MATH 237
Fall 2009
Maple Assignment 1
Preliminary Reading: Chapters 1 and 2 of the Maple manual.
Problem 1
Using the commands in the rst chapter of the manual, generate the plots of the family of functions:
cfw_f = enx , n = 1, 2, 3, 4, 5
for the values x

MATH 237
Fall 2009
Maple Assignment 3
Problem 1
Ask maple to solve the following second order differential equation:
y (2) + 2y (1) + 5y = 0,
by typing in
> Deq := diff(y(t), t, t) + 2 diff(y(t), t) + 5 y(t) = 0;
and
> sol1 := rhs(dsolve(Deq, y(t);
Note t

MTHE 237
Fall 2014
Solutions to Assignment 1
Problem 1
a) Find the order of the dierential equation:
t
d4 y
+ t2 y = e t .
dt4
Is the dierential equation linear? Is the equation homogeneous?
b) Repeat the above for the following dierential equation:
dy
dy

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MTHE 237
Solutions to Practice Midterm
There are four problems
Be as neat as possible, clearly state your results
Calculators are not allowed
Student Number:
Student Name:
Fall 2014
Problem 1 [25 points]
Consider the dierential equation dened for x R:
y (

MTHE 237
Fall 2014
Solutions to Assignment 7
Problem 1
Show that the Laplace transform of cos(t) satises
Lcfw_cos t =
s2
s
+ 2
Solution:
A
L(cos t)
est cos(t)dt
= lim
A
0
= lim est
A
sin(t) A
|0
A
0
s st
e
sin(t)dt
sin(t) A
s
= lim est
|0 + est cos(t) 2

MTHE 237
Fall 2014
Solutions to Assignment 4
Problem 1
a) Find the general solution to the dierential equation:
y (3) + y (2) + 3y (1) 5y = 0,
given that m = 1 is a solution of L(m) = m3 + m2 + 3m 5 = 0.
b) Find the general solution to the dierential equa

MTHE 237
Fall 2014
Solutions to Assignment 2
Problem 1
Solve the following dierential equation:
y y = 2te2t ,
y(0) = 1
Hint: Check if the equation is exact. Find an appropriate integrating factor.
Solution:
It can be veried that the equation is not exact.

MTHE 237
Fall 2014
Solutions to Assignment 5
Problem 1
Show that for square matrices A and B, which commute, that is
AB = BA,
it follows that
e(A+B) = eA eB .
T
Ak
k=0 k!
=
eA eB
Hint: Recall that eA = limT
=
Ak
k=0 k! .
k=0 l=0
=
k=0 u=k
u
=
u=0 k=0

MTHE 237
Fall 2014
Solutions to Assignment 6
Problem 1
Let
A=
1
5
2
5
2
5
4
5
.
a) Find two linearly independent solutions to the equation
x = Ax,
and construct a fundamental matrix.
b) Solve the dierential equation:
x = Ax,
with the initial condition x(0

MTHE 237
Fall 2014
Solutions to the Midterm
There are four problems
Be as neat as possible, clearly state your results
Calculators are not allowed
Student Number:
Student Name:
Problem 1 [35 points]
Consider the dierential equation dened for x R:
y (2) +

MATH 237
Fall 2009
Maple Assignment 2
Preliminary Reading: Chapter 1, 2, 3 and Chapter 8 of the Maple manual.
Problem 1
Consider the separable differential equation:
(1 + x)(1 + y)dx dy = 0
First obtain the solution analytically using the appropriate inte

MATH 237
Fall 2009
Maple Assignment 1
Preliminary Reading: Chapters 1 and 2 of the Maple manual.
Problem 1
Using the commands in the first chapter of the manual, generate the plots of the family of functions:
cfw_f = enx , n = 1, 2, 3, 4, 5
for the values

MATH 237
Fall 2009
Maple Assignment 2
Preliminary Reading: Chapter 1, 2, 3 and Chapter 8 of the Maple manual.
Problem 1
Consider the separable dierential equation:
(1 + x)(1 + y)dx dy = 0
First obtain the solution analytically using the appropriate integr

MATH 237
Fall 2009
Maple Assignment 3
Problem 1
Ask maple to solve the following second order dierential equation:
y (2) + 2y (1) + 5y = 0,
by typing in
> Deq := di(y(t), t, t) + 2 di(y(t), t) + 5 y(t) = 0;
and
> sol1 := rhs(dsolve(Deq, y(t);
Note that Ma

Queens University
Mathematics and Engineering and Mathematics and Statistics
MATH 237
Dierential Equations for Engineering Science
Supplemental Course Notes
Serdar Y ksel
u
August 27, 2014
This document is a collection of supplemental lecture notes used f

MATH 237
Fall 2008
Solutions to Midterm I
There are ve problems
Be as neat as possible, clearly state your results
Calculators are not allowed
Student Number:
Student Name:
Problem 1 [15 points]
Consider the dierential equation:
(x2 + 2y)dx + (2x 2y)dy =

MATH 237
Fall 2009
Solutions to Assignment 2
Problem 1
Solve the following dierential equation:
where a, b, c, d are constants.
Consider the dierential equation:
dy
(a/c)y + (b/c)
=
,
dx
y + (d/c)
where a, b, c, d are constants, which are all positive val

MATH 237
Fall 2009
Solutions to Assignment 3
Problem 1
Solve the following initial value problem
100y (4) 10y (3) + y (1) = 0
with initial conditions y(0) = y (1) (0) = y (2) (0) = y (3) (0) = 0. Find y(t) for t 0.
Solution:
By the Existence and Uniquenes

MTHE 237
Fall 2014
Homework Assignment 4
Problem 1
a) Find the general solution to the differential equation:
y (3) + y (2) + 3y (1) 5y = 0,
given that m = 1 is a solution of L(m) = m3 + m2 + 3m 5 = 0.
b) Find the general solution to the differential equa

MTHE 237
Fall 2014
Homework Assignment 2
Problem 1
Solve the following differential equation:
y 0 y = 2te2t ,
y(0) = 1
Hint: Check if the equation is exact. Find an appropriate integrating factor.
Problem 2
Solve the following differential equation:
ty 0

MTHE 237
Fall 2014
Homework Assignment 5
Problem 1
Show that for square matrices A and B, which commute, that is
AB = BA,
it follows that
Hint: Recall that eA = limT
Ak
k=0 k!
e(A+B) = eA eB .
P
k
= k=0 Ak! . Complete the following:
eA eB
=
PT
X
X
Ak B

MTHE 237
Fall 2014
Homework Assignment 1
Problem 1
a) Find the order of the differential equation:
d4 y
+ t2 y = e t .
dt4
Is the differential equation linear? Is the equation homogeneous?
t
b) Repeat the above for the following differential equation:
dy

MTHE 237
Fall 2014
Homework Assignment 6
Problem 1
Let
A=
1
5
2
5
2
5
4
5
.
a) Find two linearly independent solutions to the equation
x = Ax.
b) Solve the differential equation:
x = Ax,
1
with the initial condition x(0) =
.
0
Problem 2
With A given as

MTHE 237
Fall 2014
Homework Assignment 7
Problem 1
Show that the Laplace transform of cos(t) satisfies, for s > 0,
Lcfw_cos t(s) =
s
s2 + 2
Problem 2
Use the Laplace transform method to obtain the solution to the following differential equation:
y 2y + 4y

MTHE 237
Fall 2014
Homework Assignment 3
Problem 1
Solve the following initial value problem
2014y (4) + 10y (2) + 10y (1) + y = 0
with initial conditions y(0) = y (1) (0) = y (2) (0) = y (3) (0) = 0. Find y(t) for t 0.
Problem 2
Recall that a set of n fu

MTHE 237
Fall 2014
Solutions to Assignment 3
Problem 1
Solve the following initial value problem
2014y (4) + 10y (2) + 10y (1) + y = 0
with initial conditions y(0) = y (1) (0) = y (2) (0) = y (3) (0) = 0. Find y(t) for t 0.
Solution:
By the Existence and