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: : . - A.

04-02-2008 1o

) N , , () !!! , y = x , x > 0. ) () !!! x ) y = xH3L HtL = 2 7 sin H3 tL, xH0L = 1, x H0L = -1, x H0L = 1.1 y = x-12 , 2 1 y = - x-32 . 4

1o (0,8 )

1 x2 y = - 4

, 8a, b, c< a = b + c , , 4 a x2 y + 2 b x y + c y = 0. 2 x2 y - x y + y = 0 , 1 1 8a, b, c< = 8 , - , 1 0, y + H1 + 3 t2 L y = e-t .3 3 3 3

, y> = e-Ht+t L Hc2 + e-t et+t d tL.3 3

,

t = 0

y> = e-Ht+t L Hc2 + et d tL = c2 e-Ht+t L + e-t .3

c2 = c1 c.3

, y=9 c e-Ht+t L + 1,3 c e-Ht+t L

t0 t0

3 + e-t ,

xH0L = 1 y=9 1, e ,-t3

t0 t0

3o

) , za HtL, . t

) , z2 HtL, Picard () z + z2 - 2 p z = 0 zH0L = p. ) , z2 HtL za HtL. zn HtL = 2 p + 0 FHt, zn-1 HtLL d t, n = 1, 2, FHt, zL = FHzL := -z2 + 2 p z, z0 HtL = p,t

3o (2,0 )

) , Picard

z = zHtL.

, Picard : z1 HtL = p + 0 FHt, z0 HtLL d t = p + 0 FHz0 HtLL d t = p + 0 FHpL d t.t t

,

,

,

,

z2 HtL = p + 0 FHt, z1 HtLL d t = p + 0 FHz1 HtLL d tt t

z1 HtL = p + 0 FHpL d t = p + p2 t.t

FHpL = -HpL2 + 2 p HpL = p2 .

FHp + p2 tL = -Hp + p2 tL + 2 p Hp + p2 tL = -Hp2 + 2 p3 t + p4 t2 L + H2 p2 + 2 p3 tL = p2 - p4 t2t 2

= p + 0 FHp + p2 tL d tt

,

z2 HtL = p + 0 FHp2 - p4 t2 L d t = p + 0 Hp2 - p4 t2 L d tt 1 = p + p2 t - p4 t3 3

) z + z2 - 2 p z = 0 z + zHz - 2 pL = 0. zHtL = 0 zHtL = 2 p. z < 0, 0 < z < 2 p z > 2 p, 1pL z = -1. zHz-2 ,1 pL d z = c - t. zHz-2

z < 0, 0 < z < 2 p z > 2 p, 1pL z = -1. zHz-2

,

,

,

z-2 p ln = 2 pHc - tL z

1 1 1 1 1 z-2 p pL d z = p H - L d y = p ln zHz-2 2 z-2 p z 2 z

1 pL d z = c - t. zHz-2

,

1 z = 2 p -2 pL t 1-HC e 2p z = t . 1-k e -2 p

z-2 p = e2 pHc-tL = C e -2 p t , C > 0. z

) 2p zH0L = 1-k

, zH0L = p 2 = 1. 1-k k = -1, zH0L = p 2p z = . 1+ e -2 p t

,

e -e z = pH1 - 1 + 2-2 p L = pI1 + t M = pH1 + tanh p tL. t 1+ e e p t + e -ppt -p t

) To z2 HtL za HtL t = 0 . 3.1.

4o

4o (1,6 )

. 3.1 z2 HtL za HtL t = 0. Hx + y2 L y = 2 x - y, 2 x - y - Hx + 2 x2 - 2 x y + y2 L y = 0.

)

:

AHx, yL + BHx, yL y = 2 x - y - Hx + y2 L y = 0, y A = -1, x B = -1. u y = BHx, yL = -Hx + y2 L.

2 x - y - Hx + y2 L y = 0,

Hx + y2 L y = 2 x - y

ux = AHx, yL = 2 x - y,

u y = -Hx + y2 L f HyL = - y2 ,1 f = - y3 + c. 3

u = x2 - x y + f HyL u y = -x + f HyL

,1 u = x2 - x y - y3 + c 3

3 x2 - 3 x y - y3 = C. ) 2 x - y - Hx + 2 x2 - 2 x y + y2 L y = 0. AHx, yL + BHx, yL y = 0 AHx, yL = 2 x - y, , BHx, yL = -Hx + 2 x2 - 2 x y + y2 L. Bx Hx, yL = -H1 + 4 x - 2 yL,

. ,A -B

A y Hx, yL = -1,

-1+H1+4 y b := - x = - x-2 yL = -2. A 2 x-y

,

uHx, yL, ux = e-2 y H2 x - yL,

e-2 y @2 x - y - Hx + 2 x2 - 2 x y + y2 L y D = 0 u = e-2 y Hx2 - x yL + gHyL,

EHx, yL = FHyL = exp@ bHyL d yD = e-2 y .

u y = -e-2 y Hx + 2 x2 - 2 x y + y2 L.

,

,

e-2 y H-2 x2 + 2 x y - xL + g HyL = e-2 y Hx + 2 x2 - 2 x y + y2 L. g HyL = 0, gHyL = c.

u y = e-2 y @H-2L Hx2 - x yL - xD + g HyL.

, ,

5o

e-2 y Hx2 - x yL = C.

5o (2,0 ) y + 2 y + 5 y = 25 t. , y + 2 y + 5 y = 0. . , , y = el t . Hl2 + 2 l + 5L el t = 0.!!!!!!!!!!! !

,

-2 4-20 l = = -1 2 i. 2

, y + 2 y + 5 y = 0 f HtL = 25 t , y = a t + b. , y = a, y = 0. H y = a t + b y + 2 y + 5 y = 25 t 2 a + 5 a t + 5 b = 25 t. a = 5, b = -2. , y = 5 t - 2 y1 = e-t cos 2 t, y2 = e-t sin 2 t.

6o

y = e-t Hc1 cos 2 t + c2 sin 2 tL + 5 t - 2. uH1L = 3, u H1L = 6.

6o (2,4 ) t2 u - 2 t u + 2 u = 2 t3 ,

) . t2 u - 2 t u + 2 u = 0. Euler-Cauchy. ansatz u = ta , u1 = t, u2 = t 2 .

H Wronski WHu1 ,u2 L = tH2 tL - t2 1 = t2 . u = f HtL + gHtL u + hHtL u , uo = c1 t + c2 t2 ,

, t = 0, u = 2 t - 2 t-2 u + 2 t-1 u .

f HtL = 2 t, gHtL = -2 t-2 ,

hHtL = 2 t-1 .

( ) u HtL = aHtL u1 HtL + bHtL u2 HtL = aHtL t + bHtL t2u2 f t 2 aHtL := - d t = - t d t = -t2 , WHu ,u L t22

,

u HtL = aHtL t + bHtL t2 = -t2 t + 2 t t2 = t3 .

u1 f t2 t bHtL := d t = 2 d t = 2 t. WHu ,u L t1 2 1 2

u = c1 t + c2 t2 + t 3 . ) u = c1 + 2 c2 t + 3 t 2 . , uH1L = c1 + c2 + 1, u H1L = c1 + 2 c2 + 3, c1 + c2 = 2, c1 + 2 c2 = 3. c1 = c2 = 1, u = t + t2 + t 3 . uH1L = 3, u H1L = 6 ,

u = uo HtL + u HtL

,

: : . & . -

13-02-20009 1o ( , )

1o (0,8 ) ) () xH3L HtL = 6 - cos t, xH0L = 1, x H0L = 4, x H0L = 6.

) N , , () , y = x sin x. ) xH3L = 6 - cos t x H0L = 6, t t

x H0L = 4 t

, xHtL xH0L = 1 x = 1 + 0 H3 + 6 t + 3 t2 + cos tL d t = 1 + H3 t + 3 t2 + t3 + sin tL t0 x = 1 + 3 t + 3 t2 + t3 + sin t = sin t + Ht + 1L3 . y = sin x + x cos x, y = 2 cos x - x sin x.

x = 4 + 0 H6 + 6 t - sin tL d t = 4 + H6 t + 3 t2 + cos tL t0 = 3 + 6 t + 3 t2 + cos t.

x = 6 + 0 H6 - cos t L d t = 6 + H6 t - sin tL t0 = 6 + 6 t - sin t.

) y = x sin x

, y + b HxL y + c HxL y = 0.

y + b HxL y + c HxL y = 2 cos x - x sin x + b HxL Hsin x + x cos xL + c HxL x sin x = @2 + x bHxLD cos x - @x - b HxL - x c HxLD sin x

,

@2 + x bHxLD cos x - @x - b HxL - x c HxLD sin x = 0

, bHxL = -2 x-1 , cHxL = 1 - x-1 bHxL = 1 + 2 x-2 . , y + bHxL y + cHxL y = y - 2 x-1 y + H1 + 2 x-2 L y. y - 2 x-1 y + H1 + 2 x-2 L y = 0. x2 y - 2 x y + H2 + x2 L y = 0.

,

2o ( )

2o (1,2 ) xHtL t x - 4 x = 3 t, 3 t4 , 0 0, xH1L = -1. t x - 4 x = 0 x-1 x - 4 t-1 = 0 Hln xL = @lnHt4 LD x = exp@lnHt4 L + cD = C t4 .4 x - x = t

, t 0, 3, t1 3 t3 , t1

ln x = lnHt4 L + c

,

x - 4 t-1 x = f HtL, t 0, x = t4 HC + f HtL t-4 d tL.

, x - 4 t-1 x = 3, t < 1, x = t4 HC1 + 3 t-4 d tL = C1 t4 - t

x - 4 t-1 x = 3 t3 , t > 1,

x = t4 HC2 + t-1 d tL = t4 HC2 + 3 ln tL,

,

, t = 1 C1 - 1 = xH1- L = xH1+ L = C2 . C1 = C2 + 1 C + 1

xH1- L = C1 - 1, xH1+ L = C2

, HC + 1L t4 - t, x= 4 t HC + 3 ln tL, 0 0. 3 y = t 1C e 3

,3 y = 3 . 1+k e t

yH0L = 3 1+k , yH0L = 2 k = 1 2. , yH0L = 2 6 y = t . 2+ e 3

3 = 2. 1+k

y HtL

y2 HtL

y 3

2

1

t -1 -0.5 -1 0.5 1 1.5

6o (, Euler)

. 3.1 y2 HtL ya HtL t = 0.

6o (1,6 ) e-x y - e-2 x + Hy - e-x L y = 0, x2 + 2 y2 - 2 x y + y - x + Hx - 2 yL y = 0.

)

e-x y - e-2 x + Hy - e-x L y = 0 AHx, yL + BHx, yL y = 0

AHx, yL = e-x y - e-2 x , , A y Hx, yL = e-x , BHx, yL = y - e-x . Bx Hx, yL = e-x , u y = BHx, yL = y - e-x .

.

ux = AHx, yL = e-x y - e-2 x ,

,

1 u = -e-x y + e-2 x + f HyL. 2

u y = -e-x + f HyL

u y = y - e-x f HyL = y,1 f = y2 + c. 2

,

uHx, yL =. () e-x y - e-2 x + Hy - e-x L y = 0 , , : !!!! y = e-x C , C 0. y = k + e-x , k . e-2 x - 2 e-x y + y2 He-x - yL2 = C.

1 1 u = -e-x y + e-2 x + y2 + c. 2 2

:

. y . e-x Hy - e-x L + Hy - e-x L y = 0 Hy - e-x L He-x + y L = 0. e-x + y = 0,

y.

)

x2 + 2 y2 - 2 x y + y - x + Hx - 2 yL y = 0. AHx, yL + BHx, yL y = 0 AHx, yL = x2 + 2 y2 - 2 x y + y - x, BHx, yL = x - 2 y. Bx Hx, yL = 1.

,

. ,A -B

A y Hx, yL = 4 y - 2 x + 1,

,

H4 y-2 x+1L-1 y a := x = y = -2. B x-2

uHx, yL, ux = e-2 x Hx2 + 2 y2 - 2 x y + y - xL,

e-2 x @x2 + 2 y2 - 2 x y + y - x + Hx - 2 yL y D = 0. u = e-2 x Hx y - y2 L + f HxL,

EHx, yL = FHxL = exp@ aHxL d yD = e-2 x .

u y = e-2 x Hx - 2 yL.

,

,

e-2 x H-2 x y + 2 y2 + yL + f HxL = e-2 x Hx2 + 2 y2 - 2 x y + y - xL.1 f HxL = @- x2 e-2 x D 2

ux = e-2 x @H-2L Hx y - y2 L + yD + f HxL. f HxL = Hx2 - xL e-2 x .

,

, : 1 Hx y - y2 - x2 L e-2 x =. 2

1 f HxL = - x2 e-2 x + c. 2

,

,

x2 - 2 x y + 2 y2 Hx - yL2 + y2 = C e2 x , C > 0, 2 y2 - 2 x y + x2 - C e2 x = 0, C > 0.

y, : !!!!!!!!!!!!!!!!!!!!!! ! !!!!!!!!!!!!!!!!!!!!!! ! 1 1 y = Ix + 2 C e2 x - x2 M, y = Ix - 2 C e2 x - x2 M, C > 0. 2 2

: I

.

29-01-2010 (-) 1o (0,8 ) ) N , , () , y = t13 , t > 0. ) ( ) yH3L HtL = 27 e-3 t , yH0L = y H0L = -1, y H0L = 0 ) y = t13 1 y = t-23 , 3 2 y = - t-53 . 9

2 1 t2 y = - t13 , t y = t13 9 3

9 t2 y = -2 t13 , 3 t y = t13 9 a t2 y + 3 b t y + c y = H-2 a + b + c L t13 . , 8a, b, c< 2 a = b + c , , 9 a t2 y + 3 b t y + c y = 0. , 9 t2 y + 3 t y + y = 0 8a, b, c< = 81, 1, 1