Department

Applied MathCourse Code

APPM 2360Professor

AllStudy Guide

MidtermThis

**preview**shows half of the first page. to view the full**3 pages of the document.**APPM 2360 Exam 3 Spring 2016

ON THE FRONT OF YOUR BLUEBOOK write: (1) your name, (2) your student ID number, (3) recitation section, (4) your

instructor’s name, and (5) a grading table. Books, class notes, cell phones, and calculators are NOT permitted. A one-page

one-sided crib sheet is allowed.

1. (30 points) For the following two ODE’s, (i) ﬁnd the general solution (ignoring initial conditions) and (ii) the unique

solution satisfying the initial conditions provided.

(a) (15 points)

y′′

−5y′

−6y= 0, y(0) = 2, y′(0) = 5

(b) (15 points)

y′′ + 2y′+y= 0, y(0) = 0, y′(0) = 1

2 (30 points) Consider the harmonic oscillator system:

Assume that the mass of the block is m= 2, the spring constant k= 4, and that the equilibrium position of the block is x= 0.

Both the damping constant band the external force F(t)can vary. Let x(t)denote the position of the block at time t. The initial

conditions are x(0) = 1 and x′(0) = 1.

(a) (2 points) Write down the differential equation for x(t), governing the motion of the block, using the supplied values of

mand k.

(b) (8 points) Suppose the external force is zero and let b= 4. Write down the differential equation and ﬁnd the solution

x(t)using the supplied initial conditions. Describe the motion of the block.

(c) (14 points) Suppose no damping is present (b= 0). An external force F(t) = cos(t)is applied to the block. Write down

the differential equation and derive the solution x(t)using the supplied initial conditions. Describe the motion of the

block.

(d) (6 points) Solving for x(t)is not necessary for this question.

(I) (3 points) Suppose that F= 0 and we would like to design the damper so the block returns as fast as possible to the

equilibrium position and ceases oscillation. What must the damping constant be (or what range of values can it take) for

this to be the case?

(II) (3 points) Suppose no damping is present (b= 0) and a force F(t) = cos √2tis applied. Describe the motion of

the block in this case.

CONTINUED

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