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Московский государственный индустриальный университет

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Kurs_vysshei_matematiki_UP_Berkov_N.A._2007-2

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Φ0(Ψ1(x, y, z), Ψ2(x, y, z)) = 0.

∂z

−y

∂z

= 0

x = 0 z = y2

∂y

∂x

−y

dz− = 0.

x2 + y2 = C

z = C2.

x2 + y2 = C

z = C2,

x = 0,

z = y

y z

C1 = C2

z = x2 + y2

a ∂x∂z + b ∂y∂z = c x = 0 z = f (y)

bx−ay = = C1 az − cx = C2

x = 0 z = f (y)

x y z C1 C2

af (−C1	/a) = C2			−						C1 C2
		a		−
af		ay − bx	= az		cx				b
af			= az		cx
						c
					z =		x + f	y −		x ,
					z =	a	x + f	y −	a	x ,

x = ϕ1(t), y = ϕ2(t), z = ϕ3(t),

0(C1, C2) = 0

z ∂z

x ∂x

−

z ∂z

= 1

x = t y = t z = (2t2 − 1)1/2

∂y

xdx

−ydy

= dz

xdx =

−

ydy

−

ydy

= dz .

x2 + y2 = C1

y2 + z2 = C2.

y z

t2 + t2 = C1

t2 + (2t2 − 1) = C2.

t2 =

C1,

t2 =

(C2

+ 1).

(C2

+ 1).

12 (x2 + y2) = 13 (y2 + z2 + 1) 32 x2 + 12 y2 − z2 = 1.

P (t, x, u) ∂u∂t + Q(t, x, u) ∂u∂x = R(t, x, u).

t x uu

t = 0

P = a > 0 Q = b > 0 R = 0

a∂u∂t + b ∂u∂x = 0.

V = ab

u = f x − ab t .

u(0, x) = ϕ0(x), −∞ < x < ∞.

f x − ab t f (x)

b	t	b
		a
a

t	D	t	D
		t0
	x		x
x0	x1	0	x1

u(0, x) = ϕ0(x), 0 x ∞;

u(t, 0) = f0(t), t > 0.

t > 0

uu x (t > 0, x > 0)

x t

x − (b/a)t = C = const uu uu x t

[x0, x1] [0, x1] [0, t0] D

	[x0, x1]
[0, x1] [0, t0]	ϕ(x) = 0 ψ(t) = 0	D
uu	x − (b/a)t = x1
x − (b/a)t = x0	x − (b/a)t = x1
x − (b/a)t = x1	x − (b/a)t = −(b/a)t0
x − (b/a)t = x1

a ∂u∂t + b ∂u∂x = c,

	c			b
u =		t + f	x −		t ,
	a			a

x −(b/a)t = const u − (c/a)t uu

		4x3	+ 2x2 + x + 1						x + 1	x−3
	lim				; b)	lim
a)	lim	8x3			; b)	lim		x + 3		.
a)	x→∞	8x3		+ 4x − 2		x→∞		x + 3		.
					n
					∞	n + 1	.
				=1		n + 2n
				=1
a)y = 2x3 ln(sin x);				b)y2 + 2x3y − xy2 + x = 0; c)z = x2y sin y.

x(t) = = sin2 t + 3t − 1 t = π/12

						y = x4 + 4x3 + 4.
			a¯ = (1, −1, 2)			¯
						b = (0, −1, −1).
B(0, −2, 2) C(−1, 3, 1)					A(1, 2, −1)

a)	√	x3		b)	cos4 xdx; c) 0		e
			dx;				x ln xdx.
		2x4 + 3

a)xydx + (x + 1)dy = 0; b)y − 3y = cos x.

a) lim

x2 − 2x − 15

;

b) lim

x sin x

2x2 − 7x − 15

x→5

x→0

tg2 5x

∞

n + 1

=1 n3 + 3

x + x ln y + xy = 0;

c)z =

b)y sin

a)y = tg (2x)e ;

x + x(t) =

√

=16 − t2 t = 1

			y = x +			4	.
			y = x +				.
			¯			x
			¯	−2) c¯ = (1, 3, −1).
		a¯ = (1, −2, 1), b = (1, 0,		−2) c¯ = (1, 3, −1).
	¯
		c¯ · (¯a + b).
B(0, −2, 2)			A(1, 2, −1)
B(0, −2, 2)
	cos x							1
	cos x
a)		dx; b)	x arcsin xdx;	c) 0				x2xdx.
a)	sin3 x	dx; b)	x arcsin xdx;	c) 0				x2xdx.
	a)y + y cos x = cos x; b)y − 2y				= e2x.

√

− x ; b)

lim

+ 1

lim

−

3x − 2

x→∞

∞

(−1)n .

√3

tg3x

b)y tg x + xy2 − sin(xy) = 0;

c)z = sin x2 + y.

a)y =

;

x(t) =

1	t = 1
= t2+1

y = x3 + x2.

ABCD A(2, −3, 1), B(1, 4, 0), C(−4, 1, 1) D(−5, −5, 3)

M (1, 2, −3)

a¯ = (2, 3, 1) b =

= (3, 1, 2)

√

e− tg x

dx;

cos3 xdx; c)

2 − x2

dx.

cos2 x

a) xy = y ln

;

b) y − 6y + 9y = 9x2 − 12x + 2.

lim

x2 − 9

;

lim 9x arcctg x.

x→3 x2 − 5x + 6

x→∞

∞ ( 1)nn

−

√3

+ 1

log3 x

a) y =

;

y sin x + xy4

− 2xy = 0;

c) z = xy3 x2 + y.

√ x(t) = = 3 4 − t2 t = 1

y = x4 − x3.

C(3, 2, 1)

ABC A(2, −1, 2) B(1, 2, −1)

A(2, −1, 2) B(1, 2, −1) C(3, 2, 1)

√

+ 4

x + 1

dx; b)

sin2 x cos3 xdx; c) −1

dx.

x(x + 2)2

x − 3

a) xy = y +

;

b) y − 2y = e2x + 5.

y2 − x2

P (x, y, z) Q(x, y, z) R(x, y, z)

D		t x
u		t x
		tn, xm
t		x

			t						t
t			n						n=N
			n

n
			3						n=n
2			2						1
2			1					x
1											x
1		x
n=0		x	n=0				m		n=0
-3 -2 -1 m=0 1	2	m	m=0	1	2	3	m		m=0	1	m=m m=M

x	h	t	τ
n				2, . . .
				2, . . .
t = n · τ xm = m · h n = 0, 1, 2, . . . m = 0, ±1, ±n
u(t, x)			(n, m)	u(t , xm) =
= u(nτ, mh) = umn
				D
n	m			n = 0
n = N m = 0 m = M
n	m	h → 0 τ → 0
τ = τ (h)		h → 0 τ → 0
τ = rh		r = const
				h
Dh
			u	u(t, x)
Dh			u