Review Problems for Exam 1, Math 308-503, Fall 2014
Notes: The exam will take place in class on Friday, October 10. No calculators, notes, textbooks
or other aids will be allowed. You will have the entire class period to work on the exam, but
exams will be collected promptly at the end of the period, so please be sure to arrive on time
to ensure that you have sufficient time to complete the exam. There will be no makeup exam
unless it is for a pre-excused absence or a legitimate emergency. The exam will cover material
included on the first five homework assignments. No explicit questions will be asked concerning
material from Chapter 1, so the covered sections are 2.1-2.7 and 3.1-3.8. The exam will consist
of approximately five questions, each having length similar to that of an average homework
question. Homework questions will also serve as an excellent study guide.
You will be expected to know integrals, derivatives, and basic properties of standard functions:
polynomials and powers, exponential (ex ), logarithmic (ln x), and basic trig (sin x, cos x, tan x).
In addition, you should be able to use basic differentiation rules (product, quotient, chain rule)
and “undo” these rules in order to calculate integrals.
In addition to solving the various types of ODEs covered in the last several weeks, you may also
be asked questions that will require you to recall, understand, and apply the hypotheses and
conclusions of relevant theorems. These are:
1. Theorem 2.4.1 (existence and uniqueness of solutions to linear first-order IVP’s)
2. Theorem 2.4.2 (existence and uniqueness of solutions to general first-order IVP’s)
You don’t necessarily need to memorize these theorems word for word, but you do need to be
able to cite them to answer questions (examples are below). I will give you the above list on the
exam.
Expectations for Sections 3.7 and 3.8 are as follows. You should know the ODE that models a
spring-mass system, be able to obtain the needed parameters (m, γ, k), and solve the homogeneous problem. You should also be able to distinguish damped versus undamped and forced
versus unforced vibrations, and know what the transient and steady-state solutions to a forced
vibration are along with their behavior as t → ∞.
Euler’s method. You will not be allowed to use calculators on the exam, so I won’t expect
you to take large number of timesteps using Euler’s method. You should be able to:
1. Take one or two timesteps by hand for simple ODE’s (I’ll make sure the arithmetic isn’t too
complicated).
2. Draw a sketch illustrating how Euler’s method works.
3. Explain why Euler’s can give spurious results in some situations, as for example in homework
problem 2.7.15.
Study hint: Calculus students sometimes feel that they study hard and know the material
well, but can’t solve problems under time pressure when taking an exam. My advice is to first
do some practice problems at a more leisurely pace to help you understand the material, then do
some problems with a timer set for 10 minutes (that’s about how long you’ll have to complete
each problem on the exam).
1. For each of the following problems:
i) Identify the equation as linear, or separable, or exact. If it is exact, use an appropriate test
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to show that it is exact.
ii) Find the general solution (if no initial condition is given) or the solution to the initial value
problem (if an initial condition is given). If the equation is exact or separable and no initial
condition is given, the general solution may be represented implicitly.
a) y 0 + 2ty = t.
sin x
b) y 0 = y1+y
2 , y(0) = 1.
−x
dy
c) dx
= x−e
y+ey .
0
d) y = cos t − (cos t)y.
e) 2x + y cos xy + (2y + x cos xy)y 0 = 0.
2. Solve problems 5a and 11a in Section 2.3 of the text.
3. Consider the problem y 0 + p(t)y = g(t), y(1) = 3. State a condition on p and g which will
ensure that the above problem has a unique solution on the interval −1 < t < 4. Please cite an
appropriate theorem as justification for your answer.
4. Write down all initial values (t0 , y0 ) for which you can guarantee that the initial value problem
3t2 +1
y 0 = y−4t
2 , y(t0 ) = y0 has a unique solution. Please cite an appropriate theorem as justification
for your answer.
5. (a) Use Euler’s method with h = .1 to approximate y(0.1), where y solves the initial value
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problem y 0 = sin(y)+0.5
, y(0) = 0.
(b) Now suppose you use Euler’s method with step size h (taken to be as small as you like!) to
approximate y for 0 ≤ t ≤ T . Do you expect the approximate solution to be valid no matter
what you choose T to be, or will there be a problem at some point?
6. Find the Wronskian W (t) of two solutions of the equation
t2 y 00 − ty 0 + (cos2 t + esin t )y = 0,
assuming that W (2) = 1. Note: You should also review the homework problems from Section
3.2, as there are other skills in that section that you may be tested on. These for example include
computing the Wronskian of two given functions and using that information to check whether
they form a fundamental solution set for a given ODE, etc.
7. Find the general solution to the following two equations, and describe the behavior of the
solutions as t → ∞:
a) y 00 + y 0 − 6y = 0
b) y 00 − 2y 0 + 2y = 0.
8. Find the solution to the given initial value problem:
y 00 + 4y 0 + 4y = 0, y(0) = 0, y 0 (0) = 2.
9. Find the general solution to the following equations:
a) y 00 + y = e2t + t2 .
et
b) y 00 − 2y 0 + y = 1+t
2.
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10. A spring-mass system has a spring constant of 3 N/m. A mass of 2 kg is attached to the
spring, and the motion takes place in a viscous fluid that offers a resistance of 1 N sec/m. The
spring is set in motion by displacing it 25 cm in the positive direction and releasing it with no
initial velocity.
a) Assuming that no forcing takes place, write down an initial value problem describing the
displacement u(t) of the mass.
b) Without first solving the equation, describe what will happen to the displacement u(t) as
t → ∞. Explain your answer by referring to the characteristics of the physical situation being
modeled.
c) Find u(t).
d) Determine the steady-state response of the system when an external force f (t) = 3 cos(t)N
is also applied to the mass.
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