Mock Examination II
MTH_256 Differential Equations
• Form a group and select at least one problem from this list. Then:
• prepare an oral report that describes the solution to the selected
problem(s), and
• deliver your report as an interesting and enlightening mini-lecture
describing your solution.
• Your presentation will be peer marked and counts for 20% of your
examination score.
• Problems provided here may be similar to problems on the coming
examination.
• N.B.:
• Undefined symbols have no meaning.
• Improper mathematical notation will be misinterpreted.
• Graphs with unlabelled axes have no meaning.
• Ambiguous conclusions will be rejected.
• Approximations must be so indicated.
• Solutions with incorrect units are incorrect solutions.
11 May 2016
Kenneth Kidoguchi
Mock Examination II
1. The Brine Solution Revisited
Initially, Tank I holds 100 litres of a brine solution
containing 25 grams of dissolved salt and Tank II
holds 200 litres of brine solution with 50 grams of
dissolved salt. At t = 0, pure water flows into
Tank I at a rate of 4 L/min and the well mixed
solution flows from Tank I into Tank II at a rate of
4 L/min. The well mixed solution in Tank II flows
out of Tank II at rate of 4 L/min. Let x(t) and y(t)
represent the quantity (in grams) of salt in Tanks I
and II respectively. Present the analysis to:
a) write the initial value problems (IVPs) for dx/dt
and dy/dt, and
b) find x(t) and y(t) the particular solutions that
satisfy the IVPs from part a).
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Pure
H2 O
Tank I
Tank II
Kenneth Kidoguchi
Mock Examination II
2. The Logistic Model with Harvesting
Kimo Keali'inohomoku traded the family pua'a for
3 magic pakalolo bushes. The three bushes were
planted in Kimo’s basement and, if not harvested,
would multiply according to the logistic model:
dP
= P(11 − P)
dt
where P is the number of pakalolo bushes and t is time in months after
the initial planting.
Let h represents the (constant) rate of plants harvested per month.
Present the analysis to find the harvest rate h that will eventually provide
Kimo with a ready supply of nine pakalolo bushes forever.
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Kidoguchi, Kenneth
Mock Examination II
3. Archimedes' Buoy
A floating cylindrical buoy of height h = 490
cm, radius r = h/2 and uniform mass density
ρ = 0.5 gm/cm3 is initially at rest with its top
at the surface of the water (x = 0). Let x(t) be
the depth of the bottom of the buoy beneath
the surface at time t so x(0) = h. Assume that
the density of water is ρ0 = 1 gm/cm3 and the
acceleration due to gravity is g = 980 cm/s2.
0
r
x
h
Further, the magnitude of the resistive force associated with the vertical
motion of the buoy is FR = 4 mB dx/dt, where mB is the mass of the buoy.
a) Write the initial value problem (IVP) that describes the buoy motion
in terms of x(t) and its derivatives,
b) Solve the IVP and sketch a properly labelled graph of x(t).
c) Describe x(t) and lim x(t ).
t →∞
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Kenneth Kidoguchi
Mock Examination II
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Kenneth Kidoguchi
Mock Examination II
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Kenneth Kidoguchi
Mock Examination II
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Kenneth Kidoguchi
Mock Examination II
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Kenneth Kidoguchi
Mock Examination II
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Kenneth Kidoguchi
Mock Examination II
4. Second Order Linear ODEs With Constant Coefficients
Present the analysis to find a general solution to each of the following
ODEs.
iπ
a ) 
x1 + 6 x1 + 8 x1 =
1+ e
b) 
x2 + 6 x2 + 8 x2 =
cos(t )
 eit + e −it
c) 
x3 + 6 x3 + 8 x3 =
85 

2

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


d ) 
x4 + 6=
x4 + 8 x4
340 sin ( t + π / 2 )
e) 
x5 + 6 x5=
+ 8 x5
85 ( cos ( t ) + sin(t ) )
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Kenneth Kidoguchi
Mock Examination II
iπ
a ) 
x1 + 6 x1 + 8 x1 =
1+ e
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Kenneth Kidoguchi
Mock Examination II
b) 
x2 + 6 x2 + 8 x2 =
cos(t )
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Kenneth Kidoguchi
Mock Examination II
 eit + e −it
c) 
x3 + 6 x3 + 8 x3 =
85 

2

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


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Kenneth Kidoguchi
Mock Examination II
d ) 
x4 + 6=
x4 + 8 x4
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340 sin ( t + π / 2 )
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Kenneth Kidoguchi
Mock Examination II
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Kenneth Kidoguchi
Mock Examination II
e) 
x5 + 6 x5=
+ 8 x5
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85 ( cos ( t ) + sin(t ) )
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Kenneth Kidoguchi
Mock Examination II
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Kenneth Kidoguchi
Mock Examination II
5. Forced Harmonic Oscillator System Response (1 of 3)
A mass-spring system is described by the Harmonic Oscillator ODE ,
 + kx F (t ),=
mx + cx
=
x(0) x0 and
=
v(0) v0
where x is the position of the mass about its equilibrium position in
centimetres and v = dx/dt is the speed of the mass in cm/s. Given the
system parameters in Table 5, complete this table by matching the system
to the response depicted by the appropriate t-domain & phase plane plots.
Table 5
System
m
c
k
[x0, v0]
F(t)
I
1
5
1
[1, −1]
sin(t)
II
1
5
1
[1, −1]
sin(5t)
III
1
1
1
[1, −1]
sin(t)
IV
1
1
1
[1, −1]
sin(5t)
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t-Domain
Phase Trajectory
Kenneth Kidoguchi
Mock Examination II
5. Forced Harmonic Oscillator System Response (2 of 3)
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Figure 1
Figure 2
Figure 3
Figure 5
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Kenneth Kidoguchi
Mock Examination II
5. Forced Harmonic Oscillator System Response (3 of 3)
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Phase Trajectory A
Phase Trajectory B
Phase Trajectory C
Phase Trajectory D
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Kenneth Kidoguchi
Mock Examination II
6. Another Forced Oscillator
A mass-spring system is described by
x<0 x>0
k
the ODE ,
m F(t)


mx + cx + kx =
F (t )
where x is the displacement of the mass about
Equilibrium Position @ x = 0
its equilibrium position in centimetres and v = dx/dt is the speed of the mass
in cm/s. Given:
m = 10 = mass in grams
c = 60 = damping coefficient in gram/s
k = 80 = spring constant in dynes/cm
F(t) = 850cos(t) = forcing function in dynes
with ICs x(0) = 7 and v(0) = 6, present the analysis to:
a) Find an expression for x(t) that satisfies this initial value problem.
b) Express the steady state solution in the form: xs(t) = A cos(ωt – α) with
exact values for A, ω, and α.
c) Sketch a properly labelled graph of the steady state solution in the phase
plane (i.e., with x as the horizontal axis and v as the vertical axis).
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Kenneth Kidoguchi
Mock Examination II
7. A Pendulum Forced to Swing
The motion of an ideal pendulum can be modelled by
the ODE:
mL
θ + cLθ + mg θ = F (t )
where θ(t) in radians, is the angular position of the
pendulum bob about its natural rest position and
d θ / dt =
Ω(t ) is the rate of change of the angular
position with respect to time, t in seconds. Given:
L
θ<0
θ>0
m
= 9 / π2 is the mass of the pendulum bob in kilograms
g = π2
is the acceleration due to gravity metres/s2
c=0
is the damping coefficient grams per second
2
L = ( π / 3) is the length of the pendulum in metres
F(t) = cos(4t) is the external forcing function in Newtons
with ICs: θ(0) = Ω(0) = 0, present the analysis to:
a) Find an expression for θ(t), the solution to this IVP, as a product sinusoids.
b) Sketch a properly labelled plot θ(t) in the t-domain over one complete cycle of
the envelope functions. Ensure that extrema and zeros are clearly identified.
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Kenneth Kidoguchi
Mock Examination II
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Kenneth Kidoguchi
Mock Examination II
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Mock Examination II
8. Particle Motion and Uncle Heaviside’s Unit Step Function
A particle travels along the x-axis with acceleration a(t) = d2x/dt2 shown in
Figure 8 and velocity v(t) = dx/dt. The particle's initial velocity and
position are v(0) = -1 and x(0) = 0 respectively. Present the analysis to
find expressions for x(t) and v(t) in terms of the unit step function.
Figure 8
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