Exercises EE,sem.II
1. Find solutions of the differential equations with separable variables:
a) x(y 2 − 4)dx + ydy = 0
b) y 0 cos x =
y
ln y
c) ln(cos y)dx + x tg ydy = 0
d)
yy 0
x
+ ey = 0 ; y(1) = 0
e) y 0 tg x − y = −2
2. Find solutions of the homogenous differential equations:
a) y 0 =
y
x
ln xy
b) y 0 =
y2
xy−x2
c) y 0 =
y2
x2
−2
2. Find the solutions:
a) y 0 = 2x + y + 3
b) y 0 = (2y + 6x + 1)2
c) y 0 = cos2 (x − y)
1
Exercises EE,sem.II
1. Find the solutions of linear equations:
a) y 0 + y = e−x
b) y 0 − 2yx = x − x3
c) y 0 − 3y = e3x
d) y 0 + y = cos x
2. Find the solutions of the total differential equations:
a) 2 − ( xy )2 + 2( xy )y 0 = 0
b) (x + sin y)dx + (x cos y + sin y)dy = 0
c) yex + (y + ex )dy = 0
d) (ex+y + 3x2 )dx + (ex+y + 4y 3 )dy = 0
3. Find the integrating factor and next solve the total differential equations:
a) (y + ln x)dx − xdy = 0
b) ydx − (x + y 2 )dy = 0
2
Exercises EE,sem.II
1. Solve the Bernoulli differential equations:
a) y 0 +
2y
x
b) y 0 −
y
x−1
4
= 3x2 y 3
=
y2
x−1
c) 4xy 0 + 3y = −ex x4 y 5
2. Solve the homogeneous linear equations:
a) y 00 − y 0 − 2y = 0
b) y 00 + 5y 0 + 6y = 0 with the condition y 0 (0) = −6, y(0) = 1
c) y 00 − 4y 0 + 3y = e5x
d) y 00 + 4y = 1 + sin 2x with the condition y 0 (0) = 0, y(0) =
e) y 00 + y =
√ 1
cos 2x
3
1
4
Exercises EE,sem.II
1.Solve the system of linear equations
a)
(
z 0 = −7z + y
y 0 = −2z + 5y
general solution and special s.t.: z(0) = y(0) = 1
b)
c)
(

0

 x = −x + y + z


d)
e)
y 0 = 2y + 4z + ex
z 0 = 4y + 2z + ex
(
(
y0 = x − y + z
z0 = x + y + z
x0 + y 0 = 2(x + y)
y 0 = 3x + y
x0 + 2x
=1
t
0
y =x+y+
2x
t
−1
3. Find the orthogonal trajectories of the family of curves
a) The family of parabolas y = ax2
b) The family of circles x2 + y 2 = 2ax
c) The family of parabolas y 2 = 4(x − a)
4
Exercises EE,sem.II-5
1. Find the double integrals:
a)
b)
c)
d)
e)
R2
0
R4
3
R2
1
dy
dx
dx
R3
−3
R 2π
0
R1
0
(x2 + 2y)dx
R2
dy
1 (x+y)2
R x x2 dy
1
x
dy
y2
R5
dϕ
y 2 −4 (x
Ra
a sin ϕ
+ 2y)dx
rdr
2. Reverse the order of integration:
a)
b)
c)
R4
0
R1
0
R1
0
dx
dx
dy
R 12x
f (x, y)dy;
3x2
R 3x
2x
f (x, y)dy;
R 1−y
−
√
1−y 2
f (x, y)dx;
3. Find the integrals:
a)
b)
RR
S
RR
S
xdσ, where S is the triangle with vertices (0, 0), (1, 1), (0, 1);
√
dxdy
a2 −x2 −y 2
;S
1
4
of the circle of radious a and center at (0, 0) in the I quarter of plane;
4. Find the area bounded by the curves:
y 2 = ax, y 2 = bx, xy = α, xy = β (gdzie 0 < a < b , 0 < α < β).
Hint: use new variables y 2 = ux, xy = u
5. Find the volume of the solid figure bounded by the surfaces:
a) z = 0, y = 1, z = x2 + y 2 , y = x2
(answ:
88
)
105
b) 3x − y = 2, x + y = 6,
y = x, z = x, z = 2x + y
(answ:
28
)
3
Additional problem: Describe region D as {x1 ¬ x ¬ x2 , y1 (x) ¬ y(x) ¬ y2 (x)} and as
{y1 ¬ y ¬ y2 , x1 (y) ¬ x(y) ¬ x2 (y)}. Next using this two descriptions find
ZZ
D
Here D is an area bounded by the lines y =
2ydxdy.
√
x, y = 0, x + y = 2.
5
(answ: 56 )
Exercises EE,sem.II-6
1. Find the volume of solid bounded by the surfaces:
a) z = 2x2 + y 2 + 1, x + y = 1 and planes x = 0,y = 0,z = 0.
b) z = x2 − y 2 , y = 0, z = 0, x = 1
2. Find the volume of the part space bounded by the sphere x2 + y 2 + z 2 = a2 and the cylinder
x2 + y 2 = ay (a > 0).
3. Find the mass of the:
a) solid figure bounded by the sphere x2 + y 2 + z 2 = 4 and of a density %(x, y, z) = z 2
answ. 324π
5
b) solid regular cube of edge length 2 when a density in a given point is equal with its
distance from the base of the cube.
answ.8
a) the solid hemisphere x2 + y 2 + z 2 ¬ R2 with the density %(x, y, z) = r =
answ. 2R
5
b) the solid hemisphere x2 + y 2 + z 2 = R2 with the density %(x, y, z) = 1
6
√
x2 + y 2 + z 2
answ.z0 =
R
2
Exercises EE,sem.II-7
1. Find the oriented integrals:
a)
b)
c)
d)
R
K
xydx + (y − x)dy, where K is an arc of the curve y = x3 from (0, 0) to (1, 1)
R
ydx + xdy, where K is an arc of the circle with the center at (0, 0) and radius R od
(0, R) do (R, 0)
K
R
K
xdx + ydy + (x + y + 1)dz, where K is the interval from (1, 1, 1) to (2, 3, 4)
R
yzdx + xzdy + xydz, where K is an arc of the spiral line x = R cos t, y = R sin t, z =
from z = 0 to z = a.
K
at
2π
2. Find the integrals in the potential vector fields:
a)
b)
R (2,3)
(−1,2)
R (2,1)
(0,0)
ydx + xdy,
2xydx + x2 dy,
3. Using Greena theorem find the integral
is positively oriented ellipse:
x2
a2
+
y2
b2
=1
H
K (xy
H
+ x + y)dx + ydy + (xy + x − y)dz, where K
4. Verify Green theorem for K (x + y)2 dx − (x − y)2 dy, K = K1 ∪ K2 where K1 is a interval
of strait line from (0, 0) to (1, 1) and K2 is an arc of the parabola y = x2 from (1, 1) to (0, 0).
7
Exercises EE,sem.II-8
1. Using Stokes theorem find:
a)
R
L
x2 y 3 dx + dy + zdz, where L is the edge of hemisphere x2 + y 2 + z 2 = R2 , z ­ 0
b) circulation of the field [xy, 0, xz] over the curve being the edge of the one-eight of sphere
x2 + y 2 + z 2 = R2 , z ­ 0, y ­ 0, x ­.
2. Calculate directly and using Gauss theorem the integrals:
a)
b)
RR
y 2 zdxdy + xzdydz + xydzdx, where S is positively oriented surface of the pyramid:
x = 0, y = 0, z = 0, x + y + z = 1;
S
RR
yzdxdy + xzdydz + x2 ydzdx, where S is a surface located in first octant, positively
oriented and composed with the parts of z = x2 + y 2 , cylinder 1 = x2 + y 2 and planes
x = 0, y = 0, z = 0
S
8
Exercises EE,sem.II-9
Here im(z) and re(z) denote the imaginary and real part of z.
1. Find the lines on the complex plane C
a) z = t + jt, t ∈ R
b) z = 2(1 + ejt ), t ∈ (−π, π)
c) z = 2t2 + jt , t ∈ [0, ∞)
d) z = t(j + e−jt ), t ∈ (0, ∞)
2. Find the limits of sequences:
a) zn =
2n2
n3 +2
b) zn =
n(1+ejn )
(n+1)2
2
+ j n3n
2 +1
3.Find im(f ) and re(f )
• f (z) = z 3 + z 2 + 1
• f (z) = sin z
• f (z) = sin |z|
• f (z) = ln z
• f (z) = eaz gdzie a ∈ C
• f (z) = 2im(z) + 3re(z)
• f (z) = j
9
Exercises EE,sem.II-9
1.Check where the function is holomorphic:
a) f (z) = |z|
b) f (z) = z|z|
c) f (z) = im(z)
d) f (z) = j cos(z)
2.Check where the function is holomorphic and find the derivative :
a) f (x + jy) = x2 − y 2 + 2jxy
b) f (x + jy) = e−x cos y − je−x sin y
3.Find the holomorphic function f (x + jy) = u(x, y) + jv(x, y) that
a) u(x, y) = 6x2 y − 2y 3
b) v(x, y) = cos y cosh x
10
21-25 maja 2007
Exercises EE,sem.II-10
1. Find Laurent expansion of f (z) = zz+1
2 +1 in the rings
√
a) P (1, 0, 2),
b) P (−2j, 3, ∞),
c) P (−2j, 1, 3)
2. Find the singular points and determine its types:
a)
zez
,
(z 2 +4)3
b)
z
,
(z 2 −1)(z 2 −4)
c)
z2
,
z(z 3 +1)
d)
z
,
sin z
e)
sin z
,
z2
f) e (1−z)2
1
3. Find residua in given points:
a) res
z+1
z=0 (1−z)z 3 ,
c) res
z=0
e) res
g) res
1−cos z
,
z3
b)res
z=0
(z+2)6
,
z6
d) res
z+1
z=±j (z 2 +1) ,
ejπz
z±1 z 2 +1 ,
f) res
z=2 e z−2 ,
ez
z=1 (z−1)n
dla n = 0, 1, 2 . . .
1
Exercises EE,sem.II-11
1. Find the integrals
H
a)
b)
H
H
c)
H
d)
ez dz
K(0,2) z 2 (z 2 +1) ,
zdz
K(0,3) (z+2)(z−1)2
K(0,1)
K(0,2)
e) d)
f) d)
H
H
sin z2 dz,
z 4 cos z1 dz
K(0,1)
1
z 5 e z dz
z 3 dz
K(0,2) z 4 −1
2. Find the real integrals:
a)
b)
c)
R∞
x−2
−∞ x4 +1 dx
R ∞ 2x+1
−∞ (x2 +1)3 dx
R∞
dx
−∞ (x2 +1)n
n = 1, 2, . . .
12
1. Find Laplace transform L[f (t)] where:
1
s−ln a
a) f (t) = at
s(s2 +7)
(s2 +1)(s2 +9)
b) f (t) = cos3 t
b
s2 −b2
c) sinh(bt)
s2 +b2
(s2 −b2 )2
d) t cosh(bt)
s(s2 +2s+3
(s−1)(s2 −2s+5)
e) et cos2 t
2. Using 1(x) express a function from the graphs :
1.0
1.0
0.8
0.5
0.6
0.4
-1
0.2
-1
1
2
3
1
2
3
4
5
- 0.5
1
2
3
4
5
- 1.0
1.0
0.4
0.8
0.2
0.6
-1
- 0.2
1
2
3
4
5
0.4
0.2
- 0.4
-1
4
Answ: a) 1(x) − 1(x − 2), b) 1(x) − 21(x − 1) + 1(x − 2), c)
5
P
k
k (−1) 1(3x
− k) +
1(x)
,
2
d) −(x−4)1(x−5)+(x−4)1(x−4)−(x−2)1(x−3)+(x−2)1(x−2)−x1(x−1)+x1(x)
3. Find Laplace transform of the periodic function
a) f (t) = t mod 1
(
b) f (t) =
1 t mod 1 < 1/2
0 t mod 1 ­ 1/2
13
Exercises EE,sem.II-12
1. Find inverse Laplace transform of the functions:
a) f˜(s) = 2 s
s −2s+5
Answ: et (cos 2t + 12 sin 2t)
b) f˜(s) =
s+1
s(s−1)(s−2)(s−3)
c) f˜(s) =
s
(s−1)3 (s+2)2
Answ:
d) f˜(s) =
1
(s−1)(s2 −4)
Answ: − 31 et + 14 e2t +
e) f˜(s) =
s+3
s(s2 −4s+3)
f) f˜(s) =
s
(s4 −1)
Answ: − 61 + et − 32 e2t + 23 e3t
3t2 +2t−2 t
e
54
+
2t+1 −2t
e
27
1 −2t
e
12
Answ:1 − 2et + e3t
(use convolution formula)
Answ: 12 (cosh t − cos t)
2. Using Laplace transform find solutions of the differential equations:

00

 y − 9y = 0,
y(0) = 0,
y 0 (0) = 0.
a) 

Answ: y = 0

00
0

 y − 4y + 4y = sin 2t
c)
Answ: y (t) = − 43 te2t + 78 e2t + 18 cos 2t
y 0 (0) = 1,
y(0) = 1.
b) 


00
0

 y − 4y + 4y = 4,
Answ: y = 1 + e2t
y(0) = 2,
y 0 (0) = 2.


 0
x = x + 2y,



 y 0 = 2x + y + 1
d)

x(0) = 0,



Answ: y =
1
3
+ 2e−t + 83 e3t
y(0) = 5.
3. Solve the integral equations:
a) y =
b)
Rt
0
Rt
0
Answ: y = et
ydt + 1
y(τ ) sin(t − τ )dτ − 1 + cos t = 0
(

1 2k < t < 2k + 1

0


 y =
0 2k + 1 < t < 2k
4. 
y(0) = 0,


 0
y (0) = 0.
Answ: y = 1(t)
k∈Z
14