MATH 2410 PRACTICE PROBLEMS FOR MIDTERM EXAM I
Midterm Exam I:
Date and place: Section 001: 11-12:15 at MSB 315; Section 005: 2-3:15, Tuesday, Oct
6, 2015.
Material covered: Lectures, quizzes, homework, and practice problems below.
Policies: No calculators will be allowed.
Material for Exam: Sections 1.1, 1.2, 1.3, 1.4, 1.5, 1.6, 1.7, 1.8, 1.9
Remark: There will be in total 6 questions, of format somewhat similar to the practice
exam given below.
A Practice Exam:
Problem 1. Consider the initial value problem
dy
= y 2 − 3, y(0) = 1.
dt
Use Euler’s method with step size 1 to approximate y(5). Include all computations. Put
them down on the paper.
Problem 2. Consider the differential equation
dy
= y(1 − y) sin y.
dt
Identify all equilibrium points within y ∈ [−4, 4] and their types as source, sink, or nodes.
Draw the phase line. Consider three solutions with initial values y1 (0) = 2, y2 (1) = 2, and
y3 (0) = 1. Among y1 (10), y2 (10), y3 (10), which one is the biggest (or rank them).
Problem 3. Solve the following initial value problem
dy
1
= − y + (t2 − 2)4 , y(1) = 0.
dt
t
Problem 4. Find the general solution to the following linear differential equation.
dy
= y + te2t .
dt
(Hint: for the term te2t , try (C1 + C2 t)e2t .)
Problem 5. (problem 42 from Section 1.2) Suppose you are having a dinner party for
a large group of people, and you decide to make 2 gallons of chili. The recipe calls for 2
teaspoons of hot sauce per gallon, but you misread the instructions and put in 2 tablespoons of hot sauce per gallon. (Since each tablespoon is 3 teaspoons, you have put in 6
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MATH 2410 PRACTICE PROBLEMS FOR MIDTERM EXAM I
teaspoons per gallon, which is a total of 12 teaspoons of hot sauce in the chili.) You don’t
want to throw the chili out because there isn’t much else to eat (and some people like hot
chili), so you serve the chili anyway. However, as each person takes some chili, you fill up
the pot with beans and tomatoes without hot sauce until the concentration of hot sauce
agrees with the recipe. Suppose the guests take 1 cup of chili per minute from the pot
(there are 16 cups in a gallon), how long will it take to get the chili back to the recipe’s
concentration of hot sauce? How many cups of chili will have been taken from the pot?
Problem 6. John has initially deposited certain amount of money in the bank, which
has compounded yearly interest 5%.
(1) Assume that John withdraw money at a rate of 1, 000 dollars yearly in a continuous
manner. How much does John need to have in the bank at the beginning so that he can
continue this forever?
(2) If John has $10,000 initially deposited, what’s the maximal constant withdrawal rate
so that he can have money forever?
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Partial answers are on the next page.
MATH 2410 PRACTICE PROBLEMS FOR MIDTERM EXAM I
3
Partial answers: (with all graphs omitted)
Problem 1.
k tk yk yk2 − 3
0 0 1
−2
1 1 −1
−2
2 2 −3
6
3 3 3
6
4 4 9
78
5 5 87
Problem 2. The equilibrium points are y = −π, 0, 1, π. They are respectively source,
node, sink, and source.
Drawing the graph shows that y2 (t) > y1 (t) > y3 (t) for all t > 0. By existence and
uniqueness theorem, this works for all t.
Problem 3. The integration factor is µ(t) = e
(t2 −2)5
10
R
+ C. Plugging in the initial value gives y(t) =
1/tdt
= t. So yt =
R
t(t2 − 2)4 dt =
(t2 −2)5 +1
.
10t
Problem 4. We take y(t) = C1 et +(C2 +C3 t)e2t . Then we need to equalize the following
two lines
dy
= C1 et + 2C2 e2t + C3 e2t + 2C3 te2t ,
dt
y + te2t = C1 et + (C2 + C3 t)e2t + te2t .
For this we see that C1 can be arbitrary, and C2 = −1, C3 = 1. So the general solution is
y(t) = C1 et + (t − 1)e2t .
Problem 5. Let y(t) be the amount of hot sauce (in teaspoons) in the pot at time t (in
minutes). Then
dy
1 y
= − · , y(0) = 12.
dt
16 2
We have y(t) = Ce−t/32 . With the initial value y(0) = 12, we see that C = 12. So
y(t) = 12e−t/32 . To get to y(t0 ) = 4 (2 teaspoons/gal for 2 gals), we need 4 = 12e−t0 /32 .
So t0 = 32 ln 3 ≈ 35 minutes.
Problem 6. Let y(t) (dollars) be the amount of money at time t. Let C (dollars per
year) denote the rate of withdrawal. Then the equation is dy
dt = .05y − C.
(1) When C = 1, 000, the equilibrium point is at y = 20, 000. So if we want y(t) never
go to zero, we need the initial value y(0) ≥ 20, 000.
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MATH 2410 PRACTICE PROBLEMS FOR MIDTERM EXAM I
(2) In general the equilibrium point is at 20C. We need 10, 000 ≥ 20C so that John can
continuously withdraw the money. This means that C ≤ 500.