Οικονομικά Μαθηματικά

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Καθηγητής Συριόπουλος Κωνσταντίνος 1 ΟΙΚΟΝΟΜΙΚΑ ΜΑΘΗΜΑΤΙΚΑ Καθηγητής : Συριόπουλος Κωνσταντίνος ΚΕΦΑΛΑΙΟ 1 ο ΣΥΧΡΟΝΗ ΑΛΓΕΒΡΑ 1.1 ΣΥΝΟΛΑ ΥΠΟΣΥΝΟΛΑ Ορισμός Συνόλου Υποσύνολου Σύνολο είναι μια συλλογή από ορισμένα και εντελώς διακεκριμένα μεταξύ τους στοιχεία. Αν a είναι ένα από αυτά τα στοιχεία τότε λέμε ότι για το σύνολο S το a S . Υπάρχει ένα σύνολο το οποίο δεν έχει στοιχεία και λέγεται Κενό σύνολο και συμβολίζεται με . Ισχύει πάντα ότι το κενό σύνολο είναι υποσύνολο κάθε συνόλου. Κάθε σύνολο λέμε ότι είναι καλά ορισμένο και εννοούμε ότι αν x είναι ένα στοιχείο , τότε λέμε ότι το x ανήκει ή δεν ανήκει στο σύνολο δηλαδή συμβολίζουμε ως εξής : ή S x S x . Υπάρχουν δυο βασικοί τρόποι γραφής ενός συνόλου ο τρόπος της απαρίθμησης , όταν λέμε για παράδειγμα πως ένα σύνολο S έχει στοιχεία τους αριθμούς 2 , 3 , 4 και γράφουμε { } 2, 3, 4 S = , και ο τρόπος της περιγραφής όπου δε μπορούμε να γράψουμε όλα τα στοιχεία ενός συνόλου και το περιγράφουμε , για παράδειγμα το σύνολο των Ακεραίων αριθμών άρα { } / S x x Z = . Έστω ένα σύνολο X και ένα σύνολο Y τότε θα λέμε ότι το X είναι Υποσύνολο του Y όταν όλα τα στοιχεία του X είναι και στοιχεία του Y δηλαδή Y X . Αν ακόμα συμβαίνει όλα τα στοιχεία του X να είναι και στοιχεία του Y αλλά όλα τα στοιχεία του Y δεν είναι στο X (δηλαδή τα υπόλοιπα είναι εκτός του X ) τότε λέμε ότι το X είναι Γνήσιο Υποσύνολο του Y δηλαδή Y X . Ιδιότητες Συνόλων και Υποσυνόλων 1. Δυο σύνολα είναι ίσα αν περιέχουν ακριβώς τα ίδια στοιχεία ανεξάρτητα της διάταξη τους. Τότε θα λέμε ότι X Y = . 2. Ισχύει ότι αν X Y και επίσης Y X τότε ισχύει ότι X Y = δηλαδή X Y X Y Y X = . 3. Λέμε ότι η Τομή δυο συνόλων είναι ένα σύνολο που έχει ως στοιχεία , τα στοιχεία εκείνα που ανήκουν και στα δυο σύνολα δηλαδή : } { Y x kai X x x Y X W = = , , :
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Transcript of Οικονομικά Μαθηματικά

  • pi 1

    : pi

    1

    1.1 pi

    pi .

    a pi S a S . pi pi

    . pi pi . x ,

    x :

    SxSx

    .

    pi pi pi pi ,

    pi pi S 2 , 3 , 4 { }2,3,4S = , pi pi pi pi pi , pi

    { }/S x x Z= . X Y X pi Y X Y YX . X Y Y X ( pipi X ) X pi Y YX .

    pi

    1. pi . X Y= .

    2. X Y pi Y X X Y= X Y

    X YY X

    =

    .

    3. pi , pi :

    }{ YxkaiXxxYXW == ,,:

  • pi 2

    4. pi pi :

    { }: , _ ,C A B x x X or else x Y= = . 5. pi CX pi pi pi

    X { }:C CX x X x X= . 6. ,X Y pi

    ( pi ). 7. (power set) pi X

    ( ) { }XAAXP = : . 8. pi

    pi pi

    pi pi .

    pi . U (Universal) pi pi U pi U . O :

    n pi iX 1, 2,....i n= : . 1, 2,3, , , , , , i ji j n X X O = = , 1, 2,3, , , , , , ,iX U i n= = n pi pi U . :

  • pi 3

    .. . pi pi pi pi

    (industry). pi pi (market). pi pi pi

    pi

    pi.

    pi pi pi pi pi pi

    pi (production set).

    1

    Z+ = { x : x } { } { }

    { }10,8,6,4,22

    :

    11.....3,2,111:

    =

    =

    ==

    +

    +

    ZxAxB

    xZxA

    1.

    ++

    ZBZAkai

    AB

    2. AZ + ? pi pi.. 12x = pi A + .

    3. + ZA + ZBAB . 4. AB ? B A .

    2 ,X Y pi .

    { }

    { }

    : 2 0 , 2 , 3, 4 , 6 , 8, ....2 02

    : 1 0 2 4 , 1 0 ,1 2 ,1 4 , ... .2 42

    xX x Z x Z

    xY x Z x Z

    + +

    + +

    = =

    = =

    }2

    .2010:{}20,...12,10{

    }2

    ,24:{}24,.........6,4,2{

    ++

    ++

    ==

    ==

    ZxxZxYX

    ZxxZxYX

    3 { }1,2,3X =

  • pi 4

    X 3 823 = : ( ) {1 , 2 , 3 , (1 , 2 ) , (1 , 3 ) , ( 2 , 3 ) , , }P X X=

    4 pi

    }0,:),{( 2121 = xxxxC . pi pi pi (budget set) }:),{( 221121 MxpxpxxB += pi 1 2,X X pi , 1 2,P P M . .

    B 1 2 pi pi .

    C pi 1 2 pi .

    CB pi 1 2 pi pi .

    CB pi 1 2 pi pi .

    1.2 R

    :

    1. (natural numbers) : { },,,,,,,,,,,4,3,2,1=+Z 2. (integers numbers) : { }.........3,2,1,0,1,2,3.... =Z 3. pi pi

    ( ),+ .

    4. (rational numbers) : [ ]

    = 0,: ZbZa

    baQ .

    5. Z pi Q Q pi pi. pi pipi pi pi pi

    pi

    .

    6. (irrational numbers) : { }QxQ = . 7. (real numbers) : QQR = .

    pi pi

    . pi R . :

  • pi 5

    X Y { }XxX = { }Y y Y= , () , ( ),x y

    YX . pi :

    ( ){ }( ){ }

    { }....,.........,,:,....),,(

    ,,:,,

    ,:,

    3

    2

    RzRyRxzyxR

    RRzRyRxzyxRRRRRyRxyxRR

    n =

    ==

    ==

    :

    { }1,2,3A = { },B a b= .

    : { } { } ( ){ }),3(),,3(),,2(),,2(),,1(,,1,3,2,1 babababa =

    1.3

  • pi 6

    pi .

    .

    pi

    pi R , pi pi ,a b b a . R , pi pi (pi). pi : X pi RXXd : pi pi pi pi

    pi d pi X pi :

    1. ( , ) 0d x y ( , ) 0d x y = pipi pi x y= 2. ( , ) ( , )d x y d y x= , 3. ( , ) ( , ) ( , )d x y d x z d z y + ( )

    ( ),X d . pipi R R pi pipi

    ( )21 ,aaa = ( )1 2,b b b= :

    ( ) ( )2222

    11),( bababad += . nR :

    ( ) = n ii babad 12),(

    ( ) ( ),R d pi R 0 > a ( a ) a :

    { }

  • pi 7

    pi

    .

    nRX x X pi ( ) XxV , pi .

    pi .

    pi pi pi

    pi.

    R a R R a pi pi R ( pi pi a pi pi pi pi pi.. 2a =

    1 = 2 3 pi pi pi ).

    S S .

  • pi 8

    (Closure) pi pi pi S S ( ) : SSS = .

    pi

    pi A X A pi pi (pi) X : XAAA == . pi R . A pi ,x y pi z pi Az ( ,x y pi pi ).

    1 pi : ) 2 3 , ) (2,3) (4,1) , ) (2,3,4) (4,1,-5)

    ) (2,3) 2 3 1d = = ) [ ] ( ) ( )2 2(2,3), (4,1) 2 4 3 1 283d = + =

    ) [ ] ( ) ( ) ( )2 2 2(2,3,4), (4,1, 5) 2 4 3 1 4 5 9.43d = + + + =

    2 ) 2 , ) (2,3) , ) (2,3,1)

    ) ( )2(2) { : 2 (2 ,2 )N x R x = +

    ) ( ) ( ) [ ]2 22(2,3) {( , ) : 2 3 } (2,3),x y R x y = +

    ) ( ) ( ) ( ){ }2 2 23(2,3,1) ( , , ) : 2 3 1 {(2,3,1), }N x y z R x y z = + +

    1.4 (CONVEX COMBINATION) [ ],a a .

    x x x x< pi ,

    ( ) : xxx += )1(

    pi [ ]0,1

  • pi 9

    )( xxxx = pi x x x . x convex combination of x and x .

    2

    2.1 pi pi

    pi

    .

    pipi : 1. . 2. pi pi . 3. pi .

    pi pipi. pi

    .

    pi .

    pi pi .

    :

  • pi 10

    11 1 12 2 13 3 1 1

    21 1 22 2 23 3 2 2

    1 1 2 2 3 3

    .....

    ....

    ....................................

    .......

    n n

    n n

    m m m mn n m

    a x a x a x a x ba x a x a x a x b

    a x a x a x a x b

    + + + =

    + + + + =

    + + + + =

    pipi pi m n= b pi m n< .

    () pi

    . , ( ) , ( pi ) Input Requirement Matrix.

    1. . 2. . 3.

    .

    4. ( ).

    pi pi .

    pi pi pipi .

    pi pipi pi

    ::

    :

    A m nC m q

    B n q

    . pipi pi :

    C pipi A j B

    pi pi : 1

    n

    ij ik kjk

    C a b=

    = . A TA pi

    . :

    1. ( )T TA A= 2. ( )T T TA B A B+ = + 3. ( )T T TAB B A=

    pi TA A= .

  • pi 11

    pi

    1 2 3.....

    nA A A A A= = = = = . ()

    : 1 1 2 2 3 3( ) ....n n ntr A a a a a= + + + + :

    1. ( ) ( )tr AB tr BA= 2. ( ) ( ) ( )tr A B tr A tr B+ = + 3. ( ) ( )tr kA k tr A= i 4. ( ) ( )Ttr A tr A=

    pi pi pi pi .

    pi pi : 1 1AA A A I = = ( ) .

    pi 1TA A= .

    (determinant) pi . pi pi :

    1 32 4

    A =

    :

    4 6 2A = = .

    n n X pi pi : AX B=

    : , : 1, : 1A n n X n B n . pi X :

    1 1 1

    1nA AX A B I X A B

    X A B

    = =

    =

    pi pipi : | | 0A .

    1. A A A | | | |TA A= .

    2. B pipi pi A | | | |B A= .

    3. pi .

  • pi 12

    4. A pipi ( ) .

    5. pi () pipi () () .

    6. pi pi () pi (pi) pi

    .

    7. pipi pipi

    | | | |B k A= . 8. pipi

    pi pipi pi

    | | | | ,n n nB k A A = . 9.

    .

    22

    11 12

    21 22

    a aA

    a a

    =

    : 22 121

    21 11

    1| |

    a aA

    a aA =

    pi pi

    33

    11 12 13

    21 22 23

    31 32 33

    a a a

    A a a aa a a

    =

    . pi M .

    : 22 23

    1132 33

    a aM

    a a

    =

    11 22 33 23 32| |M a a a a=

    : 12 13

    31 31 12 23 13 2222 23

    | |a aM M a a a aa a

    = =

    ...

    pipi pi .

    ( 1)i j+ . pi ( pi) pi

    ( 1)i jij ijC M+= .

  • pi 13

    pi A pi pipi pi

    pi pi pi :

    11 11 12 12 13 13| |A a C a C a C= + + pipi pi .

    33 pi pipi. pi

    pi :

    11 12 13

    21 22 23

    31 32 33

    C C CC C C C

    C C C

    =

    A 1 1| |TA C

    A = .

    pi n n .

    1. 1 1 1( )AB B A = , 2. 1 1( )A A = 3. 1 1( ) ( )T TA A =

    A

    ( )r A .

    2.2 x AX B= pi A n n pi , X 1n B pi 1n . pi pi A pipi 1X A B= . pi pipi

    CRAMER pi pi pi .

    CRAMER X :

    1 2 3

    1 2 3

    1 2 3

    2 4 15: 3 2 5

    6 5 28

    x x x

    x x x

    x x x

    + =

    + = + + =

    pi pi A , X B :

  • pi 14

    2 4 11 3 26 5 1

    A

    =

    1

    2

    3

    x

    X xx

    =

    155

    28B

    =

    A pi . pi 5A = . 1 2 3| |,| |,| |A A A cofactors pi B . :

    | |ii

    Ax

    A=

    1 2 3| |,| |,| |A A A pi pi (pi.. 1X pi ) B

    pi 1

    15 4 15 3 2

    28 5 1A

    =

    .

    pi :

    11

    10 25

    Ax

    A

    = = =

    , 2

    215 35

    Ax

    A

    = = =

    , 3

    35 15

    Ax

    A

    = = =

    (LEONTIEF) pi pi

    pi pi

    . pi

    pi pi pi .

    pi pipi pi pi

    pi pi .

    pi

    .

    n pi pi pi n pi ix

    - pi. pi

    :

  • pi 15

    1

    2, 0

    ...

    n

    x

    xX X

    x

    =

    pi pi

    pi - pi id :

    1

    2, 0

    ...

    n

    dd

    D D

    d

    =

    ija - pi pi pi

    pi pi j- pi. pi A pi pi

    pi pi

    11 1

    1

    n

    n nn

    a a

    Aa a

    =

    pi ( pi ija pi ) . iia pi - pi pi pi

    pi pi pi.

    pi - pi pi

    pi pi 1 1 2 21

    ..

    n

    ij j i i in nj

    a x a x a x a x=

    = + + + pi ij ja x pi - pi pi

    pi jx pi pi j .

    11 1 12 2 1111 1

    2 21 1 22 2 2

    11 1 2 2

    ...

    ..

    ... ........................

    ...

    n n

    n

    n n

    n nn

    n n n nn n

    a x a x a xxa a

    x a x a x a xAX

    a ax a x a x a x

    + + + + + + = = + + +

    pi pi pi

    pi i pi pi.

    pi 1

    n

    ij j ij

    a x d=

    +

    i pi :

  • pi 16

    1( )

    n

    i ij j ij

    x a x d=

    = +

    1( )X AX D X I A D= + =

    2.3

    pi , pi. pi

    pi pi pi

    () pi .

    A pi pi + pi ( ), ,A + . pi pi pi

    pi pi pi .

    pi : A A A+ . ,

    pi , , ..

    pi pi ( ) .

    1

    2

    3

    ...

    n

    VV

    V V

    V

    =

    [ ]1 2 3, , ......, nV V V V V=

    1. pi :

    1 1 1 1

    2 2 2 2

    .. ... .........

    n n n n

    w v w v

    w v w vW V

    w v w v

    = =

  • pi 17

    2. pi/ : [ ]

    1

    21 2

    1, , ...,

    ...

    n

    n i ii

    n

    w

    wW V v v v w v

    w

    =

    = =

    pi

    .

    3. pi/ 1 2 1 2[ , ,... ] [ , ,..., ]n nkV k v v v kv kv kv= =

    4. : 2 2 21 2 .... nV v v v= + + + (norm)

    pi TV VV= .

    pi .

    pi

    pi pi

    ( ) . U ,V W , :

    1 2 1 2, ,U V W R = + pi

    .

    1. pi B V V pi B V B=< > . 2R (1,0) (0,1) .

    2. pipi

    :

    1 1 2 2 3 3

    1 2 3

    .... 0... 0

    nv v v v

    + + + + =

    = = = = =

    pi pipi pi V pi ( ) pi pi pi

    pipi .

    pi pi :

    cosTV W

    V W = .

  • pi 18

    pi

    pi :

    pi B V B : 1. (pi ) 2.

    pi pi pi pi pi..

    2R { }(1,0), (0,1)B = 2R . pi :

    1. pi B .

    2. V pi B pi pi pi B .

    3. pipi (

    pi pi 2R ).

    (dimension) .

    pi

    . : pi

    A pi AX X= X pi A ( ) . pi

    pi A pi 1 2 3, , ,... :

    1 2

    1 2

    ( ) ...| | ...tr A

    = + + +

    =

    A pi . pi pi pi P pi 1P AP :

    11

    0

    0 n

    P AP

    =

  • pi 19

    pi A pi pi () pi pi pi K .

    1 pi

    5 6 61 4 2

    3 6 4A

    =

    .

    pi pi :

    5 ___ 6 ____ 6| | 0 1____ 4 ____ 2 0 ( 1)( 2) 0

    3 ______ 6 ___ 4A I

    = = =

    pi () 1 = 2 = pi pi () 1 = :

    4 6 6 0( 1 ) 0 1 3 2 0

    3 6 5 04 6 6 0

    3 2 03 6 5 0

    x

    A I X yz

    x y zx y zx y z

    = =

    = + + = =

    i i

    33

    x yz yy y R

    =

    =

    =

    31

    3X y

    =

    1

    2 = 1 11 00 1

    y zX y y z

    z

    + = = +

    2

    2 THE LINEAR PRODUCTION TECHNOLOGY pi pi pi 1 2,y y 1 2,z z

    ija pi pi pi pi.

    ija pi 3 12 5

    A =

    pi pi pi 200 pi 1y 15

  • pi 20

    pi 2y pi pi pi Z AY= (pi) :

    1 1

    2 2

    3 1 752 5 115

    z yZ AY

    z y

    = = =

    pi 1 2,z z 10,20 pi pipi pi 1 2,y y ? Z AY= 1Y A Z= .

    3 IS-LM MODEL OF A CLOSED ECONOMY :

    15 0.8( ), 25 0.25 , 65 , 94, 5 50 , 1500C Y T T Y I R G L Y R M= + = + = = = = pi C = , = pi pi , = () , = pi pi , R = pipi pi , G = pi , L = , = pi pipi pi R .

    IS 485 2.5Y C I G R= + + = LM 300 10L M Y R= = + IS LM 10 300Y R = :

    1 2.5 485 4481 10 300 14.8

    Y YR R

    = =

    pipi pi pi 448 pi pi 14,8% . pipi pi 25 0, 25(448) 87ET = + = pi 94 87 7G T = = .

    2.4 k

    1 1 2 2 ... , 1, 2...i i i k ki iy b x b x b x U i n= + + + + =

    1 1 111 1

    2 2 2

    1... .. ...

    k

    n knn k n

    y b ux x

    y b u

    x xy b u

    = +

    pi

    ( 1) ( ) ( 1) ( 1)n n k k kY X B U = + U

    2(0, )U IID ( IID pi n pi ).

  • pi 21

    X ( ) ( )Tr X X r X k= = pi pi 1( )TX X .

    1

    2

    00( ) 0....

    0n

    u

    uE U E

    u

    = = =

    2var( ) ( )T nU E UU I= =

    OLS ( ) : 2

    1

    1

    min ( ) ( )

    2

    0 2 2 0

    ( )

    nT T T T T T T T

    ii

    T T T T T

    T T

    T T

    S u U U Y XB Y XB Y Y B X Y Y XB B X XB

    S Y Y B X Y B X XBS X Y X XBB

    B X X X Y

    =

    = = = = +

    = +

    = + =

    =

    pipi

    2 TH X X= . ( ) pi . X pi

    ( )r X k= X n k n k> pi .

    3

    3.1

    X Y pi X Y pi () pi X Y . X pi Y pi . : RyxxfyYXf = ,),(,: .

  • pi 22

    y ax= pi x 11 axy = 22 axy = (gradient) pi :

    12

    12

    xx

    yyx

    ya

    =

    =

    ),( 11 yx ),( 22 yx . ),( yx pi :

    ),( yx = ),( 11 yx + (1-) ),( 22 yx

    ),( yx = [ ]2121 )1(,)1( yyxx ++

    pi

    1 Convex combination (2, 3) (-3, 2)

  • pi 23

    )]1(2),1(32[)2,3)(1()1,2( +=+=x

    ( ) pi :

    0,,2 ++= aRxcbxaxy

    pi ( pipi 0b < ) pi ( MAX x

    pipi 0b > ). pi :

  • pi 24

    (Power Function)

  • pi 25

  • pi 26

    )ln(

    ,log

    ,0,

    xyex

    kaibxxy

    basebaaby

    y

    yb

    x

    ==

    ==

    =>=

    f CONCAVE

  • pi 27

    ]1.0[)1(

    )()1()()(

    +=

    +

    pi

    xxx

    xfxfxf

    pi strictly concave (0,1) . pi :

    f CONVEX :

    ]1.0[)1(

    )()1()()(

    +=

    +

    pi

    xxx

    xfxfxf

    pi :

  • pi 28

    2 2)( xxf = . To x x x : 0,4 = , 2x = 5x = .

    : x = 0,42 + 0,65 = 3,8 ( ) 44,148.3)8.3()( 2 === fxf pi x .

    pi pi pi

    )(),( xfxf . : 6,16)()1()( =+ xfxf pi pi (2,4) (5,25) x =3,8 strictly convex 14,44 < 16,6 .

  • pi 29

    3.2

    pi pi

    : ( ) :f n Z R .

    nX ( L ) pipi 0 >

    pi N :

    Lx

    NnLx

    nn

    n

    =

    >

    lim

    ,

    a

    n

    bn

    a L

    b L

    n :

    ) lim an nca cL = ) lim( ) a bn na b L L = ) lim( )( ) a bn na b L L=

    ) lim , 0a

    bnb

    n

    a L Lb L

    =

    pi a

    n

    n

    a Lb

    n :

    ) limn ncb = ) lim( )n na b = ) lim( )( )n na b = ) lim 0n

    n

    a

    b=

    ) lim( ) 0n

    c

    b=

  • pi 30

    pi pi : (1 )nrPn

    + pi r pi n

    P . pi n pi :

    lim(1 )nn

    r

    n+

    pi pi

    pi

    pi pi pi pi pi

    :

    ( )tt rVPV+

    =1

    0r pi T pi . pi

    pi

    pi .

    1 3 , 1,2,3.....nx n n= + = nx {4,5,6,..}. ( ) 3 ( 1)nf n = + pi {2,4,2,4} pi {2,4}.

    2

    H 1nx

    n= {1, 1/2, 1/3.1/4,.}. pi nx

    0 0nx pi n pi

    1

    nxn

    = 0 ( 0) pi :

  • pi 31

    3 pi PV pi pi 1 3 pi 12% .

    1. 1 PV = 1000000/1+0.12 = 892000 2. 2 21000000 /(1.12)PV = = 797000 3. 3o 31000000 /(1,12)PV = = 71100000

    SUM = 2401000 pi 12% pi 3 pi pipi pi pi 1 pi pi 3 ? pi SUM pi pi.

    4

    pi pi pi 12% 1000 . 1000 (1.12) 1120 = . pi pi pi 6 pi 12% : 2 = 6% : 2(1000 1.06) 1.06 1000 1.06 1123.60 = = pi : 121000 1.01 1126.82 = pi pi pi

    .

  • pi 32

    pi : (1 )nrPn

    + pi r pi n P

    pi n pi :

    lim(1 )nn

    r

    n+

    5 pi :

    r = 1 ( 100%) 1lim(1 ) 2.7nn

    en

    + = =

    n

    sr

    = 1r

    n srn s= =

    lim(1 )nn

    r

    n+ =

    1 1lim(1 ) lim[(1 ) ]sr r rs s

    es s

    + = + =

    pi P pi r rPe

    6 100P = , 0,12r = rPe = 1127 pi N :

    (1 ) (1 )t

    n ntr rV P Pn n

    = + = +

    pi P t pi :

    ( )r t rtV P e P e= =

    7 1000P = 0,12r = . pi 2 ( 2t = ) : 0.12 21000 1.270,25V e = = pi pi V pi t pi pi

    %r n X :

    (1 )[1 ( )]

    nt

    nt

    r VV X Xrn

    n

    = + =+

    pi pi :

    [1 ( ) ]t

    n t

    VP Vr

    n

    =+

    n

  • pi 33

    rt rtrt

    rtt

    VXe V X Vee

    ara

    PV Ve

    = = =

    = t

    8 pi pi pi pi 10000 pi 3% pi pipi?

    1. 6- 2. 5 6- 3. 5 4. 1 5. 5

    pi : (1 ) n trV Pn

    = + , r tV P e=

    pi :

    1. 20.0310000(1 ) 103022

    V = + =

    2. 2 50.0310000(1 ) 116052

    V = + =

    3. 5 120.0310000(1 ) 1161612

    V = + =

    4. 0.0310000 10304V e= = 5. 0.03 510000 11618V e = =

    9 pi 25000 pi 8% pi pipi :

    1. 2. 20 3. 20

    pi pi : [1 ( / )]t n tVP Vr n

    =+

    r ttP V V e

    = pi pi

    pi pipi :

    1. 25000 23.1481 0.08

    =+

  • pi 34

    2. 2025000 5363(1 0.08) =+

    3. 1.625000 5047e =

    3.3

    ,....,3,2,1, =tat pi :

    ,.........2,1,1

    ===

    naSn

    ttn

    : La

    a

    n

    n

    n =+

    1lim

    - 1L < - 1L > pi - 1L = pi

    pipi pi pi pi.

    1= tt apa ,a p :

    =

    ++++==n

    t

    ntn apapapaapS

    1

    121......

    :

    papap

    a

    an

    n

    n

    n ==

    +1

    1

    - 1p < - pi 1p > - 1p = aat = naSn = pi pi 0a .

    pipi pi n

    an1

    = :

    111

    1

    lim 1 =+=+

    n

    n

    a

    a

    n

    n

    n

  • pi 35

    pi pi

    pi pi ( )tt r

    VPV+

    =1

    , pi

    V pi pi t pi . pi pipi pi pi pi pi

    pipi 1, 2,....t T= pi pi pi .

    pipi T pi V pi pi r . pi :

    ( ) ( ) ( )TT

    ttT

    r

    Vr

    Vr

    Vra

    VP+

    +++

    ++

    =+

    == 1

    .....

    11 21

    pi pi pi pi

    : V

    aa r

    =+

    1

    1p

    r=

    + .

    ( )

    = =

    +

    +=+

    =1

    111

    11

    limt

    tTT r

    V

    r

    r

    V

    r

    VP

    pi (RR) o IRR pi pi pi pi pi pi r

    pi pi :

    ( )==

    +

    n

    tt

    t

    r

    V0

    01

    3.4 pi pi pi pi

    GDP pi pi . TOTAL OUTPUT (Real GDP) = C + I + G + NET EXPORT (Planned Aggregate Expenditure). pi pipi pi pi .

    pi 100 pi (pi.. ). pi 60% pi pi pi

    ( 0,6 60 ). pi 60% pi ( 60 x 0,6 = 36 ) ... pipi

  • pi 36

    1 ( pi) 100 2 60 3 36 4 21,6 5 12,96 ...

    :

    ( ) ( )2100 100 0,6 100 0,6 ....+ + + 100a = 0,6p = .

    2506.01

    100=

    .

    pipi KEYNES : MPC

    M

    =1

    1 pi MPC pi

    .

    :

  • pi 37

    4

    4.1

    pi

    .

    ( )f x pi x a= a pi lim ( ) l

    x af x L

    = 0 pipi

    pi 0 || ( ) | ,lf x L x pi a x a lim ( ) rx a

    f x L+

    = pi | ( ) | ,rf x L x ( )f x x a= lim ( )

    x af x

    = lim ( )

    x af x

    +

    : lim ( ) ( )x a

    f x f a

    =

    ( )f x pi pi pi x a= pi pi 0 : | ( ) ( ) | ,| | , 0f x f a x a >

    pi

    pi F .

    ( )f x ( )g x c pi :

    1

    ( )( ) ( )( )( ) / ( )( ) ( )

    ( )

    cf xf x g xf x cf x g xf x g xf x

    +

    +

    : 1. pi 2. Bonus 3. pi pi 4. pi 5. pi

    6. pi 7. pi pi

  • pi 38

    pi

    pi pi pi pi pi

    pi .

    ( )y f x= [ ], ,a b b a> pi pi pipi ( )f a ( )f b . pi :

    ( )f x [ ],a b ( ) ( )f a f b pipi y ( ), ( )f a f b pi x :

    ( ), ( , )y f x x a b= :

    pi pi

    pi pi pi p , Y pi ( )Y D p= ( )Y S p= pi .

    0ep p= pi ( ) ( )e eD p S p= pi ( , )e ey p .

    0ep = . pi ,

    ( ) ( ) ( )Z p D p S p= pi pi pi pi .

    pi ( ) 0 ( ) ( )e e eZ p D p S p= = 0ep > pi .

  • pi 39

    pi pi pi

    .

    pi pi pi

    pi (0) 0S = pi pi (0) 0D > 0p = pi pi pi :

    (0) (0) (0) 0Z D S= > . pi pi

    pi pi p : ( ) ( ) ( ) ( ) ( ) 0D p S p Z p D p S p< = <

    pi [ , ]o p :

    [0, ] ( ) [ (0), ( )]p p Z p Z Z p

    pi c pi pi : (0) 0

    ( ) 0ZZ p

    >

    (0) 0Z > - pi pi 0 : ( ) ( ) ( ) 0p S p D p Z p> <

    - pi pi pi

    pi .

    -

    pi pi .

    - pi

    ( pi) .

    - pi 2 1

    2 1

    ( ) ( )f x f xm

    x x

    = =

    (

    pi). - 1 1( , ( ))K x f x ( )y f x= pi

    pi pi 0

    limx

    m l

    = pi pi

    pi pi.

    - yx

    dy ddx

    = . pi pi

    (pi pi) . -

    x a= pi pi pi

  • pi 40

    pipi ( pi) : 0

    ( ) ( )limx

    f a x f ax

    +

    pi a . pi

    pipi

    .

    4.2 pi ( )C C y= pi pi .

    ( ) ( )C C y y C y

    y y +

    =

    pipi

    pi pi pi pi

    Y 0 pi pi pi pi .

    pi

    1

    ) ( ) ( ) 0 ___( )) ( ) ( ) _____( )) ( ) ( ) _____( )) ( ) ( ) ( ) ( ) ( ) ( ) ___( )) ( ) ( ) ( ) ( ) ___( )) ( ) ( ) ( )

    n n

    A f x c f xB f x mx b f x mC f x x f x nxD f x g x h x f x g x h xE f x cg x f x cg xF h x f x g x

    = =

    = + =

    = =

    = = +

    = =

    =

    2

    ( ) ( ) ( ) ( ) ( ) ___( )( ) ( ) ( ) ( ) ( )) ( ) ( ) _____( )( ) ( )

    ) ( ) ( ( )) ( ) ( ) ______( )1) ( ) ln ( ) _____( )

    h x f x g x f x g xf x f x g x f x g xG h x h xg x g x

    h x f g x h x f g uf x x f x

    x

    pi

    = +

    = =

    = =

    = =

    ( )y f u= ( )u g x= ( ( )) ( )y f g x h x= = ( ) ( ) ( )h x f u g u = pi :

    dy dy dudx du dx

    =

    pi pi D HOPITAL .

  • pi 41

    ay L= 0a > , L pi pi. pi TR(L). pi MP(L) :

    1( ) adyMP L aLdL

    = =

    pi pi

    pi ( )TR q pq= pi q pi p . pi p pi ( )TR q pq= . To :

    ( )( ) dTR qMR q pdq

    = =

    pi pi pi

    ( )q D p= 1( ) ( )p D q p q= = ( ) ( )TR q pq pq q= =

    :

    ( )( ) ( )dTR qMR q p q q pdq

    = = +i

    pi pipi pi pi

    ( )MR q p< pi : pi pi pi pi pipi pi 0q > . pi pi :

    pi pi pi q ( )p p q= . pi pipi pi pi q pipi pi pi p .

  • pi 42

    R = pi pi :

    ( )R p p q p q = i

    R pi pi . p pi pi q pi pi .

    pi pi pi

    pi .

    pi :

    R pp q pq q

    dpMR p qdq

    =

    = +

    i

    pi

    pi

    p

  • pi 43

    f pi pi , pi ...

    ( )( ) f xA xx

    = pi

    ( ) ( )( ) f x A xA xx

    = pi ( )f x . :

    - pi pi

    ( ) pi pi ( ).

    - pi pipi pi pi pi pi

    pi () pipi . - pi pi pipi

    pi pi

    .

    - pi

    ( pi pi : ) pi ) pi pi ) pi pi pi .

    pi q pi pi L

    ( )q q L= . ( ) dqMR LdL

    =

    pipi pi pi .

    MVP(L) pi pi pi pi ( p ) : ( ) ( )MVP L p MP L= i pipi pi pi pi pi pipi

    MRP(L) R(q) pi :

    ( ) ( ) ( )dR dR dqM RP L M R q M P LdL dq dL

    = = =i i

    pi

    pi pi ( )MR q p= pi pi ( ) ( )MVP L MRP L= . pipi pi pi pipi pi

    pi pi pipi pi .

    ( )q D p= 1( ) ( ) ( ( ))p D q p q p q L= = = pi pi pi

    pi : ( ( )) ( ( )) ( )R q L p q L q L= i

  • pi 44

    ( ) 0dp dp dqp LdL dq dL

    = =

    pi : ( ) ( ) ( )dRMRP L p L q MP L pdL

    = = +i i

    pi pi pi

    pi pi . pi pi

    pi pi .

    pi ( )q q L= pi ( ) dqMP L

    dL= . pi

    pipi pi pi pi

    . pi pipi pi

    pi pipi

    . pi

    pi pi .

    0C pi W () pi 0( )C L WL C= + pi pi L . pi pi pi pi

    0( ) ( )C q W L q C= +i dC dLWdq dq

    = .

    4.3 y a bp= pi a pi

    0p = dybdp

    = pi

    b pi p. pi b () . pi pi

    p. 1 1 2 2( , ), ( , )y p y p pi : 2 1

    1 2

    % 100/ 2

    p ppp p

    =

    +

    pi : 2 1

    1 2

    % 100/ 2

    y yyy y

    =

    +i pi

    p . () :

  • pi 45

    2 1

    1 2 1 2

    2 1 1 2

    1 2

    / 2 ( )%% ( )

    / 2

    y yy y y y yy

    p pp p p pp p

    + + = = +

    +

    pi p :

    1 2 1

    1 2 1

    ( )lim ( )( )py y y pdyp p p dp y

    + = +

    by a p =

    1 = .

    pipi pi pi .

    pi pi .

    .

    pi pi pi

    pi pi pi.

    pi pi pi

    pi

    pi pi .

    pi pi

    .

    f pi . pi () pi

    pi pi .

    pi pi pi

    pi .

    - pi [ ],a b pi x

    pi .

    - pi [ ],a b pi x

    pi ( ) .

  • pi 46

    pipi

    pi pipi pi pi

    .

    pi pi :

    1...(1 ) (1 )

    nCB n

    CFCFPy y

    = + ++ +

    Y IRR= pi pi .

    pi pi pi pi pi

    pi pi 1dp

    dy pi pi

    pi pi 1% pipi .

    pi :

    ( 1)1

    1(1 )

    (1 )( )( ) (1 ) (1 )

    Z T T

    TT

    T TT T

    P CFy

    P CF yT CFdP T CF y

    dy y +

    +

    = +

    = +

    = + =

    +

    i

    i

    pi ( ) pi 11 y+

    pipi 1/ p :

    1

    ( )1 1 11 (1 )

    ni

    ii

    i C Fd pd y p y y p=

    = + +

    i (MODIFIED DURATION)

  • pi 47

    pipi pi . H MD pi pi . pi

    pi pi . pi

    pi pi MD :

    Convexity of a bond: 2

    21d p

    dy p

    (pi) pi :

    231 2

    2 3 4 5 212 ( 1)2 6

    ....(1 ) (1 ) (1 ) (1 )n

    n

    CF n n CFCF CFd Pdy y y y y +

    += + + + +

    + + + +

    4.4 TAYLOR E pi pi

    ( ) ( )dyf x dy f x dxdx

    = = pi pi pi

    .

    pi pi

    pi pi pi TAYLOR. pi pi

    0x pi ( pi pi pi pi.. 5 pi ...). pi LOR pi pi 0x pi

    .

    pi TAYLOR pi pi pi 0x pi

    0x pi pi pi 1x

    1( )f x . pi pi TAYLOR :

    ( )10 1 0

    1 01

    ( )( )( ) ( )!

    k kn

    n

    k

    f x x xf x f x Rk

    =

    = + +

    pi :

    1 2 ( 1) 10 1 0 0 1 0 0 1 0

    1 0( )( ) ( )( ) ( )( )( ) ( ) ....

    1! 2! ( 1)!n n

    n

    f x x x f x x x f x x xf x f x Rn

    = + + + + +

  • pi 48

    pi ( ) 1 0 0 1( )( ) ,

    !n

    n

    x xR f c x c xn

    = < < pi

    0nn R pi . :

    ( )10 1 0

    1 01

    ( )10 1 0

    1 01

    ( ) ( )( ) ( )!

    ( ) ( )( ) ( )!

    k kn

    n

    kk kn

    n

    k

    f x x xf x f x Rk

    f x x xf x f x Rk

    =

    =

    = + +

    = +

    ( )10 1 0

    1

    ( )( )!

    k kn

    n

    k

    f x x x Rk

    =

    = +

    4.5 ( )Y f x= . pi

    pi x pi . pi pi pipi

    . pi pi

    R.

    pi (global max) 0x : 0( ) ( ),f x f x x ( ).

    pi

    pi pi (local max) 0x : 0 0 0( ) ( ), ( , )f x f x x x x + ( ). pi 0x pi pi

    0( ) 0f x = . pi pi

    pipi dy pi pi pi . pi

    pi pi pi pi

    pi .

    pi pipi pi pi

    pi pi pi

    (pi.. pi ) pi .

    .

    pi pi

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

  • pi 49

    pi pi

    pi .

    pi ( )X p (pi Output) p pi pi pi

    pi pi

    pi 1 (/ ) pi pi .

    pi pi :

    100X p= pi pipi pi : 25C x= . pi 2( ) 100 25x px C x x x = = pi pi :

    0

    0 0

    0

    ( ) 0 37.5

    100 62.51.4

    x x

    p x

    = =

    = =

    =

    - 2( ) 100R x pX x x= = - ( ) 25C x x= - ( ) ( ) ( )x R x C x = - 100X p= - ( )R x - ( )C x

    5C x= pi . pi p pi pi. pi pi

    pi pi

    pi pi pi .

    8p = pi pi pi ( ) 8R x px x= = ( ) 8 5 3x x x x = = . pi pi pi 3 = pi . pi pi pi

    . pi

    pi . pi pi

    pi pi pi

  • pi 50

    pi pipi pi pi

    .

    pipi

    pi .

    ; pi pi

    (pi pipi ) pi pi

    pi pi pi pi pi

    pi pi

    pi . pi pi

    pi pi pi pi pi

    pi pi .

    (=1) pi 110X p= 110p x= 5C x= .

    2 1( 10 )( ) 110dx p p

    e pdp x p

    = = =

    pi .

    pi : ( ) 10( ) 5( ) ( ) ( ) 10 5( ) 5

    R x pxC x x

    x R x C x xx

    = =

    =

    = =

    =

    pi pi pi ,

    pi . pi

    pi

    pi pi .

    pi pi pi pi

    .

    1 pi 100 , ( ) 5x p C x x= = pi pi

    .

    10%r = pi pi :

    2( ) 0.1 0.1 ( ) 0.1(100 )Y x px R x x x= = = . 2( ) ( ) ( ) 65 0.9x R x C x x x = = pi pi :

  • pi 51

    ) pi pi 0 0( ) 10 0.2 50Y x x x = = pi pi pi 0 100 50 50p = = pi ) pi pi pi :

    0 0 0 0( ) 65 1.8 36.1, 63.9x x x p = = = pi pi pi

    pi pi pi .

    : 0 1r< < , ( ) ( )( ) ( ) ( ) ( ) (1 ) ( ) ( )

    Y x rR xx R x C x rR x r R x C x=

    = =

    pi () ( ) 0AR x = pi

    : ( )( ) 0 ( )

    1p

    p p

    C xx R x

    r

    = =

    .

    pi pipi pi pi

    x pi Ax .

    pi pi

    pipi A px x> .

    pi pi pi pi pi

    pi pi pi

    pi pi pi. pi pi

    pi pi pi .

    pi ( )p p x= ( ) ( )R x p x x= i ( ) dpR x p x

    dx = +

    p dx x dp

    e ex dp p dx

    = =

    pi : 1( )R x p pe

    =

    0p > : - 1e < 0R <

    pi .

    - 1e > 0R > pi .

    - 1e = pipi pi pi pi 0R =

    pi pi pi pi

    pi pi pi pi 1e > pi 0R C = > R C > pipi 1e > .

  • pi 52

    ---

    TC=TFC+TVC pi : = + .

    - TCF = pi pi pi

    pi TFCAFC

    q= pi AFC

    pi pi pi.

    - TVC = pi pi pi . pi pi pi

    pi . pi

    pi pi pi

    TVCAVCq

    =

    - MC pi pi pi pi pipi

    pi. MC pi pi pi (pi.. ) pi pi pi pi pipi

    pi pi .

    pi pi .

    TVC TVCTVCq

    =

    . MC TVC

    pi pi pi pi 1q = . MC TVC . pi pi

    pi pi pi pi

    (pi pi ). R p y= i . pi MR pipi pi pi pi pi pi.

    MR p= pi pi pi .

    pi pi pi pipi

    pi pi min(MR) R MC pipi pi pi pi pi pi pi

    pi .

    pi MR p= MR MC p= = .

  • pi 53

    2 pi 2 3( ) 10 12f x x x x= + pi 2

    ( )( ) 10 12f xA x x xx

    = = +

    pi 2( ) 10 24 3f x x x = + . pi pi pi 0( ) 0 6A x x = = .

    (6) 46(6) 46

    Af

    = =

    pi

    pi 6 pi .

    pi

    pi . pi

    pi ( )Y f x= pi pi pi pi pipi pi

    pi pi pi pi

    pipi pi .

    pi pi pi

    /

    pi

    1 ny x= 1ny nx = ( )( ) ny f x= ( ) 1( ) ( )ny n f x f x = 2 y x=

    12

    yx

    = ( )y f x= ( )2 ( )

    f xy f x

    =

    3 siny x= cosy x = sin ( )y f x= ( ) cos ( )y f x f x = i 4 cosy x= siny x = cos ( )y f x= ( ) sin ( )y f x f x = i 5 y x=

    21

    cosy

    x =

    ( )y f x= 2( )

    cos ( )f xy f x

    =

    6 y x= 2

    1sin

    yx

    = ( )y f x=

    2( )

    sin ( )f xy f x

    =

    7 lny x= 1yx

    = ln ( )y f x= ( )

    ( )f xy f x

    =

    8 expy x= expy x = exp ( )y f x= ( ) exp ( )y f x f x = 9 xy a= lnxy a a = ( )f xy a= ( ) ( ) lnf xy a f x a =

  • pi 54

    4.7

    :f A B pi :

    - ( , pi ...) pi.. R pi

    - pi x pi

    pi

    - pi pi pi x pi

    pi

    - pipi pi pi pi

    3 :

    2( ) 5 10f x x x= + + pi R pi

    5 18( )

    42xf x

    x

    =

    +

    pi pi pi pipi 42x =R-{42}

    ( ) 1f x x= pipi 1 0x > : 1x > ( )f x x= | 3 2 | 2x pi :

    | 3 2 | 2 2 3 2 2 0 3 4 0 4 / 3x x x x pi pi x

    pi pi pi ,

    .

    pi pi pi

    pi

    .

    4 1x =

    | 2 |,| | 1( ) | 3 |,| | 1x xf xx x

    >=

    pi pi R 1x = (1) 3f = .

  • pi 55

    pi pi pi pi 1 pi . pi :

    1 1lim ( ) lim | 3 | 3x x

    f x x

    = = 1 1

    lim ( ) lim | 2 | |1 2 | 1x x

    f x x+ +

    = = = pi

    1 .

    pi pi pi pi

    .

    5 pi

    22 7 23( ) 5 ln ln

    4x xf x x x x x x

    x

    += + + +

    22

    2

    2

    7 23( ) 5 ln ln4

    1 1 (14 23)4 4(7 23 )( ) 10 ln 1162

    x xf x x x x x xx

    x x x xf x x xx xx

    += + + +

    + + = + + + +

    -

    pipi pi

    pi

    6 () 2( )f x x= (2,4) (4,16). pi .

    4 2 2 = = 16 4 12 = = 12 62

    yx

    = =

    7 () y ax b= + , 1 1

    2 2

    ( , )( , )

    K x yP x y

    .

    ?

    ( )KP

    y b x x a bx am b

    x x

    + + = = =

  • pi 56

    8 (pi) pi 2( )f x x= (4,16)

    2( )

    ( ) 2(4) 8

    f x xf x xf

    =

    =

    =

    pi

    pi : Y-16=8(X-4)

    9 (pi) 3 2( ) 2f x x x x= pi pi 7+3-1=0.

    pi pi

    pi pi = -7/3. pi. pi

    x pi 3 2 2( ) 2 ( ) 3 4 1f x x x x f x x x= = pi

    0 0( , )A x y : 2

    0 0

    0

    0

    73 4 13

    2 / 3

    2 / 27

    x x

    x

    y

    =

    =

    =

    -2/27 = -7/3(-2/3)

    TAYLOR 10 pi ( ) xf x e= 1

    ( )1

    12 3 1

    (0)( 0)( ) (0)!

    ( ) 1 ......2! 3! ( 1)!

    k kn

    n

    kn

    n

    f xf x f Rk

    x x xf x x Rn

    =

    = + +

    = + + + + + +

  • pi 57

    11

    pi : ( ) , 0( ) , 0

    D p a b p bS p c ep e

    = >

    = + >

    pi pi pipi pi a c pi pi

    pi pi.

    pi ( ) ( ) ( )Z p a c e b p= + . pipi :

    - (0) 0Z > a c> - pi p pipi ( ) 0 a cZ p p

    e b

    < >+

    pi a c> pi pi ep pi pi :

    e a cpe b

    =+

    ( pi pi pi pi pi ).

    12 ( ) 80C y y= pi pipi pi pi pi 80 pi pi ( ) . 80dC dy= . . pipi pi

    pi , pi , pi pi

    (pi) . pi . pi pi

    pi () pi pipi pi pi , pi ,

    pi pi ( pi pi pi pi). pi pi 2( )C y y= pi pi 2C y = .

    pi pi 200 pi 400 pipi pi . pi pi pi 300 pi 400 ( 100 ) :

    (400) (300) 70000C C C = = pi :

    2 2*300*100 60000dC ydy= = = 14% .

  • pi 58

    13 ( ) 40 2p q q=

    pi : ( ) ( ) (40 2 )TR q p q q q q= =

    pi pi q :

    ( ) (40 2 ) (40 2 ) 40 4dPMR q q q q q qdq

    = = + =

    pi.. q p MR .

    14 pipi pi ( )A Y ( )M Y pi pi pipi pi ( ) pi pi pi pi

    .

    2( ) 10 25TC y y y= + + . ) MC = C AC :

    2

    ( ) 2 10( ) 10 25 /

    251 0 5

    MC T y yT yAC y y

    y

    AC yy

    = = +

    = = + +

    = = =

    pi = 5 pi MC = 20 AC = 20

    B) MC AC< AC 5Y <

    15 pi pi

    pi 1/ 2q L= .

    1/ 20( ) ( ) 2C q WL q C C W L= + = .

    16 2 pi pi 5 CF pi 8% .

  • pi 59

    4

    1

    5 103.692(0.4)iiP

    =

    = = MD

    4

    1

    ( )5 386.405(0,4)iiMD = = pipi 11 Y+ :

    1 371.405 (1/ ) 3.580.4

    MD p=

    pi pi pi pi

    pipi 2 ( ) 1,7. pi (pi) pi 1,8% pipi .

  • pi 60

    5

    pi pi

    pi

    pi .

    1 2( ) ( , , .... )ny f X f x x x= = . pi , , pi ,

    pipi Lagrange .

    5.1 1 2( , , .... )ny f x x x= pipi pi pi y :

    11

    1 1 2 1 20

    1 1

    ( , , .... ) ( , , ... )lim n n xx

    f x x x x f x x xy y fx x x

    + = = =

    pi .

    ( , )Q Q K L= QK

    Q pi L . pi L pi pi pi .

  • pi 61

    pi pi pi , ...

    pi pipi .

    : 0 0xx yyf f f = + =

    0x

    pi 0x :

    1

    00

    2

    ( )

    ( )( )

    .........

    f xx

    f xf xx

    =

    pi ( )H f 2

    iji j

    fHx x

    =

    pi

    j :

    2 2

    21 1

    2 2

    21

    ( )n

    n n

    f fx x x

    H ff f

    x x x

    =

    ( , )f x y x y

    ( , )( , )

    x x u v

    y y u v=

    =

    pi f pi ( , )u v :

    x u y u

    x v y v

    f f x f yu

    k a if f x f yv

    = + = +

  • pi 62

    pi

    pi pi pi pi :

    Q a bPQ c dP=

    = +

    , , , 0a b c d > pi :

    a cPb dad bcQb d

    + = +

    = +

    pi pi P pi pipi pi pi pi P . pi pi pi .

    1 ( ) pi :

    2 3( , ) 3 2 4f x y x xy y= + +

    6 2xf x y= + 22 12yf x y= +

    2 ( pi) 2 2( , )f x y x xy y= + + 22 ,x t y t= = f t .

    3 2(2 )2 (2 )2 4 6 8t x t y tf f x f y x y y x t t t t= + = + + + = + +

    5.2 1 2( , , . . . )i nf x x x () n pi pi :

    1 1 1

    1 2

    2 2 2

    1 2

    1 2

    __ __ ... __

    __ __ .... __| | | |..................................

    __ __ .... __

    n

    in

    i

    n n n

    n

    f f fx x x

    f f d f fx x xJ

    x

    f f fx x x

    = =

  • pi 63

    1 2( , , . . . )i nf x x x ( pi pi pi

    plus ...) pi pi

    pi pipi

    .

    pi pi

    .

    3 () 2 2

    ( , ) 2 3( , ) 4 12 9

    f x y x yg x y x xy y

    = +

    = + +

    :

    23

    x

    y

    ff=

    =

    8 1212 18

    x

    y

    g x yg x y

    = + = +

    |J| = 0 .

    5.3 1 2( , ,... )kQ P x x x= pi pi pi pi pi 1 2, ,... kx x x ( , , pi ). ix pi

    pi i

    Px

    .

    0i

    Px

    >

    pi.

    pi ( pi ).

    pi pi i

    Up

    1 2( , ... )ky f x x x= pi pi x pi

    , i

    iy x

    i

    x yEy x

    =

    .

    pi U ix 1% pipi pi .

  • pi 64

    5.4 pi pi :

    [ , , ( , )] 0f x y g x y =

    pi pi pi pi pi

    pi pi

    pi .

    .

    pi pi pi pi pi

    pi .

    pi :

    )

    ( , ( )) 0f x y x = pipi 2

    2,dy d ydx dx

    pi pi pi ( )Y x . 0f fdx dy

    x y

    + =

    pi pi :

    x

    y

    fd yd x f= pi pi pi :

    2 22

    2 3

    2xx y xy x y yy xy

    f f f f f f fd ydx f

    +=

    pi pi pi pi .

    ) ( , , ) 0 ( , , ( , )) 0f x y z f x y z x y= = pi pi pi ( pi). pipi :

    x

    z

    y

    z

    fzx f

    fzy f

    = =

    ) pi pi pi

    .

  • pi 65

    4

    pi pi pi dydx

    3 3( , ) 6f x y x y xy= +

    pi pi

    pi pi 2

    23 63 6

    x

    y

    fd y d y x yd x f d x y x

    = =

    pi pipi

    ( , ) ( , )nf kx ky k f x y= . () .

    EULER EULER x yxf yf f+ .

    pi

    [ ( , )]g f x y ( , )f x y .

    ( , )z f x y= . pi pi pipi 0dz = pi 0, 0dx dy= = . pipi :

    0 0, 0x y x ydz f dx f dy f f= + = = = pi pi pi .

    pi : 2 2 22xx xy yyd z f d x f dxdy f d y= + +

    pi

    pi

  • pi 66

    1 . pi

    pi .

    pi pi pi

    .

    5 () 3 3( , ) 3f x y x y xy= +

    pi pi pi : 2

    2

    3 3 03 3 0

    x

    y

    f x yf y x

    = + =

    = + =

    (0,0) (1,-1) .

    pi :

    6 , 3, 6xx xy yyf x f f y= = = pi : (0,0) :

    0, 3, 0xx xy yyf f f= = = 2xx yy xyf f f< (0,0) . (1,-1) :

    6, 3, 6xx xy yyf f f= = = 2xx yy xyf f f> pi .

    5.5 ( LAGRANGE) pi

    pi pi (pi ) . pi . pi pipi pi pi

    :

    1 0x yf f= = 0x yf f= =

    2 2

    , 0xx yy

    xx yy xy

    f fandf f f

    i

    2

    , 0xx yy

    xx yy xy

    f fandf f f

    i

  • pi 67

    1 pipi ( , )f f x y= pi pi ( , )g x y c= . pi pi

    0x y y xf g f g = ( , )f f x y= pi ( , ) , ( )g x y c y y x= = pi :

    x x

    y y

    f gf gf g

    =

    = =

    pipi LAGRANGE pi pi .

    2 pipi pipi pi n pi m

    pi .

    pi pi pipi 2 3 . ( , , ) ( , ) ( , )F x y f x y g x y = +

    0 0( , )P x y .

    2 2

    __ __

    ( , , ) __ ____ ___ 0

    xx xy x

    xy yy y yy x xx y

    x y

    F F g

    D x y F F g F g F g

    g g

    = =

    1. D ( , )f x y pi pi .

    2. D ( , )f x y pi pi .

    6 (LAGRANGE) ( , )f x y xy=

    ( , ) : 1g x y x y+ =

    ( , , ) ( 1)F x y xy x y = + + pi pi :

  • pi 68

    1

    x

    y

    F yF x

    F x y

    = +

    = + = +

    ,

    : 1/ 2

    1/ 2x y= =

    =

    pi 01

    xx yy

    xy yx

    F F

    F F

    = =

    = = pi LaGrange :

    __ __ 0 __1__1__ __ 1__ 0 __1 2 0

    1__1__ 0__ ___ 0

    xx xy x

    yx yy y

    x y

    F F g

    D F F g

    g g

    = = = >

    pi ( , ) max 1/ 4f = .

  • pi 69

    6

    pi .

    -

    ( pi

    )

    (pi pi)

    pi

  • pi 70

    pi pi ( )tY f t= pi t , t o f . , pi , ,

    ..

    : ) pi ) pi pi

    6.1

    pi

    ( ) ( )tY f t= . pi pi pi tY .

    pi , , () pi pi pi .

    .

    pi pi tY pi pi pi .

    pi .

    pi pi

    :

    1 1 ( )t t tY a Y Y f t= =

    pi pipi () .

    6.1.1 1 pi :

    1t tY aY+ = pi pi :

    ) tY pi pipi ? pi pi .

    ) 0Y ? ) pi pi pi pi tY pi Y pi?

  • pi 71

    0Y pi

    1 02

    2 1 0

    .......

    tt

    Y aY

    Y aY a Y

    Y a Y

    =

    = =

    =

    1 : 0t

    tY a Y= pipi :

    - | | 1a (0). pi pi pi 0a pi :

    ) 01 0a pi

    ) 00 1a pi ( 0Y ) 0

  • pi 72

    - | | 1a pi explosive

  • pi 73

    - 0 1a =

  • pi 74

    - pi Malthus pi pi.

    pi 1 1( )t tDp f p = pi pi 1t tDp kp = . :

    1 1 1

    1( 1)t t t t t

    t t

    Dp kp p p kpp k p

    = =

    = +

    : 0( 1) ttp k p= + 0k > ( )1 1k + > ( pi pi) . 0k < 1 1k + < pi pi .

    6.1.2 1 pi :

    1n nY aX b+ = +

    pi pi pi ( pi n pi) .

  • pi 75

    12 2

    2 2 22 3 2

    3 33 2 4 3 2

    ( ) ( 1)( ) ( 1) ( 1)( ) ( 1) ( 1)

    n

    n n n

    n

    n n

    aX ba aX b b a X ab b a X b a

    Ya aX b b a a X b a aa aX b b a a a X b a a a

    +

    + + = + + = + +=

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

    pi

    1 20 ( ...... 1)n n nnY a X b a a a = + + + + +

    pipi

    - = 1 0nY X b n= + i

    - 1a 1 2

    0 01( .... 1) ( )1

    nn n n n

    n

    aY a X b a a a a X ba

    = + + + + + =

    1 pipi

    1

    - 0nY X b n= + i = 1

    - 01( ), 11

    nn

    n

    aY a X b aa

    =

    6.2 pi pi pi

    pi 1t tY Y Y+ = = Y pi 1 pi pi 1a

    pi pi 1t tY aY b= + pi

    1bY aY b Y

    a= + =

    pi pi

    pi

  • pi 76

    1( )t tY Y a Y Y = t tX Y Y=

    1 0t

    t t tX aX X a X

    = = pi

    0( )ttY Y a Y Y =

    0( )1 1t

    t

    b bY a Ya a

    = +

    pi pi 1ta

    pi pi t pi pi pi pi

    .

    - 1 0ta t 0

    11t

    aYa

    - 1ta t tY pi

    pipi pi 1t

    a t

    pi 1a

  • pi 77

  • pi 78

  • pi 79

  • pi 80

    6.3 pipi .

    1a

    t tY Y+ = . pi pi Y pi 1t tY Y Y+ = = . pi pi tY . - pi 0Y ( ) Y . - pi

    pi .

    1a

    t tY Y+ = pi Y 01

    aY

    Y YY =

    = =

    .

    pi pi pi.

    pi pi pi pi

    ( ) . pi pi Phase Diagram pi pi .

  • pi 81

    PHASE DIAGRAM

    pi

    E PD 1tY + tY f. Steady-state (pi) ( )tf Y 45 1t tY Y += . pi 1/ 2a = 1/ 2 pi 1Y = . pi pi tY pi pi 0Y = 1Y = pi pipi

    00 1Y . tY pi

    1Y = pi.. 0,5 pi pi 1. pi pi pipi 0Y tY = 1 pi = 0 pi pi tY pi .

    1 21t tY Y+ = 0 0.5Y = pi 1Y = 0Y = 0 0.5Y =

  • pi 82

    0 1.5Y = pi 1

    pi 1,0. __ _

    YY

    pi pi

    =

    =

    1 pi pipi 1 pi pi | ( ) | 1f Y :

    11 1( )a at t t tY Y f Y aY + += = pi

    11 | | 1 | | 1atY aY a= pi 1 0 | | 1Y a= pi

    2

    11 int

    (0) 0, 10

    ( ) (0) ,0 11 (1) 1 __ 1 1

    locallystable

    stationarya at t t t po s

    locallystable

    f aY

    Y Y f Y aY fY f a Y if a

    pi +

    = =

    = = = < pi 0tY > . ( 2 pi pi)

  • pi 83

    7

    pipi pi pi

    pi pi pi .

    pi pi

    pi pipi

    .

    pipi pi pi

    , , pi

    ..

    7.1

    ( )y y x= pi 2

    ( )2( ) , ( ) ,............., ( )

    nn

    n

    dy d y d yy x y x y xdx dx dx

    = = =

    :

    ( )( , , , ,...... ) 0nf x y y y y =

    . pi pi

    pi pi , .

    pi pi

    pi pi 1 , pi 2 ...

    . .

    : 2 5 6( ) ( ) 0y xy y y + + + = 3 2.

    pi pi 1 pi pi pi 0y y yx + + = . 2 1 . :

    ( ) ( 1)1 1 0( ) ( ) ...... ( ) ( ) ( ) 0n nn nf x y f x y f x y f x y f x + + + + + =i i i i

    f pi ( pi pi ).

  • pi 84

    pi pi pi

    pi . pipi pi

    pi pi : - pi pi ? - pi? - pi

    pi ( )?

    - pi .

    pi pi pi

    .

    pi pi pi

    pi .

    pi pi pi . pi :

    1 2( , , ,... )ny y x c c c= c . n pi pi pi pipi

    ( )( , , , ,...... ) 0nf x y y y y = .

    1 2 31 2y c x c x= +

    : 2

    1 22 3y c x c x = + 1 22 6y c c x = + pi c ( ) :

    12

    2y xy

    cx

    = 2 23

    xy yc

    x

    =

    2 3 2

    1 2 4 6 0y c x c x x y xy y = + + =

    ( )( , , , ,...... ) 0nf x y y y y =

    =() pi pi .

  • pi 85

    1( )a ax ax =

    ( )x xe e = ( )sin cosx x =

    ( )cos sinx x =

    ( )1ln xx

    =

    ( ) 21

    arctan1

    xx

    =+

    ( ) 21

    arcsin1

    xx

    =

    cos( ) 2x xe ehx

    +=

    sin( ) 2x xe ehx

    =

    ( ) lnx xa a a =

    dy dy dtdx dt dx

    =

    ( ) ( )y f x dy f x dx= =

    11

    1a ax dx x c

    a

    += ++

    1 ln | |dx x cx

    = + x xe dx e c= + cos sinx dx x c = + sin cosx dx x c = +

  • pi 86

    1 ln | |dx x a c

    x a= +

    21

    arctan1

    dx x cx

    = ++

    2

    1arcsin

    1dx x c

    x= +

    1

    lnx xa dx a c

    a= +

    ( ) ( ) ( )b

    a

    f x dx F b F a=

    ( ) [ ( ) ( )] ( )t

    a

    d df x dx F b F a f tdt dt

    = =

    pi b

    a

    f gdx fg gdf= i

    7.2

    1 ( , , ) 0F x y y = ( , )dy f x ydx

    =

    ( , ) ( , ) 0A x y dx B x y dy+ = .

    : ( )y f x = . pi pi .

    1yx

    = 1 ln | |y dx c x cx

    = + = +

    ( ) ( ) 0A x dx B y dy+ = pi ( ) ( )A x dx B y dy C+ =

    2 2( 1)y y x = +

    pi

  • pi 87

    (pi ? pipi pi pi pi pi

    pi ).

    2 2

    22

    ( 1) ( 1)

    ( 1)1

    dyy y x y xdx

    dydy y xdx xdxy

    = + = +

    = + =+

    21( ) , ( )

    1A y B x x

    y= =

    + :

    21 1arctan1 2

    dy xdx C y x Cy

    + = =+

    7.2.1 pi pi

    pi pipi

    ( , ) ( , )nf kx ky k f x y= . () . pi pi

    :

    ( )y u x x= x u(x) dy pipi pi

    .

    3 2 2( ) 2 0x y dx xydy+ =

    pi ( , ) ( , ) 0A x y dx B x y dy+ = 1 pi , ( , ) ( , )nA kx ky k A x y= ( , ) ( , )nB kx ky k B x y= pi 2 2( , ) , ( , ) 2A x y x y B x y xy= + = . pi pi ( )y u x x= :

    ( ) ( )dy d ux u x u dx= = + . : 2 2 2 2

    2 2 2

    2 2 2

    2 2

    ( ) 2 ( ) 0(1 ) 2 ( ) 0

    [(1 ) 2 2 ] 0[(1 ) 2 ] 0

    x u x dx x u u x u dxdu

    x u dx x u x u dxdx

    x u dx uxdu u dxx u dx uxdu

    + + =

    + + =

    + =

    =

  • pi 88

    pi pi

    : 2

    2

    2

    2

    (1 ) 2 021

    ln ln( 1)( 1)

    u dx uxdudx udux u

    x u c

    x u C

    =

    =

    + =

    =

    pi pi pipi 2

    2( 1)y

    x Cx

    = 2y

    x Cx = pi pi .

    1 1 1

    2 2 2

    ( )a x b y cy fa x b y c

    + + =

    + + , b ,c .

    3 pipi

    1 2 2 1 0a b a b (pi ) pipi 1 1 1a x b y c u+ + = 2 2 2a x b y c w+ + = pipi ( , ) 0F u w = .

    1 2 2 1 10 , 0a b a b b = 1 1a x b y u+ = pipi u , x .

    1 2 2 1 20 , 0a b a b b = 2 2a x b y u+ = ( pipi) pipi .

    4

    1( 1) ( 2) 02

    x yx y dx x y dy y

    x y+ +

    + + + + + = = + +

    pi 1 2 2 1 0a b a b ( -2) : { 1 , 2 }x y u x y v+ + = + + = :

  • pi 89

    { , }{ }

    1{ }1

    dx dy du dx dy dvdx dy dudx dy dv

    y duy dv

    + = + =

    +=

    ++=

    +

    ( ) ( ) 0v u du v u dv+ + = pi 2

    1ln | | arctan ln( 1)2

    u t t c= + + .

    v ut= : 22 1 2ln | 1| arctan ln[( ) 1]

    1 2 1x y x y

    x y cx y x y + + + +

    + + = + ++ + + +

    5

    2 1

    2 4 1x yyx y + =

    pi 1 2 2 1 10 , 0a b a b b = { 2 , 2 4 2 }x y u x y v = = . pi :

    { 2 ,}{ }

    2

    dx dy dudu dxdy

    =

    +=

    pi :

    2 1 1 (2 1) 3 02 4 1 2 2 1

    dy x y du dx uu du dx

    dx x y u + + +

    = = =

    pi

    . pi 2( 2 ) ( 2 ) 3x y x y x c + =

    ( ) ( )y f x y g x + = f , g pipi x.

    : :

    ( ) 0y f x y + =

    pipi : ( )( ) f x dxy x c e= i

    :

    pi ( )

    00 ( )f x dx

    y y c x e = = i

    : pipi :

    ( )0( )

    f x dxy x c e y

    = +i

  • pi 90

    6 2 xy y e + =

    pi 22 0 xy y y ce + = =

    pi pi 2 30 010 2 ( )3

    x xy y c x e y e = = = .

    2 312

    x xy ce e= +

    -

    .

    - c(x) pi pi

    ( )0 ( )

    f x d xy c x e

    = i pipi

    c pi c(x). pi

    7.2.2 BERNULLI H BERNULLI ( ) ( ) ny f x y g x y + = . n n>2 0 1 pi pipi . ernulli pi

    11 ny u =

    pipi u(x).

    7

    31y y yx

    =

    Bern. pi pi pi pipi .

    11/ 21 3y u u = =

    pi : 1/ 2 3/ 2( )

    2dy dy du d u du u

    udx du dx dx dx

    = = =

    ,y y pipi : 1 1 21 22

    u u u ux x

    = + =

    pi ( ) pipi 2 2( )

    3u x c x x

    = i

    Bernulli :

  • pi 91

    2 1/ 22( )3

    y c x x = i

    7.2.3 RICCATI H Riccati 2( ) ( ) ( )y f x y g x y h x + = + pi ( ). pi pi

    1my y

    u= + pi

    my (pi) y pi pipi ( )u x .

    8

    2 121 1{ , }my y y y xx x

    + = + =

    Riccati pi pi pi x :

    11 1my y y x

    u u

    = + = +

    pi 1 1

    2 2dy dx du y x u udx dx dx

    = + =

    ,y y pipi 3 1u ux

    = pi

    pi ( ) 3 1( )2

    u x c x x= i

    Riccati pi :

    1 3 11( )2

    y x c x x = + i

    7.2.4

    ( , ) ( , ) 0 P QP x y dx Q x y dyy x

    + = =

    .

    pi ( ) pi ( , ) ( , ) ( , )dF x y P x y dx Q x y dy= + . ( , )F x y c=

  • pi 92

    9 2 2( 2 ) ( ) 0x xy dx x y dy+ + + =

    pi

    2 2( , ) 2 , ( , )P x y x xy Q x y x y= + = + pi

    2 2P Q x xy x

    = =

    ( , )F x y c= pi pi : 2( , ) 2F P x y x xy

    x

    = = +

    2( , )F Q x y x y

    y

    = = +

    : pi pi x y pi . pi : pi pi x y .

    2

    3 2

    ( , ) ( 2 ) ( )1( , ) ( )3

    F x y x xy dx c y

    F x y x yx c y

    = + +

    = + +

    3 2 21 ( )( ( ))3

    dc yx yx c y x y y

    y dy

    + + = + =

    pi pi ( )c y pi 21( )2

    c y y k= +

    ( , )F x y : 3 2 21 1( , )3 2

    F x y x yx y k c= + + +

    7.2.5 EULER pipi pi pi pi

    ( , ) ( , ) 0 KAI P QP x y dx Q x y dyy x

    + =

    pi pi

    pipi. pi ( , )M x y pipi pi

    :

    ( , ) ( , )( , ) ( , ) ( , ) ( , ) 0 M P M QM x y P x y dx M x y Q x y dyy x

    + = =

    pipi EULER . pipi pi pipi. pi pi

    pipi , x y (pi.. u = x+y) pi

  • pi 93

    pi. pi

    ( x) pi .

    10 ( ) 2 0x Y dx xydy+ = x

    pi x

    :

    ( , ) ( , )( , ) ( , ) ( , ) ( , ) 0, M P M QM x y P x y dx M x y Q x y dyy x

    + = =

    pi

    ( , ) ( , ) ( ) 1 ( )( )M P M Q M x P Q

    y x M x Q y x

    = =

    ( 1) pi x pipi

    1 ( ) ( )P Q f xQ y x

    =

    ( 2) . 1,2 . pi pi :

    ( )( ) ( ) ( )( )f x dxM x f x M x e

    M x = =

    pi 1 f pipi 1 2( ) ( ) ( )P Q f x f xQ y x x

    = =

    ( 3)

    pi 1 3 pipi 2( ) ( ) 2( ) ( )( ) ( )

    M x M xf x M x xM x M x x

    = = =

    pipi. pipi pipi

    pi .

    21 lny x cx

    + = .

    11 2 2( ) ( ) 0y xy dx x yx dy + + = pipi

    pi . pi

    pipi . pi . pi

    pipi ( )M u pi x,y pi :

  • pi 94

    ( ( ), ) ( ( ), )( ) ( , ) ( ) ( , ) 0, M u P M u QM u P x y dx M u Q x y dyy x

    + = =

    pi (pi pi) :

    ( )( ( ), ) ( ( ), )

    ( )

    Q PdM uM u P M u Q x ydu

    u uy x M u P Qy x

    = =

    ( 1)

    ( pi x,y) 1 pi pipi . pi pi

    x

    y . u x= :

    2

    ( )4 4

    ( ) 1

    Q PdM uxy yx ydu

    u uM u x yx xyP Qy x

    = =

    pi y pipi . u y= pi pipi x. pi

    pi x y . u xy=

    ,

    u uy xx y

    = =

    1 pi pi

    ( )2

    ( )

    Q PdM ux yduu uM u uP Qy x

    = =

    pi

    pi 1 pipi pipi pi :

    2 2 2

    ( )2 1 1( )( )

    dM udu M u

    M u u u x y= = =

    pipi pi

    pi pipi

    1 ln lnx y cyx

    + =

  • pi 95

    7.2.6 CLAIRAUT ( )y xy f y = + pi ( )f y y pi y p = pipi ( )y xp f p= + . pi pi y p = pi pipi

    ( )( ) 0dp df pxdx dp

    + = pipi p c=

    ( )y xc f c= + .

    12

    s in ( )d y d yy xd x d x

    = +

    pi pi Clairaut dy pdx

    =

    s in ( )y x p p= + pi pi x : s in ( ) c o s ( ) ( c o s ) 0d y d p d p d py x p p p x x p

    d x d x d x d x= + = + + + = pi

    pi

    c o sc o s 0

    s inx p

    x py x c p=

    + = = +

    pi pi

    7.2.7 LAGRANGE

    Lagrange ( ) ( )d y d yy x f gd x d x

    = +i ( pi CLAIRAUT) ,f g . pi pi y p =

    __ __

    ( ) ( )( ) ( ) ( ( ))x

    dx df p dg py xf p g p p f p xdp dp dp

    pi pi= + = pipi

    1 ( )x x p= pi pi p

    13

    12y x yy

    = +

    (pi LaGrange) y p =

    __ __ 21 12 2

    x

    dxy xp p xp dp p

    pi pi= + + = pi 1

    ( +

    ) pipi : 2 2( ) lnOM EIx p x x cp p p = + = +

    pi :

  • pi 96

    2 2( ) ln12 12

    x p c p p py x y

    y x pyp

    = += + = +

    7.2.8 ( , )y f x y = ( )y a b= pi pi pipi . pi

    .

    pi (pi pipi pipi) pi pi c pi pi .

    pi ( ),a b , f pi .

    7.2.9 1 pi pipi pi .

    . pi pi.

    ( , , )dyF x ydx

    y p =

    pi p x,y pipi ( , , ) 0F x y pp

    =

    pi

    pi p (x,y) = 0 . P pi pi pi pi p pi pi .

    ( , , )F x y c pi c pi pi pipi pi (pi.. ...) c pi pi pi . pi pi : pi pi ( , , )F x y c pi pi ( pi ). : pi 1 pi pi pi

  • pi 97

    ( pi) tan

    1 tan

    dydy dxdx

    + (

    ). pi 1 pi .

    90 pi :

    1/

    dydx dy dx

    7.3 1 pi 1 pi pi pi

    pi 1 . pi pipi :

    1 (pi pi )

    ) ( , ) ( , ) 0) ( , )

    A P x y d x Q x y d yd yB f x yd x

    + =

    =

    2 pi pi :

    1. pi pi 2. P,Q

    y=ux

    3. pi P Qy x

    =

    3 pipi

    pi pi : 1. y pi 2. y pi pi BERNOULLI 3. y pipi

    pi RICATTI

    4 pi pi pi

    pipi ( pi)

  • pi 98

    5 pi pi Lagrange (Clairaut) pi pi y p =

    6 pipi pi pi

    7.4 pi pipi pipi pi pipi

    pi ,

    BERNULLI pi ( t). pi pi pi

    (pi.. ) pi pi pi pi

    1 ( )dy f ydt

    = .

    pi { , , }dyO ydt

    pi pi pi pi.

    1. pipi pi { 0}dydt

    y pipi

    pi pi .

    pi pipi

    y pi pi pi pi . .

    pi pi pi y pi .

    2. pi pipi pi y pi

    { 0}dydt

    = pi y .

    pi pipi

    . pi

    pi pipi pi y pi pi.

    3. pi pi pi . pi 0y pi pi pi ( ) y pi pi pi pi

    pi pi pi ,

    pi pi pi pi

    pi

  • pi 99

    pi pi pi pi.

    pi pi pi

    pi

    . pi pi

    . pi

    { }dydt

    pi ( ). pi.

    pi pi

    pi pi pi

    pi pi .

    Solow.

    14 pi pi

    : ( ) ( ) ( )K t I t K t = pi () pi pi (). pi pi , pipi

    pi (capital stoke) pi pi .

    ( ) t IK t Ce

    = + pi pi pipi

    / ( pi pi) pi pipi pi .

    0( ) ( ) tI IK t K e

    = + pi / pi

    pi .

    >0 pi ( ) IK t

  • pi 100

    WALRASIAN pi pi

    pi ( ) ( )( ) ( )

    D

    S

    q t A Bp tq t F Gp t = +

    = + A,B,F,G>0

    pi :

    ( ) ( )D S A Fq t q t A Bp F Gp pG B

    = + = + =

    pi t pi . ( )p t p t p(t). pi pi? pi pi

    pi pi pi.

    : ( )D Sdp p a q qdt

    = =

    pi a

    pi pi .

    : 0 D Sp q q ( ) 0 D Sp q q (pi ) Dq 0B < Sq

    0G > . pi. ,

  • pi 101

    pi pi G pi pipi pi? ( ) ( ) ( )D Sp a q q p a B G p a A F = = :

    ( ) ( )0( ) [ ]a B G t a G B t

    A Fp t Ce p p e pG B

    = + = +

    pi

    pi pipi . pi? pipi 0G B > 0a > . pi ? 0B < 0G > .

    SOLOW To pi SOLOW pi pi pi DOMAR pi pi pi pi pi pi pi

    pi pi.

    pi DOMAR pi pi pi pipi

    pi . pi pi

    pi

    , pi pi pi

    pi.

    SOLLOW pipi pi ( ). pi pi pi ( , ) ( ,1) ( )KY f K L L f L F k

    L= = =i i

    pi pi ( pi) pi ( pi ). pi :

    1. dK S Ydt

    = i pi S pi pi pi pi

    pi

    2. / 0dL dt cL

    = ( ) pi pi L. pi pi pi pi. pi pi k pi S , c pi SOLLOW :

    ( ) ( )d k s F k c kd t

    = i

  • pi 102

    7.5 (1) ( )0 1( , ( ) , ( ) ,.... ( ) ) 0nnF x a t y a t y a t y = pipi 2 pi pi ( ) ( ) 0y P t y Q t y + + = .

    WRONSKI 1 2,y y 2

    WRONSKI

    1 2

    1 2

    y y0

    y yW =

    ( )

    ( ) ( ) 0y P t y Q t y + + = 1 1 2 2y c y c y= + ( 1 2,y y pi ). WRONSKI 3 ...

    WRONSKI pi ( ) 0dw P t wdt

    + = pi

    1 ( )( ) P t dtw t ce= .

    pipi pi pi

    pi . pi pi

    15 ( ) ( ) 0y P t y Q t y + + = pi 1 ty e= ,

    2 (0) 1y = 2( ) tw t e= . .

    pi W pipi 1 2 2 21 2

    y yy y

    tW y y e= = =

    pi pi

    1 2 ( ) tMy c t e= . pi pi 2 2 ( ) ( )t t tM My y e c t e e c t t = = = 2 2 2

    t tM Oy y y e te= + = + ( pi pi ).

    1 1 2 2y c y c y= + .

    2 2 pipi . 0y ay by + + = a,b 1 1 2 2y c y c y= + . .

  • pi 103

    22

    2

    1ydydtd ydt

    pi 12

    2

    0b

    + + =

    .

    pipi :

    ) 1 2 pi : 1 21 2

    t ty c e c e = +

    1,2 1 2b b t = : 1

    1 2 2 2[ c o s ( ) s i n ( ) ]b ty c e b t c b t= + ) 1 2 = pi :

    1

    2

    t

    t

    y e

    y te

    =

    =

    16 3 2 0y y y + = 4 5 0y y y + =

    ) :

    22

    2

    1ydydtd ydt

    pipi 12

    2

    13 2 0

    2

    = + =

    =

    2

    1 2t ty c e c e= +

    )

  • pi 104

    22

    2

    1ydydtd ydt

    pipi 12

    2

    24 5 0

    2ii

    = + + =

    = (pi

    pi )

    2 21 2c o s ( ) s in ( )t ty c e t c e t= +

    2 ( )

    22 ( )1, , ,......

    nn

    n

    dy d y d yydt dt dx

    pipi pipi pi pi pi .

    pipi pi

    pi

    .

    pi pi ( pipi pi )

    EULER

    pi pi pi

    pi pi .

    pipi EULER pipi . pipi 2 EULER :

    22

    1 22( ) ( ) 0d y dy

    at b c at b c ydx dx

    + + + + =

    pipi

    pi pi (2 pi 2 ).

    xx e bat b e t

    a

    + = =

    pi EULER pi pi

  • pi 105

    pi

    te pi pi

    .

    17

    2

    22(2 1) 6(2 1) 4 0

    d y dyt t y

    dx dx+ + + + =

    EULER (pi pi pi) pi 12 1

    2

    xx et e t

    + = = . t

    pi pi t pi x pi y pi x .

    pi pi pipi :

    1 1

    2 2

    xxe dtt e

    dx

    = =

    2

    x

    dydy dy dx dxdt dx dt e

    = =

    2

    2 2

    /2 2

    2( ) 4d dxx x

    dy d y dyd y dx dx dxdt e e

    = =

    pipi 2

    2 2 0d y dy ydx dx

    + + = pi

    pi pi

    12

    2

    12 1 0

    1

    = + + =

    = 1 2( ) x xy x c e c e = + pi

    2 1 xt e+ = 11 2( ) ( ln | 2 1 |)(2 1)y t c c t t = + + +

    pipi pi

    . : 2

    12 ( ) ( ) 0d y dyP t Q t y ydt dt

    + + =

    1( )y u t y= pipi.

  • pi 106

    ( ) 2 .

    2

    2 ( ) ( ) ( )d y dyP x Q x y F xdt dt

    + + =

    2

    2 ( ) ( ) 0d y dyP x Q x ydt dt

    + + =

    pi

    ( 1 ). pi pi

    pi.

    pi

    2

    2 ( ) ( ) ( )d y dyP x Q x y F xdt dt

    + + = .

    pi 2

    2 ( ) ( ) 0d y dyP x Q x ydt dt

    + + = pi

    1 2,y y pi 1 1 2 2( ) ( ) ( )Oy t c y t c y t= + c .

    pi 2

    2 ( ) ( ) ( )d y dyP x Q x y F xdt dt

    + + = pi

    1 1 2 2( ) ( ) ( ) ( ) ( )My t c t y t c t y t= + c(t) t ( ) ( )My t pipi pi pi . pi ( 2 pi pi ) ( )My t ( )My t ( )My t pi :

    1 1 2 2 1 1 2 2( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 0My t c t y t c t y t c t y t c t y t = + + = ( 1 )

    1 1 1 1 2 2 2 2( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )My t c t y t c t y t c t y t c t y t = + + +

    . pi pi ( pi pi pipi )

    1 1 2 2( ) ( ) ( ) ( ) ( )c t y t c t y t F t + = ( 2)

  • pi 107

    1 2 pi 1 2( ), ( )c t c t pi CRAMER. ( pi WRONSKI pi pi ). pi 1 2( ), ( )c t c t pi 1 2( ), ( )c t c t .

    18

    2 2

    2 24 4 1

    xd y dy eydx dx x

    + =+

    pi pi

    (0) 1, (0) 0dyydx

    = = .

    2 . .

    pi 2

    2 4 4 0d y dy ydx dx

    + = pi

    ( pi pi) pi pi = 2

    2 21 2

    x x

    Oy c e c e= + .

    2 21 2( ) ( ) ( )x xMy x c x e c x xe= + pi :

    2 21 2 1 2( ) ( ) 0 ( ) ( ) 0x xc x e c x xe c x c x x + = + = ( 1) pipi pi pi

    1 2 212 ( ) (1 2 ) ( )

    1c x x c x

    x + + =

    + ( 2 ) 1 2 1 2( ), ( )c x c x CRAMER 1 2( ), ( )c x c x pi :

    21

    2

    1( ) ln ( 1)2

    ( ) a r c t a nc x x

    c x x

    = +

    =

    :

    2 2 21( ) ln( 1) arctan2

    x x

    My x x e xe x= + + pi

    2 2 2 2 21 2

    1( ) ln( 1) arctan2

    x x x xy x c e c xe x e xe x= + + +

  • pi 108

    pi pi

    (0) 1, (0) 0dyydx

    = =

    :

    2 2 2 2 21( ) 2 ln( 1) arctan2

    x x x xy x e xe x e xe x= + +

    pi ,D SQ Q P. pi pi

    pi pi

    pi pi pipi

    pi pi pi

    pi.

    pi pi

    pi , pi dPdt

    pi 2

    2d Pdt

    . pi

    pi pi pi

    pi : 2

    2

    2

    2

    [ ( ), , ]

    [ ( ), , ]

    D

    S

    dP d PQ D P tdt dt

    dP d PQ S P tdt dt

    =

    =

    2 2

    2 2

    2 2

    2 2

    [ ( ), , ]

    [ ( ), , ]

    D

    S

    dP d P dP d PQ D P t a bP m ndt dt dt dt

    dP d P dP d PQ S P t c dP u wdt dt dt dt

    = = + +

    = = + + +

    pi a , b , c , d pi pi pi m , n , u , w . pi pi pi pi

    pi. pi 0m > dQ . pi pi pipi pi

    . pi 0m < pi pi pi pi

    pi pipi pi pi pi

    pi . pi pi

    pi pi

  • pi 109

    . pi m , n

    pi . pi pipi.

    7.6

    dDdt

    = pi pi

    pi ( )dfDf f tdt

    = = (pi..

    ( )( )

    atat atd eD e ae

    dt= = ) .

    pi pi pi pi

    :

    ( ) ( )D k f k D f=i i 1 2 1 2( ) ( ) ( )D f f D f D f+ = + 1 2 1 2 2 1( ) ( ) ( )D f f D f f D f f= +i

    ( ) ( )n

    n

    n

    dD f fdt

    =

    11 ( )D f t dtD

    = =

    2 21 ( ( ) )D f t dt

    D = =

    : pi D pi.. 2 ( 1)D D D D+ = + .

    pi pi pipi

    pi

    pi pi y . pi :

    22 2

    2 2 0 2 2 0 ( 2 2) 0d y dy y D y Dy y D D ydt dt

    + = + = + =

    pi 3 pipi ( ) .

    ( )( )

    atat atd eD e ae

    dt= =

    ( ) ( )at atf D e f a e=

  • pi 110

    11/ ( ) ( )

    at atf D e ef a= F(a) =0 ( pipi ) pi

    1( ) ( )

    at atte ef D f a

    =

    19

    2

    42 5 6

    td y dy y edt dt

    + =

    24 2 4 4

    2 215 6 ( 5 6)5 6

    t t td y dy y e D D y e y edt dt D D

    + = + = = +

    4 pi pi 4 4

    21 1

    45 6 2t ty e y e

    DD D= =

    = +

    pi. pi

    .

    ) (n+1) pi n pi . ) pi . 21 1 ....

    1k k

    k= + + +

    20

    2 2dy y xdx

    =

    2 2 212 ( 1) 2 ( 2)1

    dy y x D y x y xdx D

    = = = +

    pi pi

    2 2 2 2 2 2 21 ( 2) (1 ...)( 2) ( 2) ( 2) ( 2) ....1

    y x y D D x x D x D xD

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

    pi 2( 2)x + 3 2( 2) 0D x + = pipi

    2 2 2 2

    2

    ( 2) ( 2) ( 2) ( )2 2 2

    y x D x D x

    y x x

    pi= + + + + +

    = + + +

  • pi 111

    pipi pipi.

    pi pi pipi

    pipi pi .

    pi :

    ( )( ( )) ( ) ( )at atF D e f x e F D a f t= +

    7.7 LAPLACE ( )f t pi t > 0

    0

    ( ) ( ) stF s f t e dt+

    = pi s

    LAPLACE ( )f t { ( )} ( )L f t F s= pi pi LAPLACE pi 1{ ( )} ( )L F s f t = . pi pi

    LAPLACE pi .

    LAPLACE

    HEAVISIDE () pi pi

    .

    1, 0( )0 , 0

    tH t

    t

    >=

  • pi 112

    2 2

    2 2

    1{ ( ) }1{ ( ) }

    { c o s ( ) ( ) }

    { s i n ( ) ( ) }

    a t

    i a t

    L e H ts a

    L e H ts i a

    sL a t H ts a

    aL a t H ts a

    =

    =

    =+

    =+

    3 ( )f t { ( )} ( )L f t F s= ( )f t a { ( )} ( )asL f t a e F s = (pi pi )

    4 ( )f t { ( )} ( )L f t F s=

    ( )t f ti { ( )} ( ) dFL tf t F sds

    = =

    pi pi

    pi pi

    L .

    21 L ( )a tt e H ti

    pi

    1{ ( )}atL e H ts a

    =

    pi pi pi

    21 1{ ( )} ( ) ( )

    atL t e H ts a s a

    = =

    i

    pi ( )f t pi L pi L. pi pipi pi pi pi

    pipi pi pi

    : 1. pi 2. ( )F s a 2

  • pi 113

    3. 1( )ks a

    pi pi pi

    4. pi ( )ase F s 3

    22

    2 25 2( ) ( 3) 1

    s sF s es

    += +

    pi e pi pi pipi

    2 2 2 25 2 5( 3 3) 2 3 15 17( 3) 1 ( 3) 1 ( 3) 1 ( 3) 1

    s s s

    s s s s

    + + + = = +

    + + + +

    s-3 = k :

    2 215 17

    1 1k

    k k+

    + +

    2 : 3

    25 2{ 5cos ( ) 17sin ( )} ( 3) 1

    t sL e tH t tH ts

    ++ =

    + 3

    3( 2)

    1 3( 2){ (5cos( 2) 17sin( 2) ( 2)} ( )

    { ( )} (5cos( 2) 17sin( 2) ( 2)t

    t

    L e t t H t F s

    L F s e t t H t

    + =

    = +

    LAPLACE pi L :

    f pi ( )

    1 2( ){ } ( ) (0) (0) .....

    nn n n

    n

    d fL s F s s f s fdt

    =

    pi pi pi :

    (2)2

    (2)

    { } ( ) (0)

    { } ( ) (0) (0)

    dfL sF s fdtd fL s F s sf fdt

    =

    =

    pi

    1. pipi pipi

  • pi 114

    2. pi 3. pi L pi

    23

    2 0, (0) 1dy y ydt

    + = =

    L ( ) :

    { 2 } {0}

    { } 2 { } 0( ) (0) 2 ( ) 0 ( 2) ( ) 1 0

    dyL y LdtdyL L ydt

    s y s y y s s y s

    + =

    + =

    + = + =i

    pi 1( )

    2y s

    s=

    +

    1 2( ) { ( )} ( )ty t L y s e H t = =

    24

    L

    23

    2

    23

    2

    2

    2 2

    2

    { 3 2 } { ( )}

    { } 3 { } 2 { } { ( )}1( ) 3 ( ) 3 2 ( )

    36 1 0 6 1 0 5 / 2 2 1 / 2( ) ( 3 2 )( 3) ( 3)( 2 )( 1) 1 2 3

    t

    t

    d y d yL y L e H td t d td y d yL L L y L e H td t d t

    s y s s sy s y ss

    s s s sy ss s s s s s s s s

    + =

    + =

    + + =

    + += = = +

    +

    1

    2 3

    5 / 2 2 1/ 2( ) { }1 2 3

    5 1( ) ( 2 ) ( )2 2

    t t t

    y t Ls s s

    y t e e e H t

    = +

    = +

  • pi 115

    7.7 ( ) ( ) 0y P x y Q x y + + = pi pi pi L pi . (FROBENIUS) .

    0x P , Q pi TAYLOR pi ( ). pi , , ...

    . pi P , Q 0x .

    0x 2

    0 0( ) ( ), ( ) ( )x x P x x x Q x 0x .

    TAYLOR E pi pi:

    ( ) ( )dyf x dy f x dxdx

    = = pi pi pi

    .

    pi pi

    pi pi pi TAYLOR. pi pi

    0x pi ( pi pi pi pi.. 5 pi ...). pi LOR pi pi 0x pi

    .

    pi TAYLOR pi pi pi 0x pi

    0x pi pi pi 1x

    1( )f x . pi pi TAYLOR

    ( )10 1 0

    1 01

    ( ) ( )( ) ( )!

    k kn

    n

    k

    f x x xf x f x Rk

    =

    = + +

    pi : 1 2 ( 1) 1

    0 1 0 0 1 0 0 1 01 0

    ( )( ) ( )( ) ( )( )( ) ( ) ....1! 2! ( 1)!

    n n

    n

    f x x x f x x x f x x xf x f x Rn

    = + + + + +

    pi ( ) 1 0 0 1( )( ) ,

    !n

    n

    x xR f c x c xn

    = < < pi

    0nn R

  • pi 116

    pi .

    : ( )1

    0 1 01 0

    1

    ( ) ( )( ) ( )!

    k kn

    n

    k

    f x x xf x f x Rk

    =

    = + +

    ( )10 1 0

    1 01

    ( ) ( )( ) ( )!

    k kn

    n

    k

    f x x xf x f x Rk

    =

    = +

    ( )10 1 0

    1

    ( )( )!

    k kn

    n

    k

    f x x x Rk

    =

    = +

    pi

    .

    TAYLOR ( ) pi pi TAYLOR :

    1 2 ( 1) 10 1 0 0 1 0 0 1 0

    0( )( ) ( )( ) ( )( )( ) ( ) ....

    1! 2 ! ( 1) !n nf x x x f x x x f x x xf x f x

    n

    = + + + +

    pi 0x pi C. pi pipi pi C pi .

    24 0 : 0y xy =

    2 . 1 2(0) , (0)y c y c= = pi (0) 0y = .

    pi pi pi 0 pi 1 2,c c pi TAYLOR :

    1 2 ( 1) 10 1 0 0 1 0 0 1 0

    0( )( ) ( )( ) ( )( )( ) ( ) ....

    1! 2 ! ( 1) !n nf x x x f x x x f x x xf x f x

    n

    = + + + +

    322 1

    1

    3 4

    1 2

    0( ) .......1! 2 ! 3!

    ( ) (1 ....) ( 2 ...)3! 4 !

    c x x cxy x c

    x xy x c c x

    = + + + +

    = + + + + +

    ( ) pipi :

    0( ) ( )kky x a x x= pi

    ka .

  • pi 117

    25 5 pi 2 0y xy xy + = 0

    0 pi ( ) nny x a x= pi :

    ( ) 2 ( ) 0n n nn n na x x a x x a x + =

    pi pi pi pipi

    . 0 1 1 2,a c a c= = C. pi pi

    pi pi pipi 122

    ( 1)( 2)n n

    n

    na aa

    n n

    +

    =

    + + pi pi

    pipi 5 pi .

    FROBENIOUS ( ) 0x P , Q 20 0( ) ( ), ( ) ( )x x P x x x Q x 0x :

    : 0 0( ) ( ) ( )r nny x x x a x x= : ( 1) 0r r mr v + + =

    : 0

    0

    0

    20

    lim( ) ( )

    lim( ) ( )x x

    x x

    m x x P x

    v x x Q x

    =

    =

    2 . pipi 3 pipi . 1 2r r

    pi

    1 2r r

    1

    2

    1 0 0

    2 0 0

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

    r n

    n

    r n

    n

    y x x x a x x

    y x x x b x x

    =

    =

    pi

    1 2r r r= = pipi

  • pi 118

    1 0 0

    12

    ( ) ( ) ( )

    ( )

    r n

    ny x x x a x x

    yy xr

    =

    =

    pi

    1 2r r

    1

    2

    1 0 0

    2 0 0

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

    r n

    n

    r n

    n

    y x x x a x x

    y x x x b x x

    =

    =

    1

    2

    1 0 0

    12

    ( ) ( ) ( )

    ( )

    r n

    n

    r

    y x x x a x x

    yy xr

    =

    =

    26

    FROBENIOUS 2 1 03 3

    y y yx x

    + + = 0

    0 pi pi . 20 0( ) ( ), ( ) ( )x x P x x x Q x pipi

    20 0

    2( ) ( ) , ( ) ( )3 3

    xx x P x x x Q x = = pi . pi :

    0

    0

    0

    20

    lim( ) ( ) 2 / 3

    lim( ) ( ) 0x x

    x x

    m x x P x

    v x x Q x

    = =

    = =

    pi 1

    2

    0( 1) 01/ 3

    rr r mr v

    r

    = + + =

    =

    pipi .

    : 1

    2

    1

    2

    ( )( )

    r n

    n

    r n

    n

    y x x a x

    y x x b x

    =

    =

    pipi pi ,n na b ( pi) pi pi ,n na b .

  • pi 119

    7.8 pi

    .

    ( ) ( )d Y A t Y B td t

    = +

    pi pi pi

    pipi. 22 pi pi :

    1 111 121

    21 22 2 22

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

    Y f ta t a tYa t a t Y f tY

    = +

    pipi

    pi

    pi .

    :

    d Y A Yd t

    =

    111 121

    21 22 22

    Ya aYa a YY

    =

    pi (). pi : pi pi

    . pi 1 21 2,t ty e y e pi

    : 1 2

    1 1 2 2( ) t ty t c y e c y e = +

    :

    ( ) ( )d Y A t Y B td t

    = +

    1 111 121

    21 22 2 22

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

    Y f ta t a tYa t a t Y f tY

    = +

    pi :

  • pi 120

    pi ( ).

    pi pi pi.

    pi .

    pi pi

    pi . pi

    pi pi

    . .

    { ( , )},{ ( , )}dy dxf x y g x ydx dy

    = =

    pi t. pi pi .

    pi t pi Y=Y(X) pi pi pi

    .

    pi :

    )

    )

    )

  • pi 121

    ) -

  • pi 122

    8

    pi pi

    . pipi

    pi pi.

    8.1 pi pi:

    ) pi pi pi ) pi pi ) pi pi pi

    pipi : 1. S 2.

    { , , ,0}cF A A S= . pi pi P pipi pi pi