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Review Exam 2

Section Objective(s):

• Exam Settings.• General Comments on Chapter 2, 4.• A Sample of Review Exercises.

4.4.1. Exam Settings.

Main Settings:

• 6 problems.• 55 minutes.• 42 grading attempts.• Integration table provided.• Laplace transform table provided.• In recitation labs.

Secure Browser Settings:

• Login name: computerlabs/math exams• Password: 235math• If something does not work:

– Do not reboot your computer.– Ask your TA.

Exam Covers:

• 1 question from Chapter 1, (10 points).• 5 questions from Chapters 2 and 4, (80 points).• TA grades one question, (10 points).• Before the exam starts we announce which question is graded by the TA.

Remark on Exam Starting and End Times:

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4.4.2. General Comments on Chapters 2, 4.

Remark: Be sure you check suggestions in our course webpage, Extra Help.

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4.4.3. A Sample of Review Exercises.

(1) Find y solution of y y′′ + 4(y′)2 = 0, with y(0) = 1 and y′(0) = 7.

(2) Find a solution y2 not proportional to the solution y1 = t−4 of

t2 y′′ + 11t y′ + 24 y = 0.

(3) Find the general solution of y′′ + 4 y = 3 sin(2t).

(4) Find the general solution of y′′ − 2 y′ + y =5 et

t2, for t > 0.

(5) Find the movement of m = 5 kg attached to a spring with k = 5 kg/s2, in a

medium with damping constant d = 5 kg/s and initial conditions y(0) = −√3 m

and y′(0) =√3 m/s.

(6) Use the definition of the Laplace transform to compute L[cosh(t)].

cosh(at) =eat + e−at

2, sinh(at) =

eat − e−at

2,

L[cosh(at)] =∫ ∞

0

e−st(eat + e−at

2

)dt, L[sinh(at)] =

∫ ∞

0

e−st(eat − e−at

2

)dt.

(7) Find L[f ] where f(t) =

⎧⎨

⎩

t

2, 0 ! t < 6,

3, t " 6.

(8) Solve y′′ + 3 y = g(t), where y(0) = 0, y′(0) = 0 and g(t) =

{0, t < 2,

et−5, t " 2.

(9) Solve y′′ + 3 y = cos(t) δ(t− π), where y(0) = 0, y′(0) = 0.

L[f(t) δ(t− c)] = f(c) e−cs.

(10) Given g such that L[g] = G, find f such that L[f ] = e−2s

(s2 + 3)G(s).