = 0.6(340+27.5) 164(340)11/(340+27.5)11 150 () .

70

1. 1000 100 . 5 , . , 10 ? 2. 220 , 0,25%. 14000 0,05% . , 0.10%? 3. 220 . , 200 , 6 , ( ) 11 . , 55 . () t . (b) ?

71

10

[6]. , ( ) m , FS, , FR (. 10.1). , FS ( ), FR v = dx/dt. . 10.2 :

FS = kx (10.1)

FR = cv (k, c > 0). , "" FS

x, FR v. F = ma :

mx = FS + FR. ,

2

2 0.d x dxm c kxdt dt

+ + = (10.2) , ,

x(t) m. .

, , FS FR m F(t), (10.2) :

( )22 .d x dxm c kx F tdt dt+ + = (10.3)

x, v > 0

m FRFS

. 10.2. , m. . 10.1. ,

, -

x = 0 x(t)

m

x > 0

72

F(t).

, . (. 10.3). W = mg s0, (10.1), FS = W = s0. , mg = ks0, s0 = mg/k. - . , . (10.3) .

.10.3. ,

y

s0

m

m

y=0

73

11 (10.3) ,

. , , . m, ( ) L, , . 11.1. t ( ) (t) ( ). m, , .

0

m s = L. , v = ds/dt = L(d/dt),

2 22 21 1 1

2 2 2ds dT mv m mLdt dt

= = = ,

(. 11.1). V mg h = L(l cos) , . .

V = mgL(l cos). T V , :

( )221 1 cos2

dmL mgL Cdt + = .

t, :

( )22 2 sin 0d d dmL mgLdt dt dt + = .

0

mh

L

. 11.1.

74

, mL2(d/dt), : 2

2 sin 0d gdt L

+ = . (11.1) , sin .

|| /12 ( 15) sin . , 15. , (11.1) sin . c , , (10.2):

+ c + k = 0, (11.2) k = g/L.

, m , . , sin , (11.2) .

, .

75

1 4 ,

L L +g = 0, g = GM/R2 , ( R ; M ).

1. L1 L2, R1 R2, , p1 p2. ,

1 11

2 2 2

.R Lp

p R L=

2. , R63665 (), 2 40 . 1, . 3. 2.54 , , R = 63730 (), , 2.54 , . 1, . 4. , . 10 76.2 . , ?

76

12 ,

, , (10.2) :

mx + kx = 0. (12.1)

0km

= (12.1) :

x + 02x = 0. (12.2) (12.2) :

x(t) = s0t + Bsin0t. , ,

, 2 2 ,C A B= + cos A

C = sin B

C = . (12.3)

. 12.1.

, tg = /,

, /2 < < /2. , . 0 2 , (12.3), A, . :

( )( )

( )/ , 0, 0 ( )

/ , 0 ( )2 / , 0, 0 ( )

arctg B A A Barctg B A A

arctg B A A B

> >= +

77

, , .

1. C. 2. 0. 3. . . t , 0

(/). , (). :

T = 2/0 . :

= 1/T = 0/2 (). , 1 . , , .

, , , .. , : x(t) = C(cos(0t ) = C(cos(0(t /0)) = C(cos(0(t )), = /0 .

12.1. m = 1/2 () . 100 () 2 (). 0 = 1 () v0 = 5 (/). (, , t = 0 .) , , , .

. k = 100()/2(). (12.2) (1/2)" + 50 = 0

" + 100 = 0. ,

0 = (100)1/2 = 10 (/). : T = 2/0 = 2/10 0.6283 (),

: = 1/T = 0/2 = 10/2 1.5915 ().

(0) = 1 (0) = 5 - :

x(t)=Acos10t + Bcos10t x(t) = 10Asin10t + 10Bcos10t. A=1 B= 1/2, ..

: x(t) = cos10t (1/2)cos10t.

78

, :

( ) 521

211

22 =

+=C ().

, :

( ) ( )=

= ttttx 10cos2510sin

5110cos

52

25 ,

: 0

52cos >= 0

51sin

79

MATLAB % clc clear t=0:0.005:3; C=1.118; T=0.628; sigma=0.582; omega_0=2*pi/T; alfa=sigma*omega_0; y=C*cos(omega_0*t-alfa); plot(t,y); title('x(t)=C*cos(wt-\alpha)'); xlabel('t'); ylabel('x');

80

13

: m" + cx + k = 0 " + 2px + 02 = 0, (13.1)

0 = (k/m)1/2 p =c/(2m) > 0.

r2 + 2pr + 02 = 0 (13.1):

r1,2 = p (p2 02)1/2. (13.2)

p2 02= c2/(4m2) k/m = (c2 4km)/( 4m2)

ccr ccr =

(4km)1/2. : > ccr, = ccr, < ccr.

13.1 : > ccr, (2 > 4km).

, . (13.2) r1 r2, . :

x(t) = c1exp(r1t) + c2exp(r2t). , x(t) 0 t .

- . 13.1 .

x0, v0. ( ).

MATLAB % clc clear t=0:0.005:5; C1=-4; C2=5; r1=[-4, -1, -2, -0.8]; r2=[-6, -5, -3, -0.9]; for i=1:length(r1) x=C1*exp(r1(i)*t)+C2*exp(r2(i)*t); plot(t,x); hold on

81

end title('x(t)=C1*e^r^1^*^t+c2e^r^2^*^t'); xlabel('t'); ylabel('x'); grid on

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-2.5

-2

-1.5

-1

-0.5

0

0.5

1x(t)=C1*er1*t+c2er2*t

t

x

. 13.1. : ( ) 1 21 2r t r tx t c e c e= + r1 < 0 r2 < 0. x0 .

82

13.2 = ccr, (2 = 4km). -

, (13.2), : r1 = r2 = p. :

x(t) = ept(c1 + c2). ept > 0, c1 + c2

t, . , x(t) 0 t . . 13.2.

, (. . 13.1). . , .

MATLAB % clc clear t=0:0.005:10; length_t =length(t); x=t; C1=1.5; C2=-3.0; p=[4, 1, 2, 0.8]; length_t =length(t); for i=1:length(p) for j=1:length_t x(j)=exp(-p(i)*t(j))*(C1 + C2*t(j)); end plot(t,x); hold on end title('x(t)=e^p^*^t(c1+c2*t'); xlabel('t'); ylabel('x'); grid on

83

0 1 2 3 4 5 6 7 8 9 10-1

-0.5

0

0.5

1

1.5x(t)=ep*t(c1+c2*t

t

x

. 13.2. : ( ) ( )1 2 ptx t c c t e= + r1 < 0 r2 < 0. x0 .

84

13.3 < ccr, (2 < 4km).

2 20 .p i p : x(t) = ept(Acos1t + Bsin1t), (13.3)

22 2

1 04 .

2km cp

m = = (13.4)

, (13.3) :

( ) 1 1cos sin ,pt A Bx t Ce t tC C = +

x(t) = Ceptcos(1t ),

2 2 ,C A B= + cos AC

= sin BC

= . (13.3)

. x(t) x(t) = Cept Cep, , 1t . , , , 1 (, , ), T1 = 2/1 , Cep , . , 13.3. (13.4) , 1 , 0, T1 T , , . , , , .

1. ( () ).

2. , . . .

, ( ) .

13.1. , 12.1, , , , 1 . (0) = 1 '(0) = 5, 20.

85

, , , , = 0.

. . . , m = 1/2 k = 50. = 1. (10.2) (1/2)" + ' + 50 = 0 " +2' + 100 = 0.

: r2 + 2r + 100 = (r + 1)2 + 99 = 0. r1,2 = 1 i(99), :

( ) ( )cos 99 sin 99 .tx t e A t B t= + (13.5) , 1 =

(99) 9.9499 ( 0=10 12.1). T1 = 2/1 0.6315 1 =1/T1 = 1/2 1.5836 ( T 0.6283 < T1 1.5915 > 1 12.1).

(0) = 1 '(0) = 5, - (13.5). :

( ) ( ) ( )cos 99 sin 99 99 sin 99 cos 99 .t tx t e A t B t e A t B t = + + + ,

(0) = A = 1 '(0) = A + B(99) = 5. . A = l B = 4/(99). ,

:

( ) 4cos 99 sin 99 .99

tx t e t t = :

( )2

2 4 1151 .9999

t t tCe e e = + = , :

( ) ( )1115 4 115cos 99 sin 99 cos 99 ,9999 99t tx t e t t e t = = 1 :

cos 1 = (99/115)1/2 > 0 sin 1 = 4/(115)1/2 < 0. 1 , . .:

14 / 115 42 2 5.9009,99 / 115 99

arctg arctg = + =

: 1 = /1 0.5931 .

86

( 0.5820 < 1 12.1). - , :

( ) ( )115 cos 99 5.9009 .99

tx t e t (13.6)

, , . 13.3 ( 12.1).

0 0.5 1 1.5 2 2.5 3 3.5 4-1.5

-1

-0.5

0

0.5

1

1.5

t

x

MATLAB

% : clc clear t=0:0.005:4; Csqrt115=sqrt(115); Csqrt99 =sqrt(99); Csqrt115del99 =sqrt(115/99); x1=Csqrt115del99*exp(-t); x2=-Csqrt115del99*exp(-t); x0=Csqrt115del99*(cos(Csqrt99*t)/Csqrt115del99-4*sin(Csqrt99*t)/Csqrt115); x = x0.*exp(-t);

. 13.3 x(t) = C1etcos(1t - 1) 12.1 ( ), x(t) = Ccos(0t - ) 12.1 ( ) x(t) = 1et.

87

plot(t,x); hold on plot(t,x0); hold on plot(t,x1); hold on plot(t,x2); xlabel('t'); ylabel('x'); grid on

(13.5) , (

= 0) , cos(1t - 1) = 0, . . : 1t - 1 = 3/2, /2, /2, 3/2, ,

t = 1 3/(21), 1 /(21), 1 + /(21), , 1 + 3/(21), . , 12.1 ( )

: t = 0 3/(20), 0 /(20), 0 + /(20), 0 + 3/(20), . 13.1 t1,

t2, t3, t4, . , .

3.1.

n 1 2 3 4 tn () 0.1107 0.4249 0.7390 1.0532 tn () 0.1195 0.4352 0.7509 1.0667

88

13.4 1. 4 16 /. 2. 0.75 48 /. 3. 3 , 20 15 . 0 = 0 v0 = 10 /. , . 4. 250 , 25 9 . t = 0 1 ( ) 5 /.

) x(t) Ccos(0t - ). b) .

5. r h 0.5 (, 1 /3). , , . , . t = 0 . : , mg = r2hg, , r2xg , = x(t) , t (. 28). , = ph = 2(h/g)1/2. , = 0.5 /3, h = 200 , g = 980 /2. 6. 45.5 , . , , 10 . , .

h

x

. 13.5. 6.

r

89

7. , , R = 63730 (), . , . m, r , Fr = GMrm/r2, r , r. ) , Fr = GMmr/R3. b) ,

, . m t=0 . r(t) t (. 13.6). (), , r"(t) = k2r(t), k2 = GM/R3 = g/R.

c) g = 9.81 /2. (b), , 84 .

d) ( ) , , . (). ? ? ) ( /) ? f) ( ) ,

, . (). ? ?

8. , . :

m

R

r FR

. 13.6. m, , ( 7)

90

: m = 25, = 10 k = 2. , (0) = 0 '(0) = 5.

) x(t) . b) , ,

.

91

14

:

( )22 .d x dxm c kx F tdt dt+ + = (14.1)

m, ( k) ( ), F(t). , , ( ), :

F(t) = F0cos(t) F(t) = F0sin(t), (14.2) F0 , .

, , (. 14.1).

( m0) m m0.

a . ( ). ( k) . , . x(t) ( ). . 14.1 , x :

( ) ( )0 0 0cos cos .m m x m x a t amx x tm m

+ += = + ,

,mx kx = k. x . :

k

. 14.1. ,

x

m0

t

92

mx m0a2cost = kx, ..

mx + kx = m0a2cost. (14.3)

, , F0 = m0a2. , . (14.1) (14.2) .

93

14.1

F(t) = F0cos(t) (14.1) = 0. :

mx + kx = F0cost. (14.4)

: xc = c1cos0t + c2sin0t.

0 = (k/m)1/2 (, ) , , .

, : 0. , =Acost (14.4). ( , '.) :

m2Acost + kAcost = F0cost,

A = F0/(k m2) = F0/m/( 02 2), (14.5)

xp(t) = cost F0/m/( 02 2) (14.6) , = + :

x(t) = c1cos0t + c2sin0t + cost F0/m/( 02 2) (14.7) c1 c2 (0) (0).

(14.7) : x(t) = cos(0t ) + cost F0/m/( 02 2). (14.8)

: 0, .

14.1. , m = 1, k = 9, F0 = 80 = 5. (14.4) :

x + 9x = 80cos(5t). . 0 = 3

=5 , , . = Acos5t . 25A + 9A =80, A = 5. , :

= 5cos5t.

= c1cos3t + c2sin3t, :

x(t) = c1cos3t + c2sin3t 5cos5t,

94

x(t) = 3c1 sin3t + 3c2 cos3t + 25sin5t. (0) = 0 (0) = 0 , c1 = 5 c2 = 0,

: x(t) = 5cos3t 5cos5t.

MATLAB % clc clear y=dsolve('D2x+9*x=80*cos(5*t)','x(0)=0, Dx(0)=0'); y=simple(y) ezplot(y); grid on

-6 -4 -2 0 2 4 6

-10

-8

-6

-4

-2

0

2

4

6

8

10

t

5 cos(3 t) - 5 cos(5 t)

. 14.2, x(t) 2 2/3 2/5 .

. 14.2. x(t) = 5cos3t 5cos5t 14.1

95

14.2 , (14.7)

(0)=(0)=0, : c1 = F0/(m( 02 2)) c2 = 0.

:

( ) ( ) ( )0 02 202 cos cos .Fx t t t

m = (14.9)

2sinA sinB =cos(A B) cos(A + B) A = (0 + )t/2 B = (0 )t/2, (14.9) :

( ) ( )( ) ( )0 00

2 20

2 sin sin .2 2

t tFx tm

+ = (14.10) , 0, 0 +

0 . sin((0+)t/2) , sin((0)t/2) . , (14.10) () (0 + )/2,

x(t) = A(t)sin((0 + )t/2), :

( ) ( )( )00

2 20

2 sin .2

tFA tm

=

14.2. m = 0.1, F0 = 50, 0 = 55 = 45 (14.10) :

x(t) = sin(5t)sin(50t). . 14.3

(0 + )/2 = 50, "" A(t) = sin5t (0 )/2=5.

. , ( ) , 0/(2) = 258 , /(2) = 254 , , (.. ) :

( )0 / 2 258 254 22 2

= = ().

96

0 0.5 1 1.5 2 2.5 3 3.5-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1x(t)=sin(5*t)*sin(50*t)

t

x

. 14.3.

97

14.3 (14.6) , 0

, . (14.5) :

( )0 0 0

220

/1 /

F F k pFAk m k = = = . (14.11)

F0/k k F0, p . :

( )201

1 /p = (14.12)

, p 0. ( 0) ( ) 0) 0.

, 0. , 0 ? (14.4), m :

2 00 0cos .

Fx x tm

+ = (14.13) cos(0t)

, :

xp(t) = t(Acos(0t) + sin(0t)). (14.13) , = 0

= F0/(2m0). , : xp(t) = F0t sin(0t)/(2m0). (14.14)

xp(t), . 14.4, m=1, F0=100 0=50, , () , =0. .

98

0 0.5 1 1.5 2 2.5 3 3.5-4

-3

-2

-1

0

1

2

3

4x(t)=F0/(2*m*omega0)*t*sin(omega0*t)

t

x

MATLAB

clc clear F0=100; omega_0=50; m=1; t=0:0.005:pi; x=F0/(2*m*omega_0)*t.*sin(omega_0*t); plot(t,x); title('x(t)=F0/(2*m*omega_0)*t*sin(omega_0*t)'); xlabel('t'); ylabel('x'); grid on

14.3. , ,

(. 14.1), m = 5 k = 500 /. 0=(k/m)1/2 = 10 /; . . 10/(2) 1.59 . , 1.59 60 95 .

.

. 14.4.

99

. , , . , , , , . , . : , 1831 Broughton Bridge. " ". , - 1981 , Skywalk (" ") . - , . 19 1985 . , , . , , , , , , . , ( ) (). - . , .

100

15

RLC , [10]. , . 15.1, :

- ( ), R ;

- ( ), L ;

- ( ), .

(, ), t E(t) (). , . . . 15.1 . t I(t) (), Q(t) (). Q I :

( ).dQ I tdt

= (15.1) ,

.

15.1. , . 15.1, :

() , .

C

LE

R

. 15.1. , ,

101

15.1

dIL

dt

RI 1 Q

C

, ,

, (. 15.1), :

( )1 .dIL RI Q E tdt C

+ + = (15.2) (15.1) (15.2),

:

( )1LQ RQ Q E tC

+ + = (15.3) Q(t) E(t).

I, Q. (15.3), I Q' :

( )1LI RI I E tC

+ + = . (15.4)

102

15.1 , (15.3) (15.4)

, : m" + ' + kx = F(t), (15.5)

( , ), F(t).

, . 15.2 .

15.2

m L c R k 1/, Q ( (15.3) ( I (15.4)) F E (

E) ,

, . , , . , , 15.2, , . , . , (, ) , . . .

E(t) = E0sin(t). (15.4) :

103

.cos1 0 tEICIRIL =++ (15.6)

, , , , , , (15.) ( , ) Itr, t ( , (15.6) , ), Isp. :

I = Itr + Isp. (15.7) ,

(15.5) F(t) = F0cos(t) : ( ) ( )( ) ( ) ,

cos222

0

cmk

tFtxsp +=

= 2

mk

carctga , 0 . m L, R, k 1/ F0 E0,

:

( ) ( ) ,1

cos2

2

0

+=

CLR

tEtI sp

(15.8)

:

= 21 LCRCarctga , 0 . (15.9)

104

15.2 (15.8):

2

2 1Z R LC

= + () (15.10)

, , , . :

( ) ( ),cos0 = tZE

tI sp (15.11)

a : I0 = E0/Z, (15.12)

I = E/R. (15.11)

, E(t) = E0sin(t) . Isp , , :

S = L 1/(C). (15.13) Z = (R2 + S2)1/2. (15.9) , ,

15.2, = /2. (15.11) :

( ) ( ) ( ).cossinsincossincossinsincoscos 000 ttZEt

ZRt

ZS

ZEtt

ZEtIsp +=

+=+=

,

( ) ( ),sin0 = tZE

tI sp (15.14)

.12

RCLCarctg

RSarctg

== (15.15) ( )

/ ( ) Isp. .

= - /2. 15.2.

R

S

Z

R

SZ

105

15.3 , ,

I(0) Q(0). I(0). t=0 (15.2). :

( ) ( ) ( ) ( )10 0 0 0 ,LI RI Q EC

+ + = (15.16) I(0) , .

15.1. RLC (. . , , ), R = 50 (), L = 0.1 () = 5104 (). t = 0, I(0) Q(0) , 110 60 . .

60 , = (2)(60) /, . .

377 /. E(t) = 110sin377t ( , ). (15.6) :

(0.1)I + 50I + 2000I = (377)(110)cos377t. (15.10) R, L, = 377,

: Z = 59.58 . :

I0 = 110 ()/59.58 () = 1/846 (). (15.15),

, : = arctg(0.648) = 0.575.

: / = 0.575/377 = 0.0015 c,

: Isp = (1.846)sin(377t 0.575).

(0.1)r2 + 50r + 2000 = 0 r1 44 r2 456. , :

I(t) = c1e44t + c2e456t + (1.846)sin(377t 0.575),

I(t) = 44c1e44t 456c2e456t + 696cos(377t 0.575). I(0) = Q(0) = 0, (15.16) , I(0)

= 0. , :

106

I(0) = c1 + c2 1.004 = 0, I(0) = 44c1 456c2 + 584 = 0.

c1 = 0.307, c2 = 1.311. - :

Itr(t) = (0.307)e44t + (1.311)e456t. , |Itr(0.2)| < 0.000047 A (

, ). , .

15.2. , RLC 15.1

I(0) = Q(0) = 0 t = 0 110 . .

E(t) 110, (15.16) :

I(0) = E(0)/L = 110/0.1 = 1100 (A/c), :

(0.1)I + 50I + 2000I = E(t) = 0.

, 15.1: I(t) = c1e44t + c2e456t.

: I(0) = c1 + c2 = 0, I(0) = 44c1 456c2 = 1100,

c1 = c2 = 2.670. : I(t) = (2.670)(e44t e456t).

, I(t) 0 t , .

107

15.4 (15.6),

E(t) = E0sint. , :

( ) 0 00 22

.1

E EI tZ

R LC

= =

+ (15.17)

I0 , R, L, E0 . 15.3. I0

1/m LC = , . m .

( , , ).

, ,

. . RLC (. . , , ). L R , , - .

, , E(t) = E0sint . Isp , . , , I0 Isp.

( ) 00 22

.1

EI t

R LC

=

+

I0

m . 15.3. I0 .

108

( ) "" , , , I0. , (15.17) , C. , I0 :

1 0.LC

= , :

C = 1/(L2). , ,

, . .

- . . , . , , , 455 () . 455 ( ) . , RLC- , 455 , . (, ) .

109

15.5 1-6 RL-,

R , L (. 15.4) (), . (15.2) :

LI + RI = E(t).

1. , . 15.4 L = 5 , R = 25 , 100 . , 1 , . . 4 . t = 0 2, I(0) = 4 = 0 t 0. I(t).

2. , 1. , 2, t = 0 1 , I(0) = 0 = 100 t 0. I(t) , I(t) 4 t .

3. , 2 E(t) = 100cos60t . , , I(t).

4. , . 15.4. 1 L = 2, R = 40, E(t) = 10010t, I(0) = 0. t 0.

5. , . 15.4. 1 L = 2, R = 20, E(t) = 10010tcos60t, I(0) = 0. t 0.

6. , . 15.4. 1 L = 1, R = 10, E(t) = 30cos60t + 40sin60t.

a) Isp(t) = Acos60t + Bsin60t, A B , , Isp .

b) Isp(t) = Ccos(t ).

2

1

L

E

R

. 15.4. 1-6

110

7-10 RC-, . 15.5.

R , , , . L = 0 (3), , :

( )1RQ Q E tC

+ = Q=Q(t) t. , I(t)=Q'(t)

7. a) Q(t) I(t) , E(t)=0

( , ) t = 0, .. Q(0) = 0.

b) , ( ) 0limx Q t E C = ( )lim 0.x I t = 8. , . 15.5 R = 10, = 0.02, Q(0) = 0, a

E(t) =1005t (), a) Q(t) I(t). b) t 0

, . 9. , . 15.5 R=200, =2.5104, Q(0)=0,

E(t)=100cos120t. ) Q(t) I(t). b) ?

10. t = 0 RC-, . 15.5, , E(t) = E0cost, RC-. Q(0) = 0. Qsp(t) = Acost + Bsint , , :

( ) ( )02 2 2

cos1

spE CQ t t

R C = + ,

= arctg(RC).

C

E

R

. 15.5. 7 - 10

111

16

16.1 ()

, .

, . 16.1. , ( 16.1 ). = (), . . . , . 16.2. .

,

( , ['(x)]2 ), , :

EIy(4) = F(x), (16.1) E ;

I ; F(x) , .

? , . , , [, + ], F(x). F(x) . , , , , w , F(x) w. (16.1) :

EIy(4) = w, (16.2) , I w

. 16.1. () . 16.2.

y

L x

112

x = L x = 0

()

. 16.3.

x = L x = 0

. - (16.1) (16.2). , . , . , (16.2) , , .. . , , I, . . I = a4/4 .

, (16.2) , , . (16.2) , , :

EIy(3) = wx + 1. :

EIy = wx2/2 + 1x + 2. :

EIy = wx3/6 + 1x2/2 + 2x + 3. , :

EIy = wx4/24 + 1x3/6 + 2x2/2 + 3x + 4, 1, 2, 3 4 . (16.2) :

( ) 4 3 2 ,24

wy x x Ax Bx Cx DEI

= + + + + (16.3) A, , D , .

, .. x = 0 x = L. . 16.3 , . , . 16.4 x = 0, ( ) x = L. 16.1 , , . , .

. 16.4. ,

,

113

16.1. , ,

y = y = 0 () y = y = 0 y = y(3) = 0

, . 16.4

(16.3), A, , D :

y(0) = y(0) = 0 y(L) = y(3)(L) = 0, (16.4) . . = 0 , = L ( ).

(16.4) (16.2) .

16.1.

L w , .

. : y(0) = y(0) = y(L) = y(L) = 0.

(16.3), EIy(4) = w , () . :

EIy(3) = wx + A; EIy = wx2/2 + Ax + B. "(0) = 0, = 0, y(L) = 0 , :

0 = wL2/2 + AL. A = wL/2 :

EIy = x2/2 wLx/2. . :

EIy = wx3/6 wLx2/4 + , :

EIy(x) = wx4/24 wLx3/12 + x + D. (16.5) (0) = 0 , D = 0. , (L) = 0,

0 = wL4/24 wL4/12 + L. = wL3/24. (16.5)

:

( ) ( )4 3 32 ,24

wy x x Lx L xEI

= + , .

114

, ymax , , . x = L/2, :

4 4 4max

1 1 1 .2 24 16 4 2L wy y L L L

EI = = +

,

.3845 4

max EIwLy = (16.6)

MAPLE > restart: > # E - > # II - # > # w - > f:=E*II*diff(y(x),x$4)=w;

:= f = E II

d

d4

x4( )y x w

> # > yx0:=y(0)=0,(D@@2)(y)(0)=0,y(L)=0,(D@@2)(y)(L)=0: > simplify(dsolve({f,yx0},y(x)));

= ( )y x w x ( ) + x3 2 L x2 L3

24 E II

16.2. , , , , , 6.1 , 0.03 . , = 7.75 /3, E = 21012 /c2.

( ) : = a2 = (1.5)2(7.75) 54.75 /.

w = g = (54.75 /)(981 /c2) 53713.4 /c2.

( ) a I = a4/4, :

I = (1.5)4/4 3.97 4. (16.6) , :

ymax (5)( 53713.4)(610)4/[(384)( 21012)(3.97)] 12.2 . ,

, , . , ymax L4, , , 16 .

I = a4/4, (16.6) , , .

115

16.2 . 16.5 L.

. , .

,

= () 0 L. :

EIy + Py = 0, y(0) = y(L) = 0. (16.7) () . , , I , . :

= P/(EI), (16.8) (16.7) :

y + y = 0, y(0) = y(L) = 0. (16.9) , {n} :

2 2

2 ,nn

L = n=1,2,3, (16.10)

n, yn(x) = sin(nx/L).

, , (16.8) P = EI.

Pn = nEI = n22EI/L2, n = 1, 2, 3, (16.11) . P , "", () . , ,

P1 = 2EI/L2. (16.12)

1 . , , 1 , .

y

P P

x = 0 x

y = y(x)

x = L

. 16.5.

116

. ( , .)

16.3. , L = 3.5 , , 0.015 . (--) :

E = 2 1012 /2, I = (1.5)4/4 3.97 4 . (16.12), ,

:

P1 6.4 108 .

117

17

. , , ( ). . , t, x1, x2, x3, , x, y, z, . t.

, . , :

( )( )

, , , , 0, , , , 0

f t x y x yg t x y x y

= = f u g . t x(t), y(t) t, .

. m, F, t, (x(t), y(t), z(t)) (x(t), y(t), z(t)). ma = F , :

( )( )( )

1

2

3

, , , , , ,, , , , , ,

, , , , , ,

mx F t x y z x y zmy F t x y z x y z

mz F t x y z x y z

= = =

, , z, t. F1, F2, F3 F.

118

17.1 17.1-17.3 ,

. 17.1. ,

, . 17.1, f(t) m2, . x(t) () m1 ( , , f(t) = 0), y(t) m2 . , , , .

. 17.1 , x . " ", . 17.2,

: ( )( ) ( )1 22 (17.1)mx k x k y x

my k y x f t = + = +

x(t) y(t) . . , m1=2, m2=1, k1 = 4, k2 = 4, f(t) = 40sin3t , (17.1) :

2 6 22 2 40sin 3x x y

y x y t = + = +

MAPLE

> dsolve({2*diff(x(t),t,t)=-6*x(t)+2*y(t),diff(y(t),t,t)= 2*x(t)-2*y(t)+40*sin(3*t)}, {x(t),y(t)}); = ( )x t + + + + ( )sin 3 t _C1 ( )cos t _C2 ( )sin t _C3 ( )cos 2 t _C4 ( )sin 2 t ,{

= ( )y t + + 6 ( )sin 3 t 2 _C1 ( )cos t 2 _C2 ( )sin t _C3 ( )cos 2 t _C4 ( )sin 2 t }

17.2. , . 17.3. 1 x(t) 400 , 2 y(t) 800 . , . . . 17.3. ,

f(t)

y(t)

k2

. 17.1 17.1

k1

x(t)

m1 m2

. 17.2. 17.1.

f(t) k2(y x)

k2(y x) m1

m2

k1x

119

1 80 / , 2 80 / ( , ). x/400 y/800 . , x(t) y(t):

==

+=+=

yxyyxy

yxyxx

203

103

80080

80040

400120

201

103

80040

400120

:

=+=

yxyyxx

3620620

MAPLE

> dsolve({20*diff(x(t),t)=-6*x(t)+y(t),20*diff(y(t),t)= 6*x(t)-3*y(t)}, {x(t),y(t)});

= ( )x t + _C1 e

+ /9 40

3340

t

_C2 e

/9 40

3340

t

( )y t 32 _C1 e

+ /9 40

3340

t

= ,

12 _C1 33 e

+ /9 40

3340

t 32 _C2 e

/9 40

3340

t 12 _C2 33 e

/9 40

3340

t

+ +

17.3. ,

. 17.4. I1(t) , ( ) L, I2(t) , R2. , R1 , I = I1 I2. () . ( )

80 /

80 / 120 /

40 /

. 17.3. 17.2

2 1

x(t) y(t)

120

. 17.4. , :

( )1 1 22 50 100 0,dI I Idt + = (17.2) 100 . :

125Q2 + 25I2 + 50(I2 I1) = 0, (17.3) Q2 . dQ2/dt = I2, (17.3), :

2 12125 75 50 0.

dI dIIdt dt

+ = (17.4) (17.2) (17.4) 2 25 ,

, I1(t) I2(t):

11 225 25 50,

dI I Idt

+ =

1 2 22 3 5 0.dI dI Idt dt

= MAPLE

> restart: > dsolve({diff(I1(t),t)+25*I1(t)-25*I2(t)=50,2*diff(I1(t),t)-3*diff(I2(t),t)-5*I2(t)=0}, {I1(t),I2(t)}); I1(t)=2-0.1* {e-5t [-12C1sin(5*61/2/3*t)+ 61/2C1cos(5*61/2/3*t)-12C2cos(5*61/2/3*t) - 61/2C2sin(5*61/2/3*t)]}, I2(t)= e-5t [C1sin(5*61/2/3*t)+ C2cos(5*61/2/3*t)].

E0: 100

L: 2

R2: 25

R1: 50

I

C: 0.008

. 17.4. 17.3

I2 I1

121

17.2

, . , , . , , , ( ), . , , .

() . 17.5 , V1, V2 V3 . 1, 1 2 3 3. xi(t) ( ) i t i = 1, 2 3. r ( , r ), :

1 1 1

2 1 1 2 2

3 2 2 3 3

x k xx k x k xx k x k x

= = = (17.5)

ki = r/Vi, i = 1,2,3. (17.6)

3 V3

2 V2

. 17.5.

1 V1

122

17.4. V1 =20, V2 =40, V3 =50, r =10 ( ), ( ):

x1(0) = 15, x2(0) = x3(0) = 0. t 0.

. [11]. (17.5) (17.6), ( ). :

( ) ( )0.5 0 0 15

0.5 0.25 0 , 0 00 0.25 0.2 0

x t x x = =

(17.7)

(t) = [x1(t), x2(t), x3(t)]T. 0.5 0 0

I 0.5 0.25 00 0.25 0.2

A

= (17.8)

: |A I | = (0.5 )( 0.25 )( 0.2 ) = 0.

, (17.7) 1 = 0.5, 2 = 0.25, 3 = 0.2.

1. 1 = 0.5. = 0.5. (17.8), :

|A +(0.5)I|v = 03.025.00

025.05.0000

=

cba

v = [a, b, c]T. , 0.25 0.05 , :

2a + b = 0 5b + 6c = 0.

b = 6 = 5, = 3. , :

v1 = [3, 6, 5]T 1 = 0.5.

2. 2=0.25. = 0.25. (17.8), :

|A +(0.25)I|v = 005.025.00005.00025.0

=

cba

v = [a,b,c]T. = 0, 0,05 :

5b + c = 0.

123

b = 1, = 5. , 2 = 0.25 :

v2 = [0, 1, 5]T 3. 3 = 0.2. = 0.2. (17.8),

:

|A +(0.2)I|v = 0025.00005.05.0003.0

=

cba

v = [a, b, c]T. , = 0 b = 0 , ( ). ,

v2 = [0, 0, 1]T 3 = 0.2.

, : ( ) ttt evcevcevctx 321 332211 ++= :

( ) ( ) ( ) ( )ttt ececectx 2.0325.025.01100

510

56

3

+

+

=

: x1(t) = 3c1e0.5t x2(t) = 6c1e0.5t + c2e0.25t x3(t) = 5c1e0.5t 5c2e0.25t + c3e0.2t.

x1(0) = 15, x2(0) = x3(0) = 0, : 3c1 = 15 6c1 + c2 = 0 5c1 5c2 + c3 = 0

( ) : c1 =5, c2 =30 c3 = 125. , , t

: x1(t) = 15e0.5t x2(t) = 30e0.5t + 30e0.25t x3(t) = 25e0.5t 150e0.25t + 125e0.2t.

MATLAB MAPLE. Maple

> restart: > V1:= 20: V2:= 40: V3:=50: > r1:= 10: r2:= 10: r3:=10: > k1:=r1/V1: k2:=r2/V2: k3:=r3/V3: > my_diff_yravnenie:={diff(x1(t),t)=-k1*x1(t),diff(x2(t),t)=k1*x1(t)-k2*x2(t),diff(x3(t),t)=k2*x2(t)-k3*x3(t),x1(0)=15,x2(0)=0,x3(0)=0};

124

my_diff_yravnenie = ( )x1 0 15 = ( )x2 0 0 = ( )x3 0 0 = ddt ( )x1 t

12 ( )x1 t, , , ,{ :=

= ddt ( )x2 t

12 ( )x1 t

14 ( )x2 t = d

dt ( )x3 t

14 ( )x2 t

15 ( )x3 t, }

> v2:=dsolve(my_diff_yravnenie);

v2 = ( )x1 t 15 e

t2 = ( )x2 t + 30 e

t2

30 e

t4

, ,{ :=

= ( )x3 t + 25 e

t2

150 e

t4

125 e

t5

}

> . 17.6 x1(t), x2(t) x3(t).

, 1 "" , x1(t) 0 t . x2(t) x3(t) 2 3 , , t .

0 5 10 15 20 25 300

5

10

15

. 17.6. ,

17.4.

125

17.3 , ,

. 17.7, , .

F = ma , :

( )( ) ( )( ) ( )( )

"1 1 1 1 2 2 1

"2 2 2 2 1 3 3 2

"6 6 6 6 5 7 7 6

"7 7 7 7 6 8 7

....

m x k x k x x

m x k x x k x x

m x k x x k x x

m x k x x k x

= + = + = + =

: X = [x1,x2,x3,x4,x5,x6,x7]T, :

1

2

3

4

5

6

7

0 0 0 0 0 00 0 0 0 0 00 0 0 0 0 00 0 0 0 0 00 0 0 0 0 00 0 0 0 0 00 0 0 0 0 0

mm

mM m

mm

m

=

,

: ( )

( )( )

( )( )

( )( )

1 2 2

2 2 3 3

3 3 4 4

4 4 5 5

5 5 6 6

6 6 7 7

7 7 8

0 0 0 0 00 0 0 0

0 0 0 00 0 0 00 0 0 00 0 0 00 0 0 0 0

k k kk k k k

k k k kK k k k k

k k k kk k k k

k k k

+ + + = + + + +

.

, [] 16 , m=16000 . k = 10 ( ) . , , . . 17.8. , , . 17.7, MX = X n = 7, mi = 16000 ( i) ki = 10000 (/) 1 i 7 (k8=0).

X = AX c A:

126

. 17.8. , i .

k(xi xi-1) k(xi+1 xi) m

,

m

m

m

m

m

m

m

x7(t)

x6(t)

x5(t)

x4(t)

x3(t)

x2(t)

x1(t)

. 17.7.

127

A =

1.25 0.625 0 0 0 0 00.625 1.25 0.625 0 0 0 0

0 0.625 1.25 0.625 0 0 00 0 0.625 1.25 0.625 0 00 0 0 0.625 1.25 0.625 00 0 0 0 0.625 1.25 0.6250 0 0 0 0 0.625 0.625

MATLAB (i),

(i) (Pi). 17.1.

17.1. .

i i i = i Pi = 2 ()

1 -38.2709 6.1863 1.0157 2 -33.3826 5.7778 1.0875 3 -26.1803 5.1167 1.2280 4 -17.9094 4.2320 1.4847 5 -10.0000 3.1623 1.9869 6 -3.8197 1.9544 3.2149 7 -0.4370 0.6611 9.5042

, ,

2 , - ( 1.9869 ) .

, , cost = 2cost, F = ma = m2cost . : x = A + (2cost)b, (17.9) b = [ 1 1 1 1 1 1 1], (1), E = 10.

(17.9), , :

(A + 2I)C = 2.

128

xp(t)=cpcost. (A + 2I) (), 2 = . MATLAB C = (A + 2I)12:

C = [8.2242 9.6523 9.9195 9.9760 9.9896 9.9927 9.9930]T.

, , . 19 1985 .

129

18

18.1 XVII 1

, , 2. , , .

1. , .

2. -, , () , 3.

3. 4.

(Principia Mathematical 1687) , . , , , 5.

, 6. - :

r(t) = (x(t), y(t)) = xi + yj, (18.1) i = (1,0) j = (0,1) . , r"(t) :

3 ,krrr

= (18.2) 2 2r x y= + . t (r(t), (t)), ( , -1 1609 1619 . 2 14 1546 , 24 1601 . . . , . 3 , . 4 , . 5 , , , . . 6 . , : , .

130

), ( . 18.1) 1:

ur = icos + jsin u = i sin + j cos. (18.3) ur, ,

, u ur 900.

1. (18.3) , :

rdu dudt dt

= rdu dudt dt = . (18.4)

2. (18.4),

r=rur , :

v = dr/dt = urdr/dt + rd/dtu. (18.5) 3. , ,

a= dv/dt :

a = 22

2

d r drdt dt

ur + 2

1 d drr dt dt

u. (18.6) 4.

(18.5) (18.6) , - , :

21 d drr dt dt

= 0. (18.7) , :

2 dr hdt = , (18.8)

1 , ur u .

y

x

(r(t), (t)) u

ur

. 18.1. ur u.

131

h . , A(t), . 18.2, dA = r2d/2. (18.8) , A(t) , 1.

5. (18.5) (18.6), (18.8), , ( ) r(t) :

2 2

2 3 2 .d r h kdt r r

= (18.9) 6. ,

2 r = r(), (18.8), , r = 1/z, :

.dr dzhdt d

= , (18.9)

, z() = 1/r() : 1 , , ( ) . , , : , . : . . (, , , . .

2 r2d/dt = h. , , , h 0. ( ) , , .. = (t), t = t(). , r = r().

(r(0), (0))

(r(t), (t))

. 18.2. , () .

A(t)

132

2

2 2 .d z kzd h

+ = (18.10) 7. , (18.10) :

z() = Asin + Bsin + k/h2. (18.11) 8. , (18.11) , r() =1/z()

:

( ) ( ) ,1 cosLr

e = + (18.12) e = Ch2/k, Ccos = A, Csin = B L = h2/k.

(18.12) , e. 0 e < 1, e = 1, e > 1. ( ) , , e < 1. . 18.3, = .

9. , (18.12) , . , e, L(. 18.3), , :

x(t) = r(t)cos t, y(t) = r(t)sin t, 0 t 2. e 0.0167, .

. , ( , ), 0.0068 0.0933 , 0.2056 0.2486 . . , e 0.97 (. 18.4)1.

1 . , , . , . .

133

x

y

r1

r2

L

. 18.3. ( ) ,1 cosLr

e = + r1 = L/(1 + e), r2 = L/(1 e).

. 18.4. .

134

18.2

. , , GM = 1 ( G ). , :

( ) ( )2 2

3/ 2 3/ 22 22 2 2 2, ,d x x d y y

dt dtx y x y= =

+ + (18.13)

T . , T . , GM = 1, :

T2 = 42a3. (18.14) x y x1=x,

x2 = y, x3 = x1 x4 = x2, (18.13) :

( )( )

1 3

2 4

13 3/ 22 2

1 2

24 3/ 22 2

1 2

x xx x

xxx x

xxx x

= = = + = +

a) MATLAB 44 : x(0)=1, y(0)=0, x(0)=0, y(0)=1, =1. (18.14) , T = 2.

b) : x(0)=1, y(0)=0, x(0)=0, y(0)=(3/2)1/2, , =2. (18.14) , T=4 (6)1/2. MATLAB

function solv_movement clc clear % X0=[1; 0; 0; 1]; % osci1 % long_T=4*pi; [T,X]=ode45(@osci1, [0 long_T], X0); %

135

% ( - , - ) plot(T,X(:,1),'r.-') % % - , - ) hold on plot(T,X(:,2),'k.:') % xlabel('\itt') legend('x', 'y',4) grid on hold off % function F = osci1(t,x) F=[x(3); x(4); -x(1)/(x(1)^2+x(2)^2)^(3/2); -x(2)/(x(1)^2+x(2)^2)^(3/2)];

0 2 4 6 8 10 12 14-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

t

xy

.18.5. ().

136

0 5 10 15 20 25 30 35 40 45 50-3

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

t

xy

.18.6. (b).

137

18.3

( ) 9 1986 . ( .) P0 = (0.325514, -0.459460, 0.166229) v0 = (9.096111, 6.916686, 1.305721) , ..., .. ( , . . ), . :

2 2 2

2 3 2 3 2 3, , ,d x x d y y d z zdt r dt r dt r

= = = (18.5) 24 = 2 2 2r x y z= + + .

(18.5). ( Maple.)

Maple > restart: > with(plots): > mu:=-4*Pi*Pi: > r:=(x(t)^2+y(t)^2+z(t)^2)^(3/2): >my_diff_yravnenie_0:={diff(x(t),t,t)=mu*x(t)/r,diff(y(t),t,t)=mu*y(t)/r,diff(z(t),t,t)=mu*z(t)/r,x(0)=0.325514,D(x)(0)=-9.096111,y(0)=-0.459460,D(y)(0)=6.916686,z(0)=0.166229,D(z)(0)=-1.305721}: > v2:=dsolve(my_diff_yravnenie_0,numeric): > odeplot(v2,[x(t),y(t)],0..300);

138

> odeplot(v2,[x(t),z(t)],0..300);

> odeplot(v2,[y(t),z(t)],0..300);

139

18.4 2010

. , , : 0 30 . . : P0= (x0, y0, z0) v0 = (vx0, vy0, vz0). , :

( ) ( );

; ; . , -

, (18.5). , . , , ( ).

140

,

. .

60-70- , . , . , . 1980- , , , , , .

, .

-, ( ) . , (, , , ), (, ).

-, () , , ( ) . , , ( !) .

( ) . , ( ) .

141

C

1. . . . .: , 1988. - 400 .

2. . . . .:, 1978. 418 .

3. . ., . ., . . MATLAB 7. - .: -, 2005. - 1104 : .

4. .. . Maple 9.5/10 , . .:-, 2006. 720 .: .

5. .. . 2- ., . .:, 2005. 384 .

6. .. : . : , 1999, 572 c.

7. . . . 1. : , 2002. - 232 .

8. .., .. : . .; , 2002. - 624 : .

9. .. : / - ( ). , 2006. 223

10. .. : . .: , 2006. 336 . ( )/

11. .. . .2..:-, 2004. 544 .

142

2009 , 12 , . - , 20092018 .

1945-1966 ( -

). 1945 . . 17 1945 . - , , , , , . , 11.

1951 . ..., . . , , .

. ( ) . -2, , -8 .

1951 1954 , , , .. - ( ).

11 - . , 1952-1953 -

143

.

1954 .. , , 11. . , , .

-, 1956 . ( 0705).

1970 . - () . , . : . 0705 . .

.. . 1973 ...

... .. . . - -1-. .. 1974 , . 1959 ... .. ( ..., ). .. . - . .. . .

1976 1996 .. ( , 1988 1992 .. ..,

144

, ). . . . - , .. .. .

1996 ..., ... - ( 2205), 090104 , - .

1998 , .

4500 . 50 , 10 .