EXERCISES - CALCULUS 3
Chapter 1
Series
1.1
Number series
Exercise 1.1. Test for convergence and find the sum (if exists):
∞
∞
P
1
P
9
2
a)
e)
−
10n 5n
n=1 n(n + 1)
n=1
1
1
1
+
+
+ ...
1.3 3.5 5.7
∞
P
c)
(sin n + 1 − sin n)
b)
f)
∞ (−1)n−1 .3n
P
10n+2
n=1
n=1
∞
P
1
d)
ln 1 +
n
n=1
g)
∞
P
n=1
arctan
n2
1
+n+1
Exercise 1.2. Test for convergence:
1. Divergence test
∞ (−1)n .n
P
a)
n=1 n + 1
∞ 2n + 3
P
b)
n=1 6n − 1
2. Comparison tests
√
∞
P
n2 + n + 1
√
a)
2 n+2
n=1 n
√
√
∞
P
n+2− n
b)
2n + 1
n=1
∞
P
1
c)
ln 1 +
n
n=1
1
c)
cos
n2
n=1
n
∞
P
n+1
d)
n+2
n=1
d)
∞
P
∞
P
√
n
e−1
n
n=1
e)
∞
P
arctan(2−n )
n=2
f)
∞ ln n
P
2
n=2 n
1
1
e)
(−1) cos
n
n=1
∞
P
n
∞
P
2
n=1 ln(2n + 1)
∞
P
1
1
√ − sin √
h)
n
n
n=1
g)
i)
∞
P
4 + cos n
2
−n )
n=1 n (1 + e
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3. Ratio test
a)
∞ 2019n
P
n!
n=1
b)
∞ 1 (2n + 1)!
P
n n2 − 1
n=2 3
(n!)2
c)
n=1 (2n + 1)!
∞ n!
P
d)
n2
n=1 3
∞
P
e)
∞
P
nn
n
n=1 4 .n!
f)
∞ en n!
P
n
n=2 n
4. Root test
n2
3n + 1
a)
3n + 2
n=1
n2
∞ 1
P
1
b)
1−
n
n
n=2 4
∞
P
n2 −1
n
c)
n+2
n=2
n2 +1
∞ 1
P
n−2
d)
n
n
n=1 3
∞
P
n3
1
e)
cos
n
n=2
∞
P
5. Integral test
∞ ln n
P
2
n=2 n
∞ ln n
P
b)
n=2 n
c)
a)
∞
P
1
n=4 n ln n ln(ln n)
√
∞ e−
P
√
d)
n=1
∞
P
e)
1
n=2 ln(n!)
g)
(−1)n
n
n=1 3 + cos n
h)
∞ (−1)n + cos n
P
n ln2 n
n=2
i)
∞ (−1)n
P
1
sin √
n
n=1 2n + 1
n
n
6. Series with sign-changing terms
∞
P
cos n
√
n3 + 1
n=1
√
∞ (−1)n n
P
b)
n+1
n=1
a)
c)
n2
3 − 2n
d)
2n + 5
n=1
∞ n cos(nπ)
P
e)
2
n=1 2n + 1
∞ (−1)n−1 .n3
P
f)
4
n=1 (n2 + 1) 3
∞
P
∞ (−1)n .n3
P
n
n=1 2 − 1
∞
P
Exercise 1.3. Test for absolute and conditional convergence:
∞ (−1)n n
P
2
n=2 n + 1
√
∞ (−1)n n
P
b)
n=1 n + 100
∞ (−1)n−1
P
c)
np
n=1
n
∞
P
n 2n + 100
d)
(−1)
3n + 1
n=1
a)
e)
∞
P
n=2
√
(−1)n
n + (−1)n
Exercise 1.4. Test for convergence
∞
P
n+1
a)
2
n=1 (n + 2) ln(n + 3)
2
∞ 2 − n2 .3−n
P
b)
n2
n=1
n5
c)
n
n
n=1 3 + 2
∞
P
1
1
cos
d)
− cos
n+1
n
n=1
∞
P
2
e)
∞
P
(−1)n
√
n
−1
e
n=2
f)
∞ (−1)n (n − 1)
P
n2 + 1
n=2
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1.2
Function series
Exercise 1.5. Determine the domain of convergence of the following function series:
√
∞
∞
P
P
∞
1
xn
P
n3 + 1
a)
d)
g)
−x
2n + 1
2
x
n=1 1 + n
n=1 x
n=1 (x + 1)n
∞ nx + (−1)n
P
e)
n
n=1
n
∞
P
1
f)
x+
n
n=1
∞ (−1)n
P
b)
nx
n=1
c)
∞
P
1
n
n=1 x + 1
h)
∞
P
n.e−nx
n=1
i)
∞ (n!)2
P
xn
n=1 (2n)!
Exercise 1.6. Determine the domain of convergence of the following power series:
∞ x2n
P
a)
n=1 n
b)
c)
∞ (x + 2)n
P
√
n n
n=1
∞
P
n
(x − 2)n
n=1 2n + 1
d)
enx
2
n=1 n + n + 1
e)
n2
(2x + 1)n
3
1
+
n
n=1
f)
xn
n
n
n=1 2 + 3
∞
P
∞ (−1)n .n
P
√
(1 − 3x)n
n3 + 1
n=1
n
∞
P
1 − 2n
h)
x2n+1
2n + 3
n=1
g)
∞
P
∞
P
Exercise 1.7. Test for uniform convergence on the given set of the following series:
a)
∞
P
sin nx
, on R
2
2
n=1 2x + n
e)
b)
∞ e−nx + 1
P
, on [0, ∞)
n2
n=1
f)
∞
P
x
, on [0, ∞)
4 2
n=1 1 + n x
∞
P
x
, on (0, 1]
n=1 (1 + (n − 1)x)(1 + nx)
∞
P
xn
g)
(1 − x)xn , on [0, 1]
,
x
∈
R
n
2
n=1
n=1 (4x + 9)
n
∞ 1
∞
P
P
2x + 1
(−1)n−1
,
x
∈
[−1;
1]
d)
, on R.
h)
n
2
x+2
n=1 2
n=1 x + n + 2
∞ sin nx
P
Exercise 1.8.
1. Let F (x) =
. Prove that
n3
n=1
c)
∞
P
(a) F (x) is continuous ∀x
Z
2. Prove that
0
π
(b) lim F (x) = 0
(c) F 0 (x) =
x→0
cos 2x cos 4x cos 6x
+
+
+ . . . dx = 0.
1.3
3.5
5.7
Exercise 1.9. Find the sum of the following function series or number series:
a)
∞
P
nxn , x ∈ (−1; 1)
d)
xn+1
, x ∈ (−1; 1)
n=1 n(n + 1)
e)
∞ x4n−3
P
, x ∈ (−1; 1)
n=1 4n − 3
f)
∞ (−1)n .π 2n+1
P
n=1 (2n + 1)!
n=1
b)
c)
xn
, x ∈ (−1, 1)
n=1 n + 1
∞
P
∞
P
(n2 + n)xn+1 , x ∈ (−1, 1)
n=1
3
∞
P
∞ cos nx
P
n2
n=1
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(−1)n
g)
n
n=1 (n + 1).2
i)
∞
P
(−1)n+1
h)
n
n=1 (2n − 1)3
j)
∞
P
∞ 3n + 1
P
8n
n=1
∞
P
1
n=1 (2n)!!
Exercise 1.10. Find the Maclaurin series of the following functions:
a) y = sin2 x cos2 x
b) y = sin x sin 3x
c) y = e2x + 3x cos x
d) y =
x2
2x + 1
− 3x + 2
e) y =
2x − 1
x2 + 2x − 3
1
f) y = 2
x +x+1
g) y = √
1
4 − x2
h) y = ln(1 + 2x)
i) y = x ln(x + 2)
j) y = ln(1 + x − 2x2 )
k) y = arcsin x
Exercise 1.11. Find the Taylor series of y at the given point:
a) y =
1
, x0 = 4
2x + 3
b) y = sin
πx
, x0 = 1
3
c) y =
√
x, x0 = 4
Exercise 1.12. Graph each of the following periodic functions and find compute corresponding
Fourier series
(
a) y = x, x ∈ (−π, π), T = 2π
2x, 0 ≤ x < 3,
d) y =
,T =6
0,
−3 < x < 0
b) y = |x| , x ∈ (−π, π), T = 2π
e) y = 2x, 0 < x < 10, T = 10
(
(
4,
0 < x < 2,
2 − x, 0 < x < 4,
c) y =
,T =4
f) y =
,T =8
−4, 2 < x < 4
x − 6, 4 < x < 8
In each part, find the points of discontinuity of the function. To what value does the series
converge at those points?
Exercise 1.13. Expand the function into a Fourier series
a) f (x) = x, x ∈ [0, π], f (x) is an odd and periodic function of T = 2π.
b) f (x) = 2 − x, x ∈ (0, 2), f (x) is an even and periodic of T = 4.
c) f (x) = x + 1, x ∈ [0, π).
d) f (x) = x − 1, x ∈ (0, π) into a Fourier sine series.
e) f (x) = x(π − x), x ∈ [0, π] into a Fourier cosine series. Then prove that
∞
X
1
π2
=
.
2
n
6
n=1
4
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Chapter 2
Ordinary differential equations
2.1
First order ODEs
Exercise 2.1.
1) Separable equations
a) 2y(x2 + 4)dy = (y 2 + 1)dx
e) y 0 = x2 y, y(1) = 1
b) y 0 + ey+x = 0
f) xdx + ye−x dy = 0, y(0) = 1.
√
g) y 2 1 − x2 dy = arcsin xdx, y(0) = 0
2x
h) y 0 =
, y(0) = −2.
y + x2 y
c) 1 + x + xy 0 y = 0
d) y 0 = cos2 x cos2 (2y)
2) Homogeneous equations:
y x
+ +1
x y
y
b) xy 0 = x sin + y
x
y 2
c) 2y 0 +
= −1
x
d) (x + 2y)dx − xdy = 0
y
e) xy 0 = y + e x , y(1) = 0
y
f) xy 0 = y + 2x3 sin2 , y(1) = π2
x
2
y
y
g) y 0 = 2 − + 1, y(1) = 2
x
x
h) (2x − y + 4)dx + (x + 2y − 3)dy = 0.
a) y 0 =
3) Linear equations:
a) xy 0 − 4y = 4x8
d) y 0 + y sin x = sin x, y(0) = 0
3
e) y 0 − y = 2x2 , y(1) = 2
x
f) (2xy + 3)dy − y 2 dx = 0.
b) (x2 + 1)y 0 + 2xy = ex
c) xy 0 − y = x2 cos x, y(π) = π
4) Bernoulli equations:
y3
2
a) y + y = 2
x
x
xe−2x
c) y + xy =
y
0
0
2
0
d) xy 0 =
3 2
b) xy + y = −x y , y(1) = 1
5
x3
− 2y, y(1) = 2.
y2
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5) Exact equations:
a) (x2 + y)dx = (2y − x)dy
b) ey dx = (2y − xey )dy
c) (3x2 y 2 + 2y + 1)dx + 2(x + x3 y)dy = 0
d)
y
dx + (ey + 1 + ln x) dy = 0
x
e) (ex sin y+y 2 )dx+(ex cos y+2xy)dy = 0
Exercise 2.2. In each of the following parts, find an integrating factor and solve the given
equation
y
y
− 1 dx +
+ 1 dy = 0
1.
x
x
2. (3x2 y + 2xy + y 3 ) dx + (x2 + y 2 ) dy = 0
3. ydx + (2xy − e−2y ) dy = 0
Exercise 2.3. Solve the ODE y 0 = y 2 −
2
z
.
by
change
of
function
y
=
x2
x
Exercise 2.4. Solve the ODE xy 0 − (2x + 1)y + y 2 + x2 = 0 by change of function y = z + x.
Exercise 2.5. Solve the following ODEs:
a) y 0 = (x + y)2
e) (x2 y 2 − x)dy = ydx
b) y 0 = 1 + x + y + xy
f) 3xy 2 y 0 − y 3 = x, y(1) = 3
c) (2xy 2 − 3y 3 )dx = (3xy 2 − y)dy
g) (8xy 2 − y)dx + xdy = 0, y(1) = 1
d) xy 0 = y + x3 sin x, y(π) = 0
h) x = (y 0 )2 − y 0 + 2
2.2
Second order ODEs
Exercise 2.6. Solve the following ODEs
a) xy 00 + 2y 0 = 12x2
(
(1 − x2 )y 00 − xy 0 = 2,
b)
y(0) = 0, y 0 (0) = 0
c) 2yy 00 = (y 0 )2 + 1
(
(1 + x)y 00 + x(y 0 )2 = y 0 ,
d)
y(0) = 1, y 0 (0) = 2
Exercise 2.7. Solve the following equations
1. (x − 1)2 y 00 + 4(x − 1)y 0 + 2y = 0, given a particular solution y1 =
2. xy 00 + 2y 0 + xy = 0, given a particular solution y1 =
1
.
1−x
sin x
.
x
2y
2xy 0
+ 2
= 0 given a particular solution y1 = x.
2
x +1 x +1
1
cos x
2 00
0
2
4. x y + xy + x −
y = 0, x > 0, given a particular solution y1 = √ .
4
x
3. y 00 −
6
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Exercise 2.8. Solve the ODEs with constant coefficients:
a) y 00 − 4y 0 + 3y = (15x + 37)e−2x
f) y 00 + y = 2 cos x cos 2x
b) y 00 − y = 4(x + 1)ex
g) y 00 + 2y 0 + 2y = 8 cos x − sin x
c) y 00 − 2y 0 + y = (12x + 4)ex
h) y 00 + y 0 − 2y = x + sin 2x
d) y 00 − y 0 − 2y = xex cos x
i) y 00 + 3y 0 − 4y = 3 sin2 x
e) y 00 + 2y 0 + 10y = e−x cos 3x
j) y 00 + 4y = e3x + x sin 2x
Exercise 2.9. Solve the ODEs using the method of variation of parameters:
a) y 00 − 2y 0 + y =
ex
x
b) y 00 − 3y 0 + 2y =
1
1 + e−x
Exercise 2.10. Solve the ODE (2x − x2 )y 00 + 2(x − 1)y 0 − 2y = −2, given two particular
solutions y1 = 1, y2 = x.
Exercise 2.11. Solve the following Euler equations
1. x2 y 00 − 3xy 0 + 4y = x3 , y(1) = 1, y 0 (1) = 2
2. y 00 −
2.3
y
2
y0
+
=
, x > −1.
x + 1 (x + 1)2
x+1
Systems of first order ODEs
Exercise 2.12. Solve the following systems of ODEs


y
dx


 dy = 5y + 4z

=
dt
x−y
a) dx
c) dy
dz
x



 =
= 4y + 5z
dx
dt
x−y




 dy = y + 5z
 dx = y
dt
b) dx
d) dy
dz
1




= −y − 3z
= −x +
dx
dt
cos t
2.4
Power series method
Exercise 2.13. Solve the following series by the power series method
a) (x2 + 1)y 00 − 4xy 0 + 6y = 0
b) y 00 + xy 0 + y = 0, y(0) = 0, y 0 (0) = 1.
7
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Chapter 3
Laplace transform
3.1
Laplace and inverse Laplace transforms
Exercise 3.1. Using the definition, find the Laplace transforms of the following functions:
b) f (t) = e2t+3
a) f (t) = t
c) f (t) = sin(2t).
Exercise 3.2. Find the Laplace transforms of the following functions:
√
√
π
a) f (t) = t + 3t − 2t2 t
c) f (t) = (et + e−2t )2
e) f (t) = 2 sin t +
3
b) f (t) = (t + 2)2 − 2e3t
d) f (t) = 2 sin 3t. cos 5t
f) f (t) = e−2t − 3u(t − 2)
Exercise 3.3. Find the inverse Laplace transforms of the following functions:
a) F (s) =
2
3
4
−
+
5
s4 s 2
s
c) F (s) =
5 − 3s
s2 + 9
e) F (s) =
e−2s + 5
s
b) F (s) =
3
10
+
s−4 s+2
d) F (s) =
10s − 3
s2 + 25
f) F (s) =
e−πs 2s + 3
− 2
s
s +4
3.2
Transformation of initial value problems
Exercise 3.4. Solve the following IVPs:
(
x(3) − x00 − x0 + x = e2t
a)
x(0) = x0 (0) = x00 (0) = 0
(
x(3) − 6x00 + 11x0 − 6x = 0
b)
x(0) = x0 (0) = 0, x00 (0) = 2
(
x(4) − 16x = 240 cos t
c)
x(0) = x0 (0) = x00 (0) = x(3) = 0
(
x(4) + 8x00 + 16x = 0
d)
x(0) = x0 (0) = x00 (0) = 0, x(3) (0) = 1
Exercise 3.5. Solve the following IVPs

0

y = 2y + z
a) z 0 = y + 2z


y(0) = 1, z(0) = 3

0
x

z + 2y = e
b) y 0 − 2z = 1 + x


y(0) = 1, z(0) = 2(HW − submit)
8
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3.3
Translation. Rational functions
Exercise 3.6. Find the Laplace transforms of the following functions:
b) f (t) = e−2t sin 3t
a) f (t) = t4 eπt
π
c) f (t) = e sin t +
3
t
Exercise 3.7. Find the inverse Laplace transforms of the following functions:
a) F (s) =
2+s
2
s − 3s + 2
e) F (s) =
2s − 1
s2 − 4
i) F (s) =
b) F (s) =
3s − 2
s(s2 + 4)
f) F (s) =
5 − 2s
s2 + 7s + 10
j) F (s) =
s2 (s2
g) F (s) =
1
3
s − 5s2
k) F (s) =
3s + 5
s2 − 6s + 25
1
s(s + 1)(s + 2)
h) F (s) =
1
4
s − 16
l) F (s) =
s2 + 3
(s2 + 2s + 2)2
c) F (s) =
d) F (s) =
3.4
2s + 1
+ 1)
s2 − 2s
s4 + 5s2 + 4
s2
3s + 1
+ 4s + 4
Derivatives, integrals and products of Laplace transforms
Exercise 3.8. Find the Laplace transforms of the following functions:
a) f (t) = t cos2 t
c) f (t) = te2t sin 3t
e) f (t) =
e2t − 1
t
b) f (t) = (t − e2t )2
d) f (t) = (2t − sin 3t)2
f) f (t) =
1 − cos 2t
t
Exercise 3.9. Find the inverse Laplace transforms of the following functions:
a) F (s) = arctan
1
s
b) F (s) = ln
Exercise 3.10. Solve the following IVPs:
(
tx00 + (t − 2)x0 + x = 0
a)
x(0) = 0
(
tx00 − (4t + 1)x0 + 2(2t + 1)x = 0
b)
x(0) = 0
s−2
s+2
c) F (s) = ln
s2 + 1
(s + 2)(s − 3)
(
tx00 + (4t − 2)x0 + (13t − 4)x = 0
c)
x(0) = 0
(
ty 00 − ty 0 + y = 2
d)
y(0) = 2, y 0 (0) = −4
Exercise 3.11. Solve the following IVPs:
(
x00 − 3x0 + 2x = u(t − 2)
a)
x(0) = 0, x0 (0) = 1
(
x00 + 4x = sin t − u(t − 2π) sin(t − 2π)
b)
x(0) = 0, x0 (0) = 0
(
y 00 + 2y 0 + 2y = e−(t−1) u(t − 1),
c)
y(0) = y 0 (0) = 0.
9
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(
(
x00 + 4x0 + 4x = f (t)
t, 0 ≤ t < 2
d)
where
f
(t)
=
x(0) = x0 (0) = 0
0, t ≥ 2
(
(
t
, 0≤t<6
x00 + x = f (t)
2
e)
where
f
(t)
=
0
3, t ≥ 6
x(0) = 0, x (0) = 1
(
(
x00 + x = f (t)
cos t, 0 ≤ t < 2π
f)
where f (t) =
0
x(0) = x (0) = 0
0,
t ≥ 2π.
10