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MA2051 - Ordinary Differential Equations

*Sample Exam 1 - B96*

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Second Exam - Originally Given 1993 A Term

Covers 5.1, 5.3, 6.1-6.6, 7.1, 7.2, 8.1-8.3, 11.1-11.4.
**Instructions:** Do your work on the paper provided. Put your
name and section number on that paper. Work neatly. Show your work.
**JUSTIFY YOUR ANSWERS.**

- 1
*35 pts*
- When a mass
of 200 gm is added to a spring, it deflects 8 cm.

- A
- Suppose a mass of 200 gm is suspended from this spring and
that the mass is started in motion from its equilibrium position
with a downward push of 6 cm/s. Write a model for the position
of the mass. (You don't need to derive the model from scratch.
Use the usual coordinate system: up is positive,
and the origin is the rest position of the mass.)

- B
- Find an expression for the position of the mass as a
function of time by solving the initial-value problem you wrote
in part (a).

- C
- Find the period of oscillation of the mass.

- D
- Find the maximum displacement of the mass.

- 2
*20 pts*
- Note that solutions
of
*x*'' + 4*x* = 0 are , .

- A
- Find a particular solution of .

- B
- Find a general solution of this equation.

- C
- If this equation were a model of a spring-mass system, would
you conclude that the displacement of the mass is bounded or
unbounded? Give a physical reason for the behavior you observe.

- 3
*25 pts*
- Consider the
system
y' = -y + 3z

z' = 3y - z.

- A
- Find the characteristic values of this system.

- B
- Find a general solution of this system.

- C
- Find a solution that satisfies the initial conditions
*y*(0) =
0, *z*(0) = 4.

#### This Topic will not be covered on the Exam

- 4
*20 pts*
- Define

- A
- Find .

- B
- Find the
*Laplace transform* of the solution of the
initial-value problem *y*' - *y* = *h*(*t*), *y*(0) = 4.

- C
- Find the
*solution* of this initial-value problem.