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

Kenneth Kidoguchi 11

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

December 7, 2019

December 7, 2019 12

Mock Examination III

8. Qualitative Analysis (Matching) – Time Domain Plots (3 of 3)

Figure I Figure II

Figure III Figure IV

Kenneth Kidoguchi