!"#$%&' (

!"#$%! &' ()#*$+', -./'0'1.*.,

!"#$% &'$()*+,% *-$-#',/$% *+ *-0*-1,'", * $!#',% "2-",% &$/*+ %*'

3$'+!),/$% *+ #*'+$% /* *4!,56*% /"3*'*-","% -, 3$'+,

y = f(x, y)

$-/* f 7 !+, 3!-58$ '*,) /,/,9 /* /!,% :,'";:*"% x * y9 * y 7 !+, 3!-58$ "-

!" #$%&'& '( )*+(, -"

+&.&/ 0(+1 &3'456& 43441+ '1'1/ $ .1,13%4'1 1 (748%9341 '( *:1 ;341 8&+*56& 01,1

& 0,&

!" #$%&'& '( )*+(, -.

/01*,2 !"3 4(5,(6(7%289& 1,:;2 '& =$%&'& '( )*+(,!

n xn yn f(xn, yn) h.f(xn, yn)0 0 1 5 2, 5

>&%( ?*( f(xn, yn) = 10+4.1 = 5! @1&,2A B2=&6 5,((7C(, 2 6(1*7'2 +07C2A 52,2

n = 1! D(1*7'2 2 (?*289& !"A xn+1 = xn+hA (7%9& x2 = x1+h = 0+0, 5 = 0, 5! E

B2+&, '( y2A 6(1*7'& 2 (?*289& !.A $ '2'& 5&, yn+1 = yn + h.f(xn, yn) = 1 + 2, 5 =

3, 5! )7%9&

n xn yn f(xn, yn) h.f(xn, yn)0 0 1 5 2, 51 0, 5 3, 5 14, 5 7, 3

F,((7C(7'& %&'2 2 %2G(+2A %(=&63

H&1&A & B2+&, '( y 52,2 2 ?*07%2 0%(,289& $ 01*2+ 2 96, 5!

I&=& 5&'( 6(, B06%&A & =$%&'& '( )*+(, ,(?*(, *= 7J=(,& =*0%& 1,27'( '(

0%(,28K(6 52,2 6( 2+2782, *= ,(6*+%2'& 62%06L2%M,0&! D9& 5,&5&6%26 =(+C&,026 52,2

& =$%&'&A &=& 6(,: B06%& 2 6(1*0,!

!#$%& '#(()%&!*( + ,-.!.& /!01*#&

!" #$%&'& '( )*+,(-.*%%/ 01

n xn yn f(xn, yn) h.f(xn, yn)0 0 1 5 2, 51 0, 5 3, 5 14, 5 7, 32 1 10, 7 43 21, 53 1, 5 32, 3 128, 5 64, 3

4 2 96, 5 385 192, 5

!" #$%&'& '( )*+,(-.*%%/

2 3$%&'& '( )*+,(-.*%%/ $4 56&7/7(83(+%(4 *3 '&9 3$%&'&9 3/:9 5&5*8/6(9 (

56(:9&9 5/6/ / && '( 9&8*=?(9 /56&@:3/'/9 '( (A*/=?(9 ':B(6(+:/:9! C3& !1F

yn+1 = yn +h

6(k1 + 2k2 + 2k3 + k4) G !1H

&+'(

k1 = f(xn, yn)

k2 = f(xn +h

2, yn +

h

2k1) G !0H

k3 = f(xn +h

2, yn +

h

2k2)

k4 = f(xn + h, yn + hk3)

!"#$%& '()(* /"0102,($# # !"#$# $% &'()%*+'"",3 ,1'1% ' , ,5.#60 ,78# $%

y(1, 5) 5,., , 9#1'78# $% y = 2xy3 y(1) = 13 # h = 0, 1:

;#1'78#

I/3&9 :+::/6 &3 &9 E8*8&9 '( k14 k24 k3 ( k4 / 5/6%:6 '/ (A*/=>& !JF

K/6/ n = 04

!#$%& '#(()%&!*( + ,-.!.& /!01*#&

!" #$%&'& '( )*+,(-.*%%/ 00

k1 = f(xn, yn)

= f(x0, y0)

= 2.x0.y0

= 2.1.1

= 2

k2 = f(xn +h

2, yn +

h

2k1)

= f(x0 +h

2, y0 +

h

2k1)

= f(1 +0, 1

2, 1 +

0, 1

2.2)

= f((1, 05), (1, 1))

= 2.(1, 05).(1, 1)

= 2, 31

k3 = f(xn +h

2, yn +

h

2k2)

= f(x0 +h

2, y0 +

h

2k2)

= f(1 +0, 1

2, 1 +

0, 1

2(2, 31))

= f((1, 05), (1, 1155))

= 2.(1, 05).(1, 1155)

= 2, 3425

k4 = f(xn + h, yn + hk3)

= f(x0 + h, y0 + k3)

= f((1 + 0, 1), (1 + 0, 1.(2, 3425)))

= f((1, 1), (1, 2342))

= 2.(1, 1).(1, 2342)

= 2, 7154

1,&2/3 4*54%6%*6+'& +/ (7*/89& !:

!#$%& '#(()%&!*( + ,-.!.& /!01*#&

!" #$%&'& '( )*+,(-.*%%/ 01

yn+1 = yn +h

6(k1 + 2k2 + 2k3 + k4)

y1 = y0 +h

6(k1 + 2k2 + 2k3 + k4)

= 1 +0, 1

6(2 + 2.(2, 31) + 2.(2, 3425) + 2, 7154)

= 1, 2337

)(/234/+'& & 5(65& 78&(66& /%$ x = 1, 5: &* 6(;/ y(1, 5) &

!"!#$%'() *'+,'-.#/"'() !

{y = y1/3

y(0) = 0

x [0; 0, 3]; h = 0, 1

!

{xy = x y

y(2) = 2

"#$&$' y(2, 1) () h = 0, 1*

"!

{y = 2y

x+1+ (x+ 1)3

y(0) = 3

x [0; 0, 2]; h = 0, 1

01'(%- *'22!%-1#2 3 4/,1,- 5167#'-