Midterm 2 Practice Questions
1. Were given the equation
y + 2
5
t
y +
8
4
2
t
t
y=0
Knowing that one solution to this equation is y1 (t) = t2 , nd another solution.
1
2. Write down the trial solutions for the following equations: (You dont have to solve t
Math 307H, Spring 2012
Final Exam: Solutions
Page 1 of 7
1. (8 points) A mass of 100 kg stretches a (very big) spring by 2 m. Suppose that the mass is in a medium
that exerts a viscous resistance of 100 N when the mass has a velocity of 1 m/s. An external
Math 307 I - Spring 2011
Practice Final
June 03, 2011
Name:
Student number:
1
2
3
4
5
6
7
8
9
10
Total
10
10
10
10
10
10
10
10
10
1
91
Complete all questions.
You may use a scientic calculator during this examination. Other electronic devices (e.g.
cell
Lecture 14. Damped Mechanical Vibrations. July 29, 2013
14.1. Deduction of the Basic Equation
Assume we have a spring with constant k. On this spring, there is a mass m. There is a damping
(resistance force) proportional to speed, with damping coecient .
Lecture 17. Laplace Transforms I. August 9, 2013
17.1. Denition
For a function y : [0, ) R, let
est y(t)dt, s 0,
(Ly)(s) :=
0
be another function Ly : [0, ) R. This new function Ly is called the Laplace transform of y.
17.2. Examples
Let y = 1, then
(Ly)(
Lecture 16. Electrical Circuits. August 7, 2013
16.1. Physics
An electrical circuit consists of impressed voltage (=battery) giving voltage E(t), a resistor, a capacitor and an inductor.
An electrical current is a ow of electrons along the conductor. They
Lecture 15. Forced Mechanical Vibrations. August 5, 2013
15.1. No Damping
The same setting as in the last lecture. Assume we have an external force F (t). Then by Newtons
Second Law we have: mu = ku + F (t), so
mu + ku = F (t).
Let F (t) = F0 cos 0 t. Let
Lecture 18. Laplace Transforms II. August 12, 2013
18.1. Laplace Transform of Trig Functions
Let y(t) = cos(bt). Let us nd (Ly)(s). We have:
(Ly )(s) = s2 (Ly)(s) sy(0) y (0).
But y (t) = b2 cos(bt) = b2 y(t), so (Ly )(s) = b2 (Ly)(s), and y(0) = 1, y (0)
Lecture 13. Mechanical Vibrations. July 26, 2013
13.1. Basic Models
Assume we have a spring-mass system. The mass m moves horizontally in one directions. Let u be
the elongation of the spring; u = 0 corresponds to the equilibrium. The spring force, which
Math 307K, Winter 2012
Practice Final Exam: Solutions
Page 1 of 7
1. (10 points) Solve the initial value problem
y + 2y + y =
1
e
et
0t 1
,
t >1
y(0) = y (0) = 0.
Before we apply the Laplace transform, we need to rewrite the right hand side in a suitable
Math 307 C - Spring 2011
Mid-Term Exam 1
April 20, 2011
Name:
Student number:
1
2
3
4
5
Total
10
10
12
8
10
50
Complete all questions.
You may use a calculator during this examination. Other electronic devices are not allowed, and should be turned off f
Math 307B Midterm 1
April 24, 2013
name
Honor Statement
I arm that my work upholds the highest standards of honesty and integrity, and that I have neither
given nor received any unauthorized assistance on this exam.
Signature
1
2
3
4
5
total
Instructions:
Spring 2012 Final Exam, Section F, page 1 of 6
1. (5 points) Solve the initial value problem
y + 2y = te2t ,
y(1) = 0.
Spring 2012 Final Exam, Section F, page 2 of 6
2. (5 points) Let y(t) be a solution to the differential equation
y = (y 1)(y 2)2 ,
y(1)
Math 307F Final
March 21, 2013
name
Honor Statement
I afrm that my work upholds the highest standards of honesty and integrity, and that I have
neither given nor received any unauthorized assistance on this exam.
Signature
1
2
3
4
5
6
7
total
Instructions
Math 307 E - Summer 2011
Practice Midterm 2
August 17, 2011
Name:
Student number:
1
2
3
4
5
6
Total
10
10
10
10
10
3*
50
Complete all questions.
You may use a scientic calculator during this examination. Other electronic devices (e.g.
cell phones) are n
Information youll have for the nal:
Table of Laplace Transforms
f
L[ f ]
f
L[ f ]
1
1
s
1
s a
n!
s n +1
n!
( s a ) n +1
cos bt
s
s2 + b2
b
s2 + b2
(s a)
( s a )2 + b2
b
( s a )2 + b2
e at
tn
tn e at
sin bt
e at cos bt
e at sin bt
Acceleration Due to Gravi
Math 307K, Winter 2012
Practice Final Exam
Your Name
University of Washington
Student ID #
This exam is closed books. No aids are allowed for this exam. You can use any information on the
note sheet on the last page without justication.
In order to rece
Math 307 E - Summer 2011
Practice Midterm 2
August 17, 2011
Name:
Student number:
1
2
3
4
5
6
Total
10
10
10
10
10
3*
50
Complete all questions.
You may use a scientic calculator during this examination. Other electronic devices (e.g.
cell phones) are n
Lecture 19. Discontinuous Right-Hand Side. August 14, 2013
19.1. Multiplication by Exponent
Let y(t) = e2t cos(3t). Let us nd (Ly)(s). We have:
e
st 2t
e
e(s+2)t cos(3t)dt.
cos(3t)dt =
0
0
But we know that
est cos(3t)dt =
L[cos(3t)](s) =
0
s
.
s2 + 9
Just
Lecture 11. Second-Order Nonhomogeneous Equations II. July 22, 2013
11.1. Case of Repeated Roots
Consider the equation y 2y + y = et . Its characteristic equation is 2 2 + 1 = 0, which has a
double root = 1. The general solution to the homogeneous equatio
Lecture 6. Eulers Method. July 5, 2013
6.1. Example 1
Consider the equation y = t + y 2 . We cannot directly solve it using the method of separation of
variables. Also, it is not linear, so we cannot use the method of variation of constants. But we can
so
MATH 307D
Midterm 1
July 12, 2013
Student ID #
Name
Your exam should consist of this cover sheet, followed by 5 problems. Check that you have
a complete exam.
Unless otherwise indicated, show all your work and justify your answers.
Unless otherwise ind
MATH 307D
Midterm 1
July 12, 2013
Student ID #
Name
Your exam should consist of this cover sheet, followed by 5 problems. Check that you have
a complete exam.
Unless otherwise indicated, show all your work and justify your answers.
Unless otherwise ind
MATH 307D
Midterm 2 Solution
August 2, 2013
Student ID #
Name
Your exam should consist of this cover sheet, followed by 5 problems. Check that you have
a complete exam.
Unless otherwise indicated, show all your work and justify your answers.
Unless oth
MATH 307D
Midterm 2
August 2, 2013
Student ID #
Name
Your exam should consist of this cover sheet, followed by 5 problems. Check that you have
a complete exam.
Unless otherwise indicated, show all your work and justify your answers.
Unless otherwise in
Math 307 E - Summer 2011
Pactice Mid-Term Exam
June 18, 2011
Name:
Student number:
1
2
3
4
5
6
Total
10
10
10
10
10
10
60
Complete all questions.
You may use a scientic calculator during this examination. Other electronic devices (e.g.
cell phones) are
Math 307 E - Summer 2011
Pactice Mid-Term Exam
June 18, 2011
Name:
Student number:
1
2
3
4
5
6
Total
10
10
10
10
10
10
60
Complete all questions.
You may use a scientic calculator during this examination. Other electronic devices (e.g.
cell phones) are
Math 307 E - Summer 2011
Mid-Term Exam
July 20, 2011
Student number:
Name:
1
10
2
10
3
10
4
10
5
10
6
10
7
3*
Total 60+
Complete all questions.
You may use a scientic calculator during this examination. Other electronic devices (e.g.
cell phones) are no