MATH 267, Final Exam , ISU, Fall 2020

Student name: Section:

Instructions:

• This exam has 10 problems, each worth 10 points.

• Print this exam or use your notebook paper to write solutions. Combine your solutionpages to a single pdf and name it as final yourname. This first page of the exam with thesigned pledge (see below) should be the first page of your final exam solution. Follow theFInal exam link in Canvas to submit it. Download and doublecheck your solution to seeif it is the correct file.

• Discussion among students about exam problems in piazza common forum, in-personmeeting, email, canvas discussion board, or online in websites like chegg is strictly pro-hibited. Reach out to your instructor via email or private message in piazza if you haveany questions about the exam.

• Your exam will be graded based on your step by step work, explanation, and mathemat-ical correctness. A solution that only has the correct answer will get a score of 0. Unlessotherwise stated, you may use any valid method to solve these problems.

• Use all the resources available for this class, including the internet and calculator. We useTurnitin software to find out if your work is plagiarized or copied from somewhere. Anyacademic misconduct will be reported to the dean of students office immediately.

• Laplace Transform table is attached with this exam.

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• Read the following pledge, sign and date it. If you do not have a printer, copy the pledge,sign and date it. This signed page should be the first page of your exam 3 solution.

“I pledge on my honor that I have not violated the Iowa State University Academic

Integrity Code while taking this exam.”

Signature.................................... Date........................

1. Consider the differential equation y 0 = y2 - y- 2.

(a) Find all critical points and use a phase diagram to determine their stability.

(b) Use any method to find the solution that passes through the point (0, 0).

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2. Suppose the population P(t) of an ant colony grows at a rate proportional to the popula-tion. Suppose that the initial population doubles in 2 weeks.

a) How long will it take for the population to become 10 times the initial population?

b) If the initial population is P0, write down an initial value problem for P(t).

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3. Solve the following initial value problem.

xyy 0 = 4x2 - y2, y(1) = 1.

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4. Solve the IVP:y 00 - 5y 0 + 6y =

2

1+ ex; y(0) = 0, y 0(0) = 1

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for the exam

5. A mass of 3 kg is attached to the end of the spring that is stretched 20 cm by a force of 15N.It is set in motion with initial position x0 = 0 and initial velocity v0 = -10m/s. Findthe equation of motion. Also find the amplitude, period, and the frequency of the motion.

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6. Solve the following system of ODEs using eigenvalue method.

x 0 = -4x+ 8y+ 8z

y 0 = -4x+ 4y+ 2z

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7. Find the general solution using the eigenvalue method:

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8. Use either the definition of Laplace transform or the Laplace Transform table to findL{f(t)} for

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9. Use Laplace Transform to solve the initial value problem

9y 00 + y = sin✓t

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1

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3

⌘, y(0) = 0, y 0(0) = 0

where � is the Dirac delta function.

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10. Derive a recurrence relation giving cn, n > 2 for the following differential equation. Usethe recurrence relation and the initial values to determine cn. Finally use cn to find theseries solution of the differential equation.

y 00 - y 0 - 2y = 0,y(0) = 0,y 0(0) = 3

.

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Table of Laplace Transforms.

f(t) L[f(t)] = F (s)

11

s(1)

eatf(t) F (s� a) (2)

U(t� a)e�as

s(3)

f(t� a)U(t� a) e�asF (s) (4)

g(t)U(t� a) e�asL{g(t+ a)} (5)

�(t� a) e�as (6)

tnf(t) (�1)ndnF (s)

dsn(7)

x0(t) sX(s)� x(0) (8)

x00(t) s2X(s)� sx(0)� x0(0) (9)

x(3)(t) s3X(s)� s2x(0)� sx0(0)� x00(0) (10)

f ⇤ g(t) F (s)G(s) (11)

tn (n = 0, 1, 2, . . . )n!

sn+1(12)

sin ata

s2 + a2(13)

cos ats

s2 + a2(14)

eat1

s� a(15)

f(t) L[f(t)] = F (s)

teat1

(s� a)2(16)

tneatn!

(s� a)n+1(17)

eat sin ktk

(s� a)2 + k2(18)

eat cos kts� a

(s� a)2 + k2(19)

eat sinh ktk

(s� a)2 � k2(20)

eat cosh kts� a

(s� a)2 � k2(21)

t sin at2sa

(s2 + a2)2(22)

t cos ats2 � a2

(s2 + a2)2(23)

sinh ata

s2 � a2(24)

cosh ats

s2 � a2(25)

t sinh at2as

(s2 � a2)2(26)

t cosh ats2 + a2

(s2 � a2)2(27)

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