Final exam preparation
Go over all the homework, sample, and midterm questions.
Attempt all the questions in this handout.
There will be no calculators or any other aids allowed for the nal exam.
Final exam topics
1. Root nding (chap 1, chap 2.7.1): bi
MATH 2120 Quiz 2
Tuesday October 7, 2014
1. For the rst order ODE
(3t + y) y = t 2y ,
make the substitution v(t) = y(t)/t to obtain a separable equation for v(t). Write the equation for v in the form
dv/dt = G(v)/t. Do not solve the equation, and you do n
Math 2120
Final Exam
Saturday, December 12, 2015
Answer the questions in the space provided on the question sheet.
A basic calculator is allowed.
The test will run from 8:30-11:30.
The included exam booklet is for scrap paper only. I will not grade an
1.2.3
Exercises
Exercise 1.2.1: Sketch slope field for y0 = e x y . How do the solutions behave as x grows? Can you
guess a particular solution by looking at the slope field?
Exercise 1.2.2: Sketch
slope field
for Homework
y0 = x 2 .
MATH
2120,
1
0
2
Exer
MATH 2120, Homework 7
1. A two-mass, three-spring system consists of two masses at positions (x (t) , y(t) attached by a
spring to each other and to two adjacent walls. An example of such a system (with a particular
choice of masses and spring constants t
MATH 2120, Homework 5
1. Find the general solution to the system x0 = Ax, where A is as specified below. Make sure to write
the solution in purely real form.
1 0
(a) A =
.
4 3
1 5
(b) A =
.
1 3
1
0 0
(c) A = 2 1 2
2 2 1
5 3
(d) A =
3
1
2. A 3x3 real
has
MATH 2120, Homework 6
1. Compute eAt where A is one of the following matrices.
1 2
(a) A =
2 1
1 1
(b) A =
2 1
1 1
(c) A =
1 3
1 0 0
(d) A = 0 1 2
0 2 1
2. (a) Find the general solution to ~x0 = A~x + f~(t), where A =
1
1
(b) Solve the system in (a) subj
MATH 2120, Homework 4
1. (a) Write down a second order linear ODE that has the following particular solutions: y1 = ex , y2 =
e2x .
(b) Write down a second order linear ODE with real coefficients that has the following particular
solution: y1 = ex sin(2x)
MATH 2120, Homework 3
1. Consider the dierential equation y 0 (t) = y(y 1)(y 2). (a) Draw the phase diagram, find all
steady states, and classify the critical points stable or unstable. (b) Find lim y(t) for the solution
t!1
with the initial condition y(0
Practice questions for midterm
1. Review all the homework questions (up to and including hw4).
2. Solve the following first-order ODEs.
y
2x
2y = 3e 3x , y(0) = 3
(a) y 0 =
(b) y 0
(c) 2x y 0
y = 2x1/2 , y(1) = 2
(d) 2xyy 0 + 2x + y 2 = 0.
(e) 2yy 0 + 2x
MATH 2120, Homework 4
1. (a) Write down a second order linear ODE that has the following particular solutions: y1 = ex , y2 =
e2x .
Answer. The exponents are 1, 2 so that the characteristic equation is ( 1) ( + 2) = 0 or
2 + 2 = 0,
y 00 + y 0 2y = 0.
(b)
MATH 2120, Homework 1
dy
= x2 + 1 subject to initial condition y(1) = 2. What is y(3)?
dx
Answer. Integrating we have
Z
y(x) =
x2 + 1 dx
1. (1.1) Solve
=
x3
+x+C
3
for some constant C. Plugging in x = 1, y = 2 we have
1
+1+C
3
C = 2/3
1.2. SLOPE FIELDS
21
MATH 2120, Homework 2
1. Consider a pond that initially contains 10 million gal of fresh water. Water containing an undesirable chemical flows into the pond at the rate of 5 million gal/yr, and the mixture in the pond
flows out at the same rate. The conce
MATH 2120, Homework 3
1. Consider the differential equation y 0 (t) = y(y1)(y2). (a) Draw the phase diagram, find all steady
states, and classify the critical points stable or unstable. (b) Find lim y(t) for the solution with
t
the initial condition y(0)
MATH 2120 Quiz 1
1.
Thursday September 25, 2014
(a) Find the solution of the initial value problem
2
dy
y = et/3 ,
dt
y(0) = a .
(b) There exists a critical value a0 for which a < a0 and a > a0 produce qualitatively different behaviors as
t . Determine t
MATH 2120 HW 3
Due date: 14 October (Thursday)
1.
(a) Determine the real and imaginary part of
(b) Using the identity cos t =
eit +eit
,
2
e1+2i
3+4i .
show that
cos3 =
1
(cos 3 + 3 cos ) .
4
3
Hint: recall that (a + b) = a3 + 3a2 b + 3ab2 + b3 .
(c) Deri
HW4 due Oct 21 (Thursday)
1. Use the method of variation of parameters to nd particular solutions to
y 2y + y =
ex
x
2. (a) Find the general solution to the ODE
x2 y 2xy + 2y = 0.
Hint: Try the solution of the form y = xr for some r.
(b) Use the method of
Homework 6
Due Tues, 16 Nov
1. Use Laplace transform and the convolution to show that the solution to
x + 2x + x = f (t); x(0) = 0 = x (0)
is given by
t
e f (t )d.
x(t) =
0
2. Page 314 #8
3. Page 314 #26
4. Page 315 #29 and 30
5. Page 315 #32
6. Solve th
Sample Math 2120 nal exam from 2009.
You have 3 hours to complete this exam. No calculators or cheatsheets allowed
A table of Laplace transforms is provided.
There are 7 questions, including a bonus question. Each question is worth 100/6 percent.
Please w
Math 2120
Test 1
Monday, May 16, 2016
You only need something to write with. No formula sheets of any kind.
If I suspect you of cheating you will have your test taken from you.
Make sure there is a seat between you and the people on either side of
you.
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