Math 2410 Practice Exam II March xth, 2019

Name:

Instructions:

• Answer each question to the best of your ability. Show your work or receive no credit.

• All answers must be written clearly. Be sure to erase or cross out any work that you do not want

graded. Partial credit can not be awarded unless there is legible work to assess.

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indicate when you have done this so that I do not miss any of your work.

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by anyone besides myself to their personal advantage. Further, I assert that this examination was taken in

accordance with the academic integrity policies of the University of Connecticut.

Signed:(full name)

Questions: 1 2 3 4 5 TotalScore:

1

1. (5 points) Use the method of undetermined coe�cients to find the general solution of

y00 � 16y = 2e4x (1)

2

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8Ae4k zen

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c gen sol is y yetyp

2. (5 points) A mass of 1 slug, when attached to a spring, stretches it 32/9 feet and then it comes to rest

in the equilibrium position. Starting at t = 0, an external force equal to f(t) = sin(3t) is applied to the

system. There is no damping on the system.

(a) Set up the IVP governing this spring-mass system.

(b) Using variation of parameters, find the corresponding equation of motion.

(c) Explain why or why not the mass will ever come back to rest.

3

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32 Mse pre'tKx fCtlI

springconstant k wagth 34g 9

stretched x t 9k sinottKG o we 0

µ I o y 1 0

fe I sendtip o Cmodamping

b aux eg m2 9 o s m 132 acct c coset 1 CzsinCst case er a

particularsolution Np UimtunesWCseno I W3t

seizethirst 3wya I 3

u since u f t t Izsinktwoman s

uz Kif wk34tik3 u Lycos twheatUp fteas t t sina.tl cus H Y cos HSindH

It Cosette zzncztgwzzctl zn nz.twsGtltsin3Ct4

ftcosC3t1tfsaidHi U c coset Canis a ft coset c stories H

C cos H tCzhis341 It cos t c relabeledUco o s C o

x Cti 3CzcosEst f wet 1 I this cIµ i o s o 3Cz f s Cz it

x t i Ipsindel f ccus Hc longteens system oscillates

uncontrollably

3. (5 points) Consider the following linear homogeneous di↵erential equation:

a(x)y00 + b(x)y0 + c(x)y = 0.

(a) If y1, y2 form a fundamental set of solutions, then what can we say about the value of their Wronskian

W (y1, y2)?

(b) If f and g are solutions, show by direct computation that f + g is also a solution.

(c) True or False: if Y1, Y2, Y3 are three solutions, then we may write one of them as a linear combination

of the remaining two. Justify by using the definition of linear dependence and the theory of linear

di↵erential equations.

4

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b af tbf te f o

org tby tcg o

act g t bCftgl'teffigy af e b f't of tagbg'tog

0 to

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fty is a solution

c True orderofODE i 2Yi Y Y has the lies dcp

GY1642141 0 fr some G ca c not allzeroi Withoutlossofgenerality assume Lito Thea

Y Yz t Y

4. (5 points) Consider the following 100 gallon tanks containing brine. As done in class, construct a system

of di↵erential equations which model the salt contents xA(t), xB(t), and xC(t) of tanks A,B, and C,

respectively, given the indicated information.

What values should xA(t), xB(t), and xC(t) respectively approach as t ! 1?

5

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Kc 77 x 7

In at For k

5. (5 points) Solve the following nonlinear ODE

y000(y00 + y)(y00 � y) = 0. (2)

Note: there will be three di↵erent solutions.

6

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I L m 1 0m4l o

m o JdI m t ln _m mz o m I O

ti t

Y _citcautsse y cei4qe in y c e 165or CcossetGuinn