Методическое пособие по высшей математике

:

2-

.

6 .

1, 2, 3;

4,5,6.

1.,

.

2.,

,

, .

3..

.

. ,

.

.

4.( , )

.

.

. (

)

, .

, .

:

1.. “ ” – .: (

).

2.. “ ” – .:

), .1, 2.

3.. “ ” – .: (

).

4.. “ ” – .:

( ).

5.. “

” – .: ( ).

.

4, 5, 6.

4.

:

1..

2..

3..

:

1.. .

2..

3..

4..

5..

6..

7..

8..

9..

10..

.

1.

z = 3x 4 − 7 y 5 + 2x3 y 4 + 9 .

:

.

, , .

z′ “ y ” .

x

,

:

z′ = 12x3 + 6x 2 y 4 .

x

z′y . “ x ” , :

z′ = −35 y 4 + 8x3 y 3 .

y

, :

z′′2

= 36x 2 + 12xy 4 ;

z′′

= 24x 2 y3 ;

z′′2

= −140 y 3 + 24x3 y 2 .

x

xy

y

2. z = 2x 2

− xy + 3y 2

− 2x − 11y + 1 .

:

z = f (x, y ( ,

) :

1) z = f (x, y

z′

= 0; z′ = 0 .

x

y

, ,

. M 0 (x0 ,

y0 .

z = f (x, y

2)

M 0 (x0 ,

y0 .

A = z′′2

(x

0

, y

0

; ) B = z′′ x

0

, y( ;

C

= )z′′2

x

0

,

y

0

.

x

xy

0

y

3) D = AC - B 2 .

D > 0 , z = f (x, y M 0 (x0 , y0 A < 0

A > 0 .

D < 0 , M 0 (x0 , y0 .

D = 0 ,

.

.

:

z¢x = 4x - y - 2; z¢y = -x + 6 y -11.

. :

ì4x - y - 2 = 0,

í

î- x + 6 y -11 = 0

x = 1; y = 2 . , M (1, 2 - .

M (1, 2 .

A = z¢¢2	(1; 2 )= 4; B = z¢¢			1; 2 =( -1; ) C = z¢¢2 1; 2 = 6 .
x			xy			y
	:
D = AC - B 2 = 23 .									M (1, 2 .
	D > 0			A > 0 , ,
zmin = z(1, 2 = -11.
		3.				U (x, y
du = (4 + xy 2 dx +) x 2 y + 2(y dy .
	P(x,			y dx)	+ Q x, y(	:
						dy du
						¢	(x, y =)	¢	x, (y .
U (x, y , Py								Qx
,	du = P(x, y dx)				+ Q x, y(	dy - U (x, y .
:
U x¢ (x, y =) P x,(y
“ y ” :
¢
òU x (x, y dx) = òP x, y( dx +j) y
U (x, y	=)	òP x, (y dx +)j y ,							(1)
j(y		- “ y ”.
j(y
( P(x, y dx) +j y (¢y =) Q(x, y )
ò
(òP(x, y dx) ¢y +j¢y =)Q(x, y )
j¢y = Q(x, y -) (òP x, (y dx ¢y)					)
j(y (1).
:
P(x, y =) 4 + xy 2 ; Q x, y = x 2 y + 2 y .
Py¢ = 2xy;		Q¢x = 2xy. Py¢ = Q¢x . ,

du = (4 + xy 2 dx +) x 2 y + 2(y dy - U (x, y . :

U x¢ = 4 + xy 2 ; U (x, y = ò(4 + xy 2 dx + j(y ;

U (x, y =) 4x + 1 x 2 y 2 + j(y ). 2

æ

1

2

2

ö

¢

2

U ¢y = ç4x +

x

y

+j(y )÷

= x

y +j¢y ; U ¢y

= Q(x, y

)

2

è

øy

,

j(y = y 2 + c , c - .

x 2 y + j¢y

= x 2 y + 2 y; j¢y

= 2 y;

,

U (x, y =) 4x +

1

x 2 y 2 + y 2 + c .

2

4.

ò(2xy + y dx + (3x 2

-1 dy ,

L y = 3x 2 + x ,

L

M1 (1; 4 M 2 (2; 4 .

:

. ,

.

:

y = 3x 2 + x - .

“ x ”

dy = (6x +1 dx ;																		M1 (1, 4 M 2 (2, 4 “ x ”
x1						= 1 x2						= 2 .
,
ò(2xy + y dx)+ (3x 2 -1 dy =) ò2 [2x 3x 2 + x +( 3x 2 + x - 3)x 2 -( 1 (6x +1 ]dx)) = (																					)
L					1
2	3		2		æ	4	8			3		5		2	ö		2		1
ò(24x		+ 8x		- 5x -1 dx =ç6x			+)		x		-		x		- x ÷		1	= 90		.
												2							6
1					è		3								ø
									3			2 x−1
5. òòdxdy = òdx													òdy .
					D				1			( x−1 2)+1									D ; 2)
:			1)												XOY

; 3) D

.

:

1) “ x ” 1 3 – ,

“ D ” x = 1 x = 3 .

“ y ” ,

“ D ” y = (x -1 2 +1 y = 2x -1 .

:

ìy = (x -1 2 +1

í

î y = 2x -1

, D .

A(1; 1 B(3; 5 . [1; 3 , D . y

B

1 A

1 3

x

2) .

“ y ” : 1 ( ) 5

).

“ x ”.

y = 2x -1 x =

1

(y +1 )- .

2

+1 - (x > 0 .

y = (x -1 2 +1 x =

y -1

,

y−1+1

2

2 x

1

5

−

òòdxdy = òdx

òdy = òdy

òdx .

D

1

(x−1 2)+1

1

1

( y+1 )

2

3) D :

3

2 x−1

3

2x -1

3

[2x -1 - (x -1 2)-1 dx =

3

[- x 2 + 4]x - 3 dx =

]

S =

ò

dx

dy =

ò

dxy

=

ò

ò

ò2

(x -

1 2)+

1

æ

1

x

3

( x−1

)+1

1

ö

3

4

1

1

2

ç

+ 2x

÷

=

.

= ç-

3

- 3x ÷

1

3

è

ø

D :

	5				y−1+1				5					y -1 +1 5							é								1				1 ù					5	æ					1		1	ö
S = òdy					òdx = òdyx							1						= òê				y	-1 +1 -							y -					údy = òç							y -1 -			y +		÷dy =
	1			1	( y+1)				1					(y +1 )					1		ë								2				2 û					1	è					2		2	ø
												2
	2		2														5
æ									1					1	ö			æ	2				25				5	ö		æ				1			1	ö		4
			(y			3	)-				2																															.
= ç					-1					y		+			y ÷		=	ç			* 8	-				+		÷	- ç		0	-				+		÷	=
	3								4										3																					3
è														2 ø			1	è					4 2					ø è						4 2				ø

5.

:

.

2..

3..

4..

I. .

,

: 1) :

f1 (x )f 2 y(

dx) +j1 x j(2

y) dy = 0 y¢ = f1 (x )f2 y( .

) f1 (x )f 2 y( dx) +j1 x j(2 y) dy = 0 .

: .

f1 (x )f 2 y( dx) + j1 x j(2 y) dy = 0

: f2 (y )j1 x( Þ

f1 (x )

dx +

j2 (y

dy = 0

Þ ò

f1 (

x )

dx + ò

j2 (y

dy = c ,

c - –

j1 (x )

f 2 (y )

j1 (x )

f 2 (y )

:

dy

) y¢ = f1 (x )f 2 y( ; )Þ

= f1 (x )f 2 y(

*) dx Þ dy = f1 x f 2 (y ()dx) : f 2 y ( Þ)

dy

= f1 (x )dx Þ

dy

dx

f1 (x )dx + c - .

Þ

=

f2 (y )

ò f 2 (y )

ò

2) .

) :

y¢ + P(x )y = q x (

(1)

) :

y¢ + P(x )y = q x (yα		(2)
a ¹ 1;	P(x ), q x( - .	y = u *n , u = u(x ),n =n x ( -

. ( ) ,

, y = u *n .

y = u *n	; y				¢	¢	¢	(3)
						= u n + un
(3) (1),
¢	¢		+ unP(x )= q x(					(4)
u n + un
un ¢ +n (u¢ + uP(x ) = q) x
u(x ) ,
ìu¢ + uP(x ) = 0,
í
î un ¢ = q(x )
.
(4) :
¢		¢		+nP(x ) = q) x
u n + u(n

n (x ) ,

ìn ¢ +nP(x = 0, íî u¢n = q(x )

.

3) :

y¢ =

f (x, y ,

f (x, y

f (tx, ty

=) f x, (y

, t ¹ 0

y = xu

.

1. :

y¢ - ytgx = 2x

1

;

cos x

:

.

y = uν , u, ν

“ x ”; y

¢

¢

¢

. :

= u n + un

¢

¢

- untgx

=

2x

;

u n + un

cos x

(1)

un ¢ +n (u¢ - utgx

=)

2x

cos x

“ u ” , u¢ - utgx = 0 ; .

du

- utgx = 0;

du - utgxdx = 0;

du

-

sin x

dx = 0;

u

dx

cos x

du

=

sin x

dx;

ò

du

= ò

sin xdx

+ c;

u

u cos x

cos x

“ u ” , c = 0 ; ,

ò

du

=

ò

sin x

dx

Þ ln u = -ln cos x; ln u = ln

1

;

u =

1

;

cos x

cos x

u

cos x

1

u =

(1),

1

2x

cos x

n

¢

=

;

n

¢

=

¢

cos x

cos x

2x; òn dx = ò2xdx + c ;

n = x 2

+ c ; ,

y = u *n ;

y =

(x 2 + c

- .

cos x

.

	¢	¢¢	=)0;			¢	¢¢	= 0
4) F (x, y , y					F y, y ,( y
			¢	¢¢		= 0 .
	) F (x, y , y
	y¢ = P(x ); y¢¢ = P¢.
F (x, P, P¢ = 0 .
			¢		¢¢	= 0 .
	) F (y, y , y
					y¢ = P(x );			y¢¢ = P	dp	.

dy

	æ	dp	ö
	Fç y, P, P		÷	= 0 .
	ç	dy	÷
	è		ø

2. (x 2	+1 y′′ = 2xy′ . ,
: y(0 )= 1;	y¢ 0 = 3.

:

“ y ”.

y¢ = P ;

y′′ =

dp

(x 2 + 1

dp

= 2xP .

dx

dx

P

x . .

dp

=

2xdx

;

dp

=

2x

dx + ln c ;

ln P = ln(x 2 + 1 + ln c .

P x 2 + 1 ò

P ò x 2 + 1

1

1

P = c1 (x 2 +1 y¢ = c1 (x 2 +1 .

, c1 = 3 . , y¢ = 3(x 2 +1 .

.

y¢ = 3(x 2 + 1 - .

dy = 3(x 2 + 1 ; dy) = 3 x 2 + 1 dx(; òdy = ò)3 x 2 + 1 dx + c2 ; y = 3(ò x2 + 1 dx + c2 ; y = x3 + 3x + c2 dx

c2 :

c2 = 1; , y = x3 + 3x +1 - ,

.

5) .

: y¢¢ + Py¢ + qy = 0 ,

p, q - .

: k 2 + pk + q = 0

) k1 ¹ k2 - ,

y = c1e k1x + c2e k2 x (c1 , c2 = const - .

) k1 = k2 - ,

y = ek1x (c1 x + c2

.

) k1 = a + b i; k2 = a - b i -

,

y = eαx (c1 cos bx + c2 sin bx)

.

3. y¢¢ - 2 y¢ - 3y = 0 .

:

: k 2 - 2k - 3 = 0

k1 = -1; k2 = 3 - .

, y = c1e − x + c2 e3x

.

6) .

:

y¢¢ + Py¢ + qy = f (x ,	(1)
P, q - , f (x	-

.

(1) :

y = y . + y .

y - (1), y . -

			y¢¢ + Py¢ + qy = 0 ,			y . -		–
(1).
	) f (x )= P x(eαx , P (x			= a	n	x n + a	n−1	x n−1 + ... + a	0	- n .
		n	n
,	y .	= Qn (x eαx x r ,	Qn (x -	“ n ”

, , r -

k 2 + pk + q = 0 , “α ”,

r= 0; 1; 2;...

) f (x )= eαx a0 cos βx + b0 sin βx). , y . = eαx (a cos βx + b sin βx x r ,

a, b - . r -

k 2 + pk + q = 0 , a + b i . a, b -

.

4. y¢¢ - 2 y¢ + 5 y = 6x + 7 .

:

:

y = y . + y .

y . - . y¢¢ - 2 y¢ + 5 y = 0 .

. k 2 - 2k + 5 = 0

k1 = 1 + 2i ; k2 = 1 - 2i - .

,

y . = e x (c1 cos 2x + c2 sin 2x) .

y . - . .

f (x )= 6x + 7 = eox 6x +(7 .
a = 0; a ¹ k1 ; a = k2 . , y . = ax + b .						a, b
.
y	.	= ax + b; y′	= a;	y′′	= 0 .	(1)
		.		.
(1) ,
− 2a + 5ax + 5b = 6x + 7 .						(2)

(2),

:

5a = 6; a = 6 ; − 2a + 5b = 7; b = 47 .

	5			25
,
y .	=	6	x +	47	.
	5			25

y = e x (c cos 2x + c	2	sin 2x) +	6	x +	47	.
1		5		25

II. .

( ,

, , , ,

).

. ,

.

.

∞

n

5. å

.

n=1

2n + 1

n

:

an =

;

n

1

2n +1

lim an

= lim

=

¹ 0 .

,

,

n→∞

n→∞ 2n +1

2

.

∞ n

6. å .

n=1 3n

:

.

an

=

n

. ,

an+1

=

n +1

.

n

n 1

:

3

3

+

lim

an+1

= lim

(n + 1 * 3n

=

1

lim

n + 1

=

1

< 1.

n→∞ an

n→∞ 3n+1 * n

3 n→∞

n

3

, .

∞

1

7. å

.

(n + 1 ln)

2

n +(

1

)

n=1

:

.

1

“ n ”

“ x ”.

(n + 1 ln) 2

n +(

1

)

f (x )=

1

. :

(x + 1 ln)

2 x +(

1

)

+∞

dx

t

dx

1

t

æ

1

1

ö

1

ò

= lim

ò

= - lim

= - lim ç

-

÷

=

.

2

2

(x +1 ln)

x

+( 1

t →+∞

(x +1 ln)

x +(

1 )

t →+∞

ln(x +1

1)

t →+∞ç

)

ln 2

÷

ln 2

1

)

1

è ln(t +1

ø

.

,

, .