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EXERCISES - CALCULUS 3
Chapter 1
Series
1.1 Number series
Exercise 1.1. Test for convergence and find the sum (if exists):
a)
b)
n=
(sin (n + 1) − sin n)
c)
n=
ln
n
d)
n=
n(n + 1)(n + 2)
e)
n=
ln
n^2
f)
n=
10 n^
5 n
g)
n=
(−1)n−^1. 3 n 10 n+
h)
n=
arctan
n^2 + n + 1
Exercise 1.2. Test for convergence:
- Divergence test
a)
n=
(−1)n.n n + 1
b)
n=
2 n + 3 6 n − 1
c)
n=
cos
n^2
d)
n=
n + 1 n + 2
)n
e)
n=
(−1)n^ cos
n
f)
n=
2 n − 1 2 n + 3
)n− 1
- Comparison tests
a)
n=
n^2 + n + 1 n^2
n + 2
b)
n=
2 n +
n + 2 √ n^4 + 2
c)
n=
n + 2 −
n 2 n + 1
d)
n=
ln
n
e)
n=
√ ne − 1
n
f)
n=
arctan(2−n)
g)
n=
ln n n^2
h)
n=
ln(n + 1)
i)
n=
√^1
n − sin
√^1
n
- Ratio test
a)
n=
2019 n n!
b)
n=
3 n
(2n + 1)! n^2 − 1
c)
n=
(n!)^2 (2n + 1)!
d)
n=
n! 3 n^2
e)
n=
nn 4 n.n!
f)
n=
enn! nn
- Root test
a)
n=
2 n^2 + 1 3 n^2 + 2
)n
b)
n=
2 n + 1 5 n + 2
) 2 n
c)
n=
4 n
n
)n 2
d)
n=
n + 1 n + 2
)n (^2) − 1
e)
n=
3 n
n − 2 n
)n (^2) +
f)
n=
cos
n
)n 3
- Integral test
a)
n=
n(ln n + 1)
b)
n=
n ln^3 n
c)
n=
n ln n ln(ln n)
d)
n=
e−
√n √ n
e)
n=
ln(n!)
- Series with sign-changing terms
a)
n=
cos n √ n^3 + 1
b)
n=
sin(2n^2 ) n^2 + 2
c)
n=
(−1)n−^1 .n (n^2 + 1)
d)
n=
(−1)n
n n + 1
e)
n=
(−1)n^ ln n n
f)
n=
n cos(nπ) 2 n^2 + 1
g)
n=
(−1)n.n^3 2 n^ − 1
h)
n=
3 − 2 n 2 n + 5
)n 2
i)
n=
(−1)n 2 n + 1 sin
√^1
n
Exercise 1.3. Test for absolute and conditional convergence:
a)
n=
(−1)nn n^2 + 1
b)
n=
(−1)n
n n + 100
c)
n=
(−1)n−^1 np
d)
n=
(−1)n
2 n + 100 3 n + 1
)n
e)
n=
(−1)n √ n + (−1)n
f)
n=
e
(− √1)n n (^) − 1
Exercise 1.4. Test for convergence
a)
n=
n + 1 (n^2 + 2) ln(n + 3) b)^
n=
n^5 3 n^ + 2n^
c)
n=
4 + cos n n^2 (1 + e−n)
g)
n=
(−1)n (n + 1). 2 n
h)
n=
(−1)n+ (2n − 1)3n
i)
n=
3 n + 1 8 n
j)
n=
(2n)!!
Exercise 1.10. Find the Maclaurin series of the following functions:
a) y = sin^2 x cos^2 x b) y = sin x sin 3x
c) y = e^2 x^ + 3x cos x
d) y = 2 x + 1 x^2 − 3 x + 2
e) y = 2 x − 1 x^2 + 2x − 3
f) y =
x^2 + x + 1
g) y =
4 − x^2
h) y = ln(1 + 2x)
i) y = x ln(x + 2)
j) y = ln(1 + x − 2 x^2 )
k) y = arcsin x
Exercise 1.11. Find the Taylor series of y at the given point:
a) y =
2 x + 3 , x 0 = 4 b)^ y^ = sin^
πx 3 , x 0 = 1 c)^ y^ =^
x, x 0 = 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
b) y = |x| , x ∈ (−π, π), T = 2π
c) y =
4 , 0 < x < 2 , − 4 , 2 < x < 4
, T = 4
d) y =
2 x, 0 ≤ x < 3 , 0 , − 3 < x < 0
, T = 6
e) y = 2x, 0 < x < 10, T = 10
f) y =
2 − x, 0 < x < 4 , x − 6 , 4 < x < 8
, T = 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 ∑^ ∞
n=
n^2
π^2 6
Chapter 2
Ordinary differential equations
2.1 First order ODEs
Exercise 2.1. 1) Separable equations
a) 2y(x^2 + 4)dy = (y^2 + 1)dx b) y′^ + ey+x^ = 0 c) 1 + x + xy′y = 0 d) y′^ = cos^2 x cos^2 (2y)
e) y′^ = x^2 y, y(1) = 1 f) xdx + ye−xdy = 0, y(0) = 1. g) y^2
1 − x^2 dy = arcsin xdx, y(0) = 0
h) y′^ = 2 x y + x^2 y , y(0) = −2.
- Homogeneous equations:
a) y′^ =
y x
x y
b) xy′^ = x sin y x
c) 2y′^ +
( (^) y x
d) (x + 2y)dx − xdy = 0
e) xy′^ = y + e yx , y(1) = 0 f) xy′^ = y + 2x^3 sin^2 y x , y(1) = π 2
g) y′^ = y^2 x^2
y x
- 1, y(1) = 2 h) (2x − y + 4)dx + (x + 2y − 3)dy = 0.
- Linear equations:
a) xy′^ − 4 y = 4x^8
b) (x^2 + 1)y′^ + 2xy = ex c) xy′^ − y = x^2 cos x, y(π) = π
d) y′^ + y sin x = sin x, y(0) = 0
e) y′^ −
x y = 2x^2 , y(1) = 2 f) (2xy + 3)dy − y^2 dx = 0.
- Bernoulli equations:
a) y′^ +
x
y = y^3 x^2
b) xy′^ + y = −x^3 y^2 , y(1) = 1
c) y′^ + xy = xe−^2 x^2 y
d) xy′^ = x^3 y^2
− 2 y, y(1) = 2.
Exercise 2.8. Solve the ODEs with constant coefficients:
a) y′′^ − 4 y′^ + 3y = (15x + 37)e−^2 x b) y′′^ − y = 4(x + 1)ex c) y′′^ − 2 y′^ + y = (12x + 4)ex
d) y′′^ − y′^ − 2 y = xex^ cos x e) y′′^ + 2y′^ + 10y = e−x^ cos 3x
f) y′′^ + y = 2 cos x cos 2x g) y′′^ + 2y′^ + 2y = 8 cos x − sin x h) y′′^ + y′^ − 2 y = x + sin 2x
i) y′′^ + 3y′^ − 4 y = 3 sin^2 x j) y′′^ + 4y = e^3 x^ + x sin 2x
Exercise 2.9. Solve the ODEs using the method of variation of parameters:
a) y′′^ − 2 y′^ + y = ex x
b) y′′^ − 4 y′^ + 4y = e^2 x x^2 + 1
c) y′′^ + 4y =
cos 2x
d) y′′^ − 3 y′^ + 2y =
1 + e−x
Exercise 2.10. Solve the ODE (2x − x^2 )y′′^ + 2(x − 1)y′^ − 2 y = −2, given two particular solutions y 1 = 1, y 2 = x.
Exercise 2.11. Solve the following Euler equations
a) x^2 y′′^ − 3 xy′^ + 4y = x^3 , y(1) = 1, y′(1) = 2 b) x^2 y′′^ − 2 xy′^ + 2y = 2x ln x
c) x^2 y′′^ − 3 xy′^ + 5y = 8 sin(ln x)
d) y′′^ − y′ x + 1
y (x + 1)^2
x + 1 , x > −1.
2.3 Systems of first order ODEs
Exercise 2.12. Solve the following systems of ODEs (unknown functions of x)
a)
y′^ = 2y + z z′^ = 3y + 4z
b)
y′^ = 2y − z z′^ = 2z − 9 y
c)
y′^ = 2y − 3 z z′^ = 3y + 2z
d)
y′^ = 2y − 4 z + 3ex z′^ = 2y − 2 z
e)
y′^ = 2y + z z′^ = 5y − 2 z + 2e^3 t
f)
y′^ = 2y − 4 z + 3ex z′^ = 2y − 2 z
g)
y′^ =
y 2 y − 3 z z′^ = z 2 y − 3 z
h)
y′^ = z z′^ = − 4 y +
cos^2 2 x
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:
a) f (t) = t b) f (t) = e^2 t+3^ c) f (t) = sin(2t).
Exercise 3.2. Find the Laplace transforms of the following functions:
a) f (t) =
t + 3t − 2 t^2
t
b) f (t) = (t + 2)^2 − 2 e^3 t
c) f (t) = (et^ + e−^2 t)^2
d) f (t) = 2 sin 3t. cos 5t
e) f (t) = 2 sin
t + π 3
f) f (t) = e−^2 t^ − 3 u(t − 2)
Exercise 3.3. Find the inverse Laplace transforms of the following functions:
a) F (s) =
s^4
s^52
s
b) F (s) =
s − 4
s + 2
c) F (s) = 5 − 3 s s^2 + 9
d) F (s) = 10 s − 3 s^2 + 25
e) F (s) = e−^2 s^ + 5 s
f) F (s) = e−πs s
2 s + 3 s^2 + 4
3.2 Solving initial value problems with Laplace trans-
form
Exercise 3.4. Solve the following IVPs:
a)
x(3)^ − 6 x′′^ + 11x′^ − 6 x = 0 x(0) = x′(0) = 0, x′′(0) = 2
b)
x(3)^ + x′′^ − 6 x′^ = 0 x(0) = x′(0) = 0, x′′(0) = 3
c)
x(3)^ − x′′^ − x′^ + x = e^2 t x(0) = x′(0) = x′′(0) = 0
d)
x′′′^ − 2 x′′^ + 16x = 0 x(0) = x′(0) = 0, x′′(0) = 20
e)
x(4)^ − 16 x = 240 cos t x(0) = x′(0) = x′′(0) = x(3)^ = 0
f)
x(4)^ + 8x′′^ + 16x = 0 x(0) = x′(0) = x′′(0) = 0, x(3)(0) = 1
Exercise 3.5. Solve the following IVPs
a)
tx′′^ + (t − 2)x′^ + x = 0 x(0) = 0
b)
tx′′^ − (4t + 1)x′^ + 2(2t + 1)x = 0 x(0) = 0
c)
tx′′^ − (3t + 8)x′^ + 3x = 0 x(0) = 0
d)
tx′′^ + (5t − 12)x′^ + 5x = 0 x(0) = 0
e)
tx′′^ + (4t − 2)x′^ + (5t − 4)x = 0 x(0) = 0
f)
ty′′^ − 2 ty′^ + 2y = 1 − e^2 t y(0) = 0
g)
ty′′^ − ty′^ + y = 2 y(0) = 2, y′(0) = − 4
Exercise 3.11. Solve the following IVPs:
a)
x′′^ − 3 x′^ + 2x = u(t − 2) x(0) = 0, x′(0) = 1
b)
x′′^ + 4x = sin t − u(t − 2 π) sin(t − 2 π) x(0) = 0, x′(0) = 0
c)
y′′^ + 2y′^ + 2y = e−(t−1)u(t − 1), y(0) = y′(0) = 0.
d)
x′′^ + 4x′^ + 4x = f (t) x(0) = x′(0) = 0
where f (t) =
t, 0 ≤ t < 2 0 , t ≥ 2
e)
x′′^ + x = f (t) x(0) = 0, x′(0) = 1
where f (t) =
t 2 ,^0 ≤^ t <^6 3 , t ≥ 6
f)
x′′^ + x = f (t) x(0) = x′(0) = 0
where f (t) =
cos t, 0 ≤ t < π 2 0 , t ≥ π 2.
g)
x′′^ + x = t[1 − u(t − 2)] x(0) = x′(0) = 0
h)
x′′^ + 4x = f (t) x(0) = x′(0) = 0
where f (t) =
cos t, 0 ≤ t < π 0 , t ≥ π.
i)
x′′^ + 2x′^ + 5x = f (t) x(0) = x′(0) = 0
where f (t) =
20 cos t, 0 ≤ t < 2 π 0 , t ≥ 2 π.
j)
x′′^ − 2 x′^ + 10x = f (t) x(0) = x′(0) = 0
where f (t) =
255 sin t, 0 ≤ t < 2 π 0 , t ≥ 2 π.
