Department of Mathematics and Statistics
MATH 375
Handout # 1 - ANSWERS, HINTS, SOLUTIONS
Dierential equations in general; linear equations
1. For each of the following ordinary dierential equations determine the order, the dependent
variable (or the unkn

MATH 375
Review Handout - ANSWERS, HINTS, SOLUTIONS
Integration, Complex numbers, Eigenvalues and eigenvectors
Part I.
(4x3 + 6x2 ) dx Z d(x4 + 2x3 + 8)
4
3
+C
x
+
2x
+
8
=
=
ln
1)
4
3
4
3
x + 2x + 8
x + 2x + 8
Z
Z
2
2u3/2
2
2)
+ C = u u + C = cos t u

MATH 375 MIDTERM - Answers, Hints, Solutions
Monday November 2, 2015, 18:00-19:30
1. The equation y + ky (y 2 1) + 3y = 2 cos(t)y 2 is
A) second order, linear B) second order, nonlinear
C) third order, linear D) third order, nonlinear
E) nonlinear, and th

THE UNIVERSITY OF CALGARY
DEPARTMENT OF I\IATHEI\IATICS AND STATISTICS
FINAL EXAMINATION
AMAT 307 L(20) - SPRING, 2011
Time: 3 hours
Scum 29V Kilii’
STUDENT IDENTIFICATION
Each candidate must sign the Seating List conﬁrming presence at the e

University of Calgary
Department of Mathematics and Statistics
AMAT 307 (L20) SPRING 2011
Date of exam MIDTERM EXAM Duration of exam
June 6, 2011 120 minutes
Same as gay
STUDENT’S ID NUMBER:
INSTRUCTIONS: Non—graphing scientiﬁc calculator is allow

Math 375
Spring 2016
Higher Order Linear Differential Equations
Worksheet # 2
Part 1
May 16 - 20
The problems on this worksheet refer to material from section 4.1 of your text. Solutions to
all problems will be available on the courses D2L website Friday,

Math 375
Spring 2016
First Order Differential Equations
Worksheet # 1
Part 1
May 09 - 13
The problems on this worksheet refer to material from sections 1.3, 2.1, 2.2, and 2.3 of your
text. Solutions to all the problems will be available on the courses D2L

Math 375
Spring 2016
First Order Differential Equations
Worksheet # 1
Part 4
May 16 - 20
The problems on this worksheet refer to some applications of first order differential equations.
Solutions to all the problems will be available on the courses D2L we

Math 375
Spring 2016
First Order Differential Equations
Worksheet # 1
Part 2
May 09 - 13
The problems on this worksheet refer to material from sections 2.2, and 2.6 of your text.
Solutions to all the problems will be available on the courses D2L website F

Math 375
Spring 2016
Higher Order Linear Differential Equations
Worksheet # 2
Part 2
May 23 - 27
The problems on this worksheet refer to material from sections 4.2 of your text. Solutions to
all problems will be available on the courses D2L website Friday

Math 375
Spring 2016
Higher Order Linear Differential Equations
Worksheet # 2
Part 3
June 06 - 10
The problems on this worksheet refer to material from sections 4.3, and 4.4, of your text.
Solutions to all problems will be available on the courses D2L web

Math 375
Spring 2016
Laplace Transform
Worksheet # 3
Part 3
June 15
The problems on this worksheet refer to material from sections 6.1, 6.2, 6.3, and 6.4 of
your text. Please report any typos, omissions and errors to
aiffam@ucalgary.ca
The Second Shift Fo

Math 375
Spring 2016
First Order Differential Equations
Worksheet # 1
Part 3
May 09 - 13
The problems on this worksheet refer to material from sections 2.6 of your text. Solutions to
all the problems will be available on the courses D2L website Friday, Ma

Math 375
Spring 2016
Laplace Transform
Worksheet # 3
Part 1
June 10
The problems on this worksheet refer to material from sections 6.1, 6.2, 6.3, and 6.4 of
your text. Please report any typos, omissions and errors to
aiffam@ucalgary.ca
Basic Transforms
01

Math 375
Spring 2016
Laplace Transform
Worksheet # 3
Part 2
June 13
The problems on this worksheet refer to material from sections 6.1, 6.2, 6.3, and 6.4 of
your text. Please report any typos, omissions and errors to
aiffam@ucalgary.ca
Division by t Formu

Math 375
Spring 2016
Systems of First Order Linear Differential Equations
Worksheet # 3
Part 1
June 17
The problems on this worksheet refer to material from sections 7.1, and, 7.4 of our text.
Please report any typos, omissions and errors to
aiffam@ucalga

MATH 375
Review Handout
Integration, Complex numbers, Eigenvalues and eigenvectors
Part I.
Compute the integrals:
(4x3 + 6x2 ) dx
x4 + 2x3 + 8
1)
4)
cos3 x dx
sin2 x
7)
(3x2 + 2x 3) dx
x3 x
10)
13)
16)
2)
5)
cos t sin t dt
3)
3 2x x2 dx
8)
x
x
cos dx
2
3

MATH 375
Handout # 8: Answers, Hints, Solutions
Partial Dierential Equations
1. Find the eigenvalues and eigenfunctions of the Sturm-Liouville problem
X (x) + X(x) = 0,
X (0) = 0, X (/2) = 0.
Solution. The characteristic equation r2 + = 0 has solutions r1

Department of Mathematics and Statistics
MATH 375
Handout # 1 - ANSWERS, HINTS, SOLUTIONS
Dierential equations in general; linear equations
1. For each of the following ordinary dierential equations determine the order, the dependent variable (or the unkn

Department of Mathematics and Statistics
MATH 375
Handout # 2
First Order Ordinary Differential Equations
1. Find the general solution of the following differential equations
a) xy 0 y = y 3 b) xyy 0 = 1 x2
c) y 0 tan x = y
d) y 0 = 10x+y
2. Find the solu

Department of Mathematics and Statistics
MATH 375
Handout # 2 - ANSWERS, HINTS, SOLUTIONS
First Order Ordinary Dierential Equations
1. Find the general solution of the following dierential equations
a) xy y = y 3 b) xyy = 1 x2
c) y tan x = y
d) y = 10x+y

Department of Mathematics and Statistics
MATH 375
Handout # 3
Applications of First Order Equations
1. Radium decomposes at a rate proportional to the amount present. If the half-life of
Radium is 1600 years , nd percentage lost in 4800 years.
2. Iodine-1

Department of Mathematics and Statistics
MATH 375
Handout # 3 - Answers, Hints, Solutions
Applications of First Order Equations
1. Radium decomposes at a rate proportional to the amount present. If the half-life of
Radium is 1600 years , nd percentage los

MATH 375
Handout # 5
Higher Order Linear Differential Equations
1. In each case , determine whether the set S is linearly independent on the interval
(, ). Use the Wronskian test or the definition.
a) S = cfw_x2 + x, x2 + 1, x2 1
b) S = cfw_cos(2x), 3, 2

Department of Mathematics and Statistics
MATH 375
Handout # 4
Second Order Linear Equations
1. Solve the initial value problem
d2 y
1
= ex
, y(0) = 1, y (0) = 2.
2
dx
(1 + x)2
2. Prove that the following functions are linearly independent:
a)
b)
c)
e2x a

Department of Mathematics and Statistics
MATH 375
Handout # 4 - ANSWERS, HINTS, SOLUTIONS
Second Order Linear Equations
1. Solve the initial value problem
d2 y
1
= ex
, y(0) = 1, y (0) = 2.
2
dx
(1 + x)2
Hint: integrate twice
Answer. y = ex + ln(1 + x) 4

MATH 375
Handout # 5 - Answers, Hints, Solutions
Higher Order Linear Dierential Equations
1. In each case , determine whether the set S is linearly independent on the interval
(, ). Use the Wronskian test or the denition.
a) S = cfw_x2 + x, x2 + 1, x2 1
b

MATH 375
Handout # 7
Fourier Series
1. Find the Fourier series of each of the following functions
a) f (x) = x, x [, ]
b) f (x) = 3 2 + 5x 12x2 , < x <
c) f (x) = 3x2 + 1, x [, ]
1
1,
2 < x 0
d) f (x) =
1
1, 0 < x < 2
1, 0 < x
2
0, < x
2
2. Find the Fo

MATH 375
Handout # 7 - Answers, Hints, Solutions
Fourier Series
1. Find the Fourier series of each of the following functions
a) f (x) = x, x [, ]
b) f (x) = 3 2 + 5x 12x2 , < x <
c) f (x) = 3x2 + 1, x [, ]
1
1,
2 < x 0
d) f (x) =
1
1, 0 < x < 2
Solution

AMAT 375
Handout # 6
Systems of linear dierential equations
1. Find the general solution of the following systems of dierential equations
a) X (t) =
2 1
3 4
X(t)
1 3
3
1
b) X (t) =
c) X (t) =
3
1
5 3
X
1 5
1
1
1 1
1
1 1 X(t)
d) X = 1
2 1
0
1 2 1
1
1 X
e)

Math 375
Spring 2016
Systems of First Order Linear Differential Equations
Worksheet # 4
Part 2
June 20
The problems on this worksheet refer to material from sections 7.5, and, 7.6 of your text.
Please report any typos, omissions and errors to
aiffam@ucalg