SM 212 Final Examination
11 December 1996
 (a)
 Find the general solution to
(i)
,
(ii)
,
 (b)
 Solve the initial value problem
.
 (a)
 A tank contains gallons of water with lbs. of salt dissolved in it.
Saltwater with a concentration of lb/gal is pumped into the tank at a
rate of gal/min.
The wellstirred mixture is pumped out at the rate of gal/min.
At what time does the tank
contain the largest amount of salt? How much salt is in the tank at that time?
 (b)
 Use the improved Euler method with a step size of to
approximate for the initial value problem.
 Find the general solution to:
 (a)

,
 (b)

,
 (c)

.
 Use undetermined coefficients (or annihilators) to solve the initial value problem
 (a)
 An lb weight stretches a spring ft upon coming to rest at
equilibrium. From equilibrium the weight is raised ft and
released from rest.
The motion of the weight is
resisted by a damping force that in numerically equal to times the
weight's instantaneous velocity.
(i) Find the position of the weight as a function of time.
(ii) What type of damping does this massspring system possess?
 (b)
 The position of a weight in a massspring system subject to
an external force is given by
.
(i) What are the amplitude and period of the
steadystate part of the solution?
(ii) Write the transient part of the solution in the form
.
(iii) Find the time past which the magnitude of the transient
part of the solution is
less than onepercent of that of the steadystate part of the solution.
 (a)
 Find:
(i)
,
(ii)
,
(iii)
.
 (b)
 Find
for the periodic function given over one period by
 (a)
 Find
(i)
,
(ii)
.
 (b)
 Use the convolution theorem to find
.
 (a)
 For the circuit in the circuit above show that the charge
on the capacitor
and the current
in the right branch satisfy the system of differential equations
 (b)
 When the switch in the circuit is closed at time , the current is
0 amps and the charge on
the capacitor is coulombs. With , , use Laplace
transforms
to find the charge on the capacitor.
 Use eigenvalues/eigenvectors to solve the initial value problem
 (a)
 Find the Fourier sine series of the function
 (b)
 The temperature of a thin bar of length
satisfies the following conditions
where is given in (10.a).
(i) Use your answer in (10.a) to find . Show all steps of the
separation of variables process
clearly. Write your answer in summation notation.
(ii) Use the first two nonzero terms of your answer in ((10)(b)1) to
approximate .
Answers for SM 212 Final Examination
11 December 1996
1. a. i.
,
,
,
.
1. a. ii.
,
,
,
,
1.b.
,
,
,
,
,
.
2.
,
,
,
gives
. The IC
, implies
2.b.
, ,
, ,
3.a.
,
,
3.b.
,
<
,
,
3.c.
,
,
(double roots),
,
4.
, ,
,
,
,
,
,
,
,
,
,
,
,
, ,
5.a. 1.
,
, slugs,
,
,
,
,
,
,
5.a.2. critically damped.
5.b.1. Amplitude:
,
Period = ,
5.b.2.
,
,
5.b.3.
,
,
.
6.a.1.
,
6.a.2
,
6.a.3
6.b.1
,
7.a.1
,
,
,
,
7.a.2
,
,
7.b. Use the convolution theorem to find
,
.
8.a.
,
8.b. Substituting one DE into the other gives
, so
.
Taking LTs:
,
,
.
Now take inverse LTs to get:
.
9.
,
10.a.
,
,
b.
,
,
.
David Joyner
20030810, last modified 1122004