MATH 2410 PRACTICE PROBLEMS FOR FINAL EXAM

Date and place: Saturday, December 16, 2017.

• Section 001: 3:30-5:30 pm at MONT 225• Section 012: 8:00-10:00am at WSRH 112.

Material covered: Lectures, quizzes, worksheets, homework, and practice exams/problems.Policies: No calculators will be allowed.Format of the Exam: The format of the final exam will be similar to the practice test below.The total is 105 points, but you cannot earn more than 100 points. The types of problems will besimilar to the practice test and the additional practice problems below.

The distribution of points roughly breaks down as follows.

• Before Midterm 1 (23 points) Section 1.1–1.9, 2.1–2.2• Between two midterms (23 points) Section 2.3–2.6, 3.1–3.5, 3.7 and 5.1 (no equal eigenvaluescase)

• Second order differential equations (29 points) Section 3.6, 4.1–4.3• Laplace transform (30 points) 6.1–6.3

Be prepared to have a full two-hours test.

Practice Exam:

Short questions (3 points ×8 = 24 points)

Question A. Set up the differential equation for the following word problem. A 30-gallon tankinitially contains 15 gallons of salt water containing 6 pounds of salt. Suppose salt water containing1 pound of salt per gallon is pumped into the top of the tank at the rate of 2 gallons per minute,while a well-mixed solution leaves the bottom of the tank at a rate of 1 gallon per minute.

Question B. Give the system of differential equations that models the following ecologicalsystem.

On a small island, there are two species: rabbits and feral cats. Let R(t) and C(t) be thepopulation of the two species (in thousands). Suppose that both rabbits population R(t) and Catspopulation C(t) satisfy logistic model with capacity 2 and growth rate 3. Suppose furthermore thatthe cats will attack the rabbits causing the rabbit population to drop at the rate of 1

2RC. Supposesuch attack will not help the population growth of the Cats.

Question C. Consider the system of differential equations

dY

dt=

(

1 α

2 3

)

Y.

For which value of the parameter α is the system a source?

Question D. Consider the differential equation

y′ = −2ty + 2t.

Suppose that we know that y1(t) = 1 and y2(t) = 1+ e−t2

are solutions to this differential equation(with initial values y1(0) = 1 and y2(0) = 2). Consider the solution y0(t) with initial valuey 3

2

(0) = 32 . What is limt→∞ y 3

2

(t)?

1

Question E. What special properties do the slope field of differential equations of the typedydt = f(t) have? Suppose that we have one solution curve. Can we get other solutions?

Question F. Find the equilibrium solutions to the following system.{

dxdt = 3x(2− x)dydt = 2y(4− y) + 4xy.

Question G. Find the following inverse Laplace transform.

L−1

[ 2s+ 3

s2 + 6s+ 13

]

.

Problem H. Find the following Laplace transform

L[

u2(t)ea(t−2)].

Problem 1. (9 points) Consider the differential equation

dy

dt= (y − 2)2y cos y.

Sketch the phase line when y ∈ [−5, 5] and classify the equilibrium points in that range as sinks,sources, or nodes. Draw a rough sketch of the solution of with initial value y(0) = 3.

Problem 2. (8 points)Solve the following initial value problem

dy

dt= −(2t+ 1)y + e−t

2

, y(0) = 0.

Problem 3. (7 points) Find the general solution to the following linear system, and sketch itsphase portrait.

dY

dt=

(

1 −12 3

)

Y.

Problem 4. (7 points) Find the general solution to the following linear system, and sketch itsphase portrait.

dY

dt=

(

1 02 3

)

Y.

Problem 5. (7 points) Solve the initial value problem

y′′ + 4y′ + 5y = e−2t, y(0) = y′(0) = 0.

Sketch the graph of the solution. Moreover, if we start with another initial value, what does thesolution look like when t→∞, and WHY?

Problem 6. (8 points) Give the general solution to the following differential equation

y′′ + 4y′ + 3y = 10 cos t.

Find the solution with initial value y0(0) = y′0(0) = 0. Draw the graph to indicate both theparticular solution and the solution to the initial value above. Discuss their long-term behavior.

Problem 7. (11 points) Solve the initial value problem

y′′ + 100y = cos 9t, y(0) = y′(0) = 0.2

Determine the frequency of the beats and the frequency of the rapid oscillation. Sketch the solutionto the given initial value. (In the actual final exam, if a problem like this is given, the followingformula will be available.)

cosα− cosβ = −2 sin α+β2 sin α−β

2 .

Problem 8. (6 points) Find the following inverse Laplace transform.

L−1

[ 7e−2s

(2s+ 3)(s− 2)

]

.

Problem 9. (8 points) Use Laplace transform to solve the following initial value problem.

y′ = y + u5(t), y(0) = 3.

Problem 10. (10 points) Use Laplace transform to solve the following initial value problem.

y′′ + 9y = 6, y(0) = 3, y′(0) = 3.

Additional practice problems

Problem I. Beth initially deposits $6, 000 in a savings account that pays interest at the rate of1% per year compounded continuously. She arranges for $20 per week to be deposited automaticallyinto the account. Assume that weekly deposits are close enough to continuous deposits so that wecan reasonably approximate her balance using a differential equation. Write an initial-value problemfor her balance over time. Approximate Beth’s balance after 4 years.

Problem II. Consider the following eco-system. On a small island, there are two species: rabbitsand foxes. Let R(t) and F (t) be the population of the two species (in thousands). Suppose thatthe rabbits population R(t) satisfies the logistic model with capacity 2 and growth rate 4. Supposewithout rabbits, the fox population will decline at the rate 1

2 . Moreover the foxes will eat rabbitat the rate of 5RF , and on average eating five rabbits will increase the population of fox by 1.

(1) Use a system of differential equations to model the population dynamics of the two species.(2) Find the equilibrium points of the system.(3) Give the linearization at the equilibrium point(s) where neither R nor F is zero.(4) determine the type of the system at the equilibrium point considered in (3).

Problem III. Consider the differential equation with parameter α

dy

dt= y2 − 4y + α.

(1) Draw the phase line of the system when α = 3. Classify the equilibrium points as sinks,sources and nodes. Draw typical solutions with initial values in each intervals.

(2) Draw the bifurcation diagram and compute the bifurcation value. Draw the phase lines forthe system when α is slightly smaller than, slightly larger than, and equal to the bifurcation value.

Problem IV. Solve the following initial value problem.

dy

dt=

1

et + ety, y(0) = 1.

Problem V. Find the general solution to the following systems and sketch the phase portrait.

(1)

(dxdtdydt

)

=

(

−1 −1−4 −1

)(

x

y

)

.

(2)dY

dt=

(

1 32

2 3

)

Y.

3

Problem VI. Solve the following initial value problem.

y′′ − 6y′ + 5y = 3e2t, y(0) = y′(0) = 0.

Problem VII. Give the general solution to the following differential equation describing adamped oscillator with sinusoidal forcing.

y′′ + 2y′ + 2y = 5 sin t.

Problem VIII. Find the following Laplace transform or inverse Laplace transform.

(1) L −1[ e−4s

s2 − 1

]

.

(2) L[

u2(t)e3(t−2) sin(4(t− 2))

]

.

Problem IX. Use Laplace transform to solve the following initial value problems.

(1) y′ = −5y + et, y(0) = 1.(2) y′′ + 4y = sin 3t, y(0) = 1, y′(0) = 1.(3) y′′ + 6y′ + 5y = 10, y(0) = 2, y′(0) = 4.

4

Problem 8. We use partial fractions. Set

7

(2s+ 3)(s− 2)=

A

s+ 32

+B

s− 2.

So we have7 = 2A(s− 2) +B(2s+ 3).

Setting s = 2 gives 7 = B · 7 and B = 1; setting s = −32 gives 7 = 2A · (−3

2 − 2) and A = −1. So

L−1

[ 7

(2s+ 3)(s− 2)

]

= −L−1

[ 1

s+ 32

]

+ L−1

[ 1

s− 2

]

= −e−3t/2 + e2t.

From this we deduce that

L−1

[ 7e−2s

(2s+ 3)(s− 2)

]

= −u2(t)e−

3

2(t−2) + u2(t)e

2(t−2).

Problem 9. y′ = y + u5(t), y(0) = 3.We apply Laplace transform to get

L [y′] = L [y] + L [u5(t)].

sL [y]− 3 = L [y] +e−5s

s.

(s− 1)L [y] = 3 +e−5s

s.

L [y] =3

s− 1+

e−5s

s(s− 1).

We know that1

s(s− 1)=

1

s− 1−

1

s(by partial fractions). So we have

L [y] =3

s− 1+

e−5s

s− 1−

e−5s

s.

y(t) = L−1

[ 3

s− 1

]

+ L−1

[ e−5s

s− 1

]

−L−1

[e−5s

s

]

= 3et + u5(t)et−5− u5(t)

= 3et + u5(t)(

et−5 − 1)

.

Problem 10. y′′ + 9y = 6, y(0) = 3, y′(0) = 3.Take the Laplace transform of the given equation, we have

L[

y′′]

+ 9L [y] = L [6].

Rewrite everything in terms of L [y]:

(

s2L [y]− sy(0)− y′(0))

+ 9L [y] =6

s.

Plugging in the initial values, we have(

s2L [y]− 3s− 3)

+ 9L [y] =6

s.

(s2 + 9)L [y] =6

s+ 3s+ 3.

L [y] =6 + 3s+ 3s2

s(s2 + 9).

11