MTH 256 Mock Exam II - Portland Community Collegespot.pcc.edu/~kkidoguc/m256/m256_x3_mock.pdf ·...
Transcript of MTH 256 Mock Exam II - Portland Community Collegespot.pcc.edu/~kkidoguc/m256/m256_x3_mock.pdf ·...
7 December 2019 Kenneth Kidoguchi
MTH_256 Differential Equations
Mock Examination III
• Form a group and select at least one problem from this list.
• Prepare a report that describes a solution to your selected
problem(s).
• Deliver your report as an interesting and enlightening mini-lecture.
• Your presentation will be peer marked and counts for 20% of your
examination score.
• Criteria for marking Mock Exam Presentations are:
• Preparation: The lecture was well prepared and thoughtfully
organised.
• Report Format: The analysis satisfied the course notation
standards and was clear, logical, and easy to follow.
• Technical Merit: Analysis conclusions were clear, concise, and
technically correct.
• Pedagogical Value: This report prepared me for this type of
problem should it appear on the coming examination.
a) Find the ODE that models the quantity of salt (in grams) in Tank C as a
function of time.
b) Solve the initial value problem to determine the concentration of salt in
Tank C as a function of time.
c) Find the exact value of t at which the quantity of salt in Tank C is a
maximum. December 7, 2019 Kenneth Kidoguchi 1
Tank C holds 100 litres of a brine solution
that contains 100 grams of salt. At t = 0, a
valve at the base of Tank A is opened for
exactly ten minutes and brine from Tank A
with a concentration of 100 gram per litre is
poured into the Tank C at a flow rate of 1
litre per minute. At exactly t = 10 minutes,
1. The Briny Solution Revisited
Ali; Abdullahi | Jamison; Tyler | Pyper; Will | Simon; Nathan
Tank B Tank A
Tank C
the flow from Tank A stops and pure water from Tank B begins to flow
into Tank C at a flow rate of 1 litre per minute. Throughout this process,
i.e., t > 0, the well-mixed solution in Tank C drains at a rate of 1 litre per
minute. Present the analysis to:
2. Archimedes' Buoy
Not Selected
7 December 2019 2 Kenneth Kidoguchi
A cylindrical buoy floats in a frictionless fluid
with mass density r0 = 1 gram/cm3. The buoy
has height h = 490 cm, radius r = h/p2 cm and
uniform mass density r = 0.5 gram/cm3. Let
x(t) be the depth of the bottom of this buoy
beneath the surface at time t in seconds. The
buoy is initially motionless at its equilibrium
position, i.e., x(0) = 245 cm and the
acceleration due to gravity is g = 980 cm/s2.
An ideal “hammer” exerts a vertical force in dynes on the buoy given by:
Present the analysis to:
write the IVP that describes the buoy motion in terms of x(t),
solve the IVP and.
sketch a properly labelled graph of x(t) on the interval 0 < t < 2p.
r
h
2
1
( ) 20 , where mass of the buoy2
B B
n
f t m t n m=
p = =
0
x
3. Uncle Heaviside & Inverse Laplace Transforms
Thomsen; Lorenzo| Suryadevara; Nitin | Wagner; Garret| Zeng; Kai
a) Express a(t) in terms
of the unit step
function and find
L{a(t)}.
b) Express v(t) in terms
of the unit step
function and find
L{v(t)}.
c) Express x(t) in terms
of the unit step
function and find
L{x(t)}.
December 7, 2019 Kenneth Kidoguchi 3
A particle travels along the x-axis with acceleration a(t) = d2x/dt2 as shown
in Figure 3 and velocity v(t) = dx/dt. The particle's initial velocity and
position are v(0) = 0 and x(0) = 1 respectively. Present the analysis to:
Figure 3
December 7, 2019
4. Convolution (Faltung)
Fleming; Amos | Hladik; Gabrielle | Ross; Jason| Ziegler; Andrew
A system's impulse response is z(t) = u(t – 0) cos(t + p/2). Present the
analysis to find x(t), this system’s response to the forcing function:
Assume t > 0 and sketch a properly labelled graph of x(t) in the t-domain.
SYSTEM
SYSTEM x(t)
( ) ( 0)cos( / 2)t u t tz = p ( ) 0f t t=
4
1
( )n
f t t n=
= p
4
1
( )n
f t t n=
= p
Kenneth Kidoguchi 4
5. An RLC Circuit
Not Selected
December 7, 2019 Kenneth Kidoguchi 5
( )q
Lq Rq tC
=
An RLC circuit is described by the ODE:
where q(t) is the charge on the capacitor in
Coulombs at time t in seconds and i = dq/dt is
the current in Ampères.
a) Present the analysis to find q(t) in simplified form.
b) Sketch a properly labelled graph of q(t) in the t-domain for 0 < t < 5p.
c) Sketch a properly labelled phase portrait of the response, i.e., a graph
with of q(t) on the horizontal axis and 𝑞 (𝑡) on the vertical axis.
Switch R
C L (t)
4
1
( ) 2 Forcing Function in Volts, 1jn
n
t e u t n jp
=
= p = =
Given a system that is initially quiescent with:
L = 1 Henry = inductance,
R = 0 W = resistance,
C = 1/4 Farads = capacitance
6. Plug-n-Chug
Cannucci; Nick | Reeser; Jack
December 7, 2019 6 Kenneth Kidoguchi
1 1
2 2
3 3
5
5 5
0
4 4
a) 4 0
b) 4 2
c) 4
d
e
2 2
) 4 2
) 4 4 0
n
in
x x t
x x t
x x t t
x x e t n
x x u t
=
p
=
= p
= p p
= p
=
Given initial conditions x(0) = v(0) = 0-, where v(t) = dx/dt, present the
analysis to find x(t) and v(t) in simplified form that satisfy each of the
following initial value problems.
0 0
7. A Whacked Pendulum
Taylor; Nathan | Whitsell; Lewis
The motion of an ideal pendulum can be modelled
by the ODE: 𝑚𝐿θ + 𝑐𝐿 θ + 𝑚𝑔𝜃 = 𝑓(𝑡) where (t) in radians, is the angular displacement of
the pendulum bob about its natural rest position and
d/dt = W(t) is the rate of change of the angular
displacement with respect to time, t in seconds. For:
m = 2 kg = mass of the pendulum bob
g = p2 m/s2 = acceleration due to gravity
c = 0 gram per second = damping coefficient
L = (p/2)2 metres = pendulum length
𝑓 𝑡 = 𝜋2 −𝛿 𝑡 − 𝜋 − 𝛿 𝑡 − 3𝜋/2 Newtons = forcing function
and ICs: (0) = W(0) = 0- , present the analysis to:
a) Find (t), the solution to this IVP.
b) Plot (t) and W(t) in the t-domain on the interval 0 < t < 2p.
c) Plot (t) vs. W(t) in the phase plane on the interval 0 < t < 2p .
8. Phase Portrait and t-Domain Matching (sheet 1 of 2)
Ayala; David |Casler; Alexander | Prado; Miguel | Tolbert; Rodston
Phase Trajectory t-domain
A
B
C
D
A direction field for a system of
ODEs is shown with selected phase
trajectories for a linear system
where Y 𝑡 = 𝑥 𝑡 , 𝑦(𝑡) .
Complete the table by matching
Figures 1 through 4 to its
corresponding phase trajectory labelled A, B, C, and D. A
B C
D
8
8. Phase Portrait and t-Domain Matching (sheet 2 of 2)
Ayala; David |Casler; Alexander | Prado; Miguel | Tolbert; Rodston
9
Mock Examination III
8. Qualitative Analysis (Matching) – Sheet(1 of 3)
December 7, 2019 10 Kenneth Kidoguchi
k
m
Equilibrium Position @ x = 0
x > 0 x < 0
f(t)
Match each Initial Value Problem to its resultant
phase trajectory (A, B, C, or D) and resultant
time-domain plot (I, II, III, or IV).
ODE w/ Initial Conditions:
𝑥 0 = 𝑥 0 = 𝑣 0 = 0− t-Domain
Phase
Trajectory
𝑥 + π2𝑥 = 𝑒𝑖π 𝛿 𝑡 − 2 + 𝛿 𝑡 − 3
𝑥 + π2𝑥 = 𝑒𝑖2π 𝛿 𝑡 − 2 + 𝛿 𝑡 − 3
𝑥 + π2𝑥 = 𝑢(𝑡 − 1)sin π 𝑡 − 1
𝑥 + π2𝑥 = 𝑢(𝑡 − 1)cos 2π 𝑡 3
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Mock Examination III
8. Qualitative Analysis (Matching) – Phase Trajectories (2 of 3)
Ph
ase
Po
rtra
it B
P
has
e P
ort
rait
D
Ph
ase
Po
rtra
it A
P
has
e P
ort
rait
C
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Mock Examination III
8. Qualitative Analysis (Matching) – Time Domain Plots (3 of 3)
Figure I Figure II
Figure III Figure IV
Kenneth Kidoguchi