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Mock Examination III
May 30, 2009 (^111) Kenneth KidoguchiKenneth Kidoguchi
MTH_256 Differential Equations
- Break up into at least six groups.
- Your group will select at least one problem that may be similar to a problem on the coming examination. You are to: - prepare an oral report that describes a solution to the given problem, and - deliver your report as an interesting and enlightening mini-lecture describing your solution.
- Your presentation will be peer marked and will count for 20% of your examination score.
- 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.
Mock Examination III
May 30, 2009 (^222) Kenneth KidoguchiKenneth KidoguchiKenneth Kidoguchi
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,
- The Briny Solution Revisited
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.
Tank A^ Tank B
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:
Mock Examination III
- A Mass-Spring System
May 30, 2009 (^3) Kenneth Kidoguchi
mx ^ + cx + kx = f t ( )
A mass-spring system is described by the ODE:
where x ( t ) is the position of the mass in centimetres at time t in seconds and v = dx / dt is the speed in cm/s.
present the analysis to find an expression for x ( t ) in simplified form and sketch a properly labelled graph of the phase trajectory ( i. e ., with x as the horizontal axis and v = dx / dt as the vertical axis for 0 < t < 5π ).
4
1
( ) 2 1^ n Forcing Function in dynes n
f t u t n
= − (^) ∑− − π =
Given a system that is initially quiescent with: m = 1 = mass in grams, c = 0 = damping coefficient in gram/sec, k = 4 = spring constant in dynes/cm,
k m
Equilibrium Position @ x = 0
x < 0 x > 0 F ( t )
Mock Examination III
May 30, 2009 (^444) Kenneth KidoguchiKenneth Kidoguchi
- Forced Oscillator Qualitative Analysis – Phase Trajectories (1 of 3)
Given the IVP:
where f ( t ) is one of the forcing functions below.
Match each of the forcing functions to its resultant phase trajectories (A, B, C, or D) and match each of the forcing functions to its resultant time-domain plot (I, II, III, or IV).
k m
Equilibrium Position @ x = 0
x < 0 x > 0
x ^ + π^2 x = f t ( ), x (0) = x ^ (0) = 0 − and v t ( ) = x t ( )
f 1 (^) ( ) t = e i^ π( δ ( t − 2 ) + δ ( t − 3 ))
f 4 ( ) t = u t ( − 1) cos 2( π t / 3)
f 2 (^) ( ) t = ei^2 π( δ (^) ( t − (^2) ) + δ (^) ( t − (^3) ))
f (^) 3 ( ) t = u t ( −1) sin (^) ( π (^) ( t − (^1) ))
f ( t )
Mock Examination III
May 30, 2009 (^777) Kenneth KidoguchiKenneth Kidoguchi
4. Convolution (Faltung)
A system's impulse response is ζ( t ) = sin( t ). Present the analysis to find x ( t ), this system's response to the forcing function: f ( t ) = δ( t – π) + δ( t – 2π) and sketch properly labelled graphs of x ( t ) in the t -domain and dx / dt vs x ( t ) in the phase plane.
SYSTEM
SYSTEM x ( t )
ζ ( ) t =sin( ) t
δ (^) ( t − (^0) )
f t ( ) = δ (^) ( t − π + δ) ( t − 2 π)
Mock Examination III
May 30, 2009 (^888) Kenneth KidoguchiKenneth Kidoguchi
- Uncle Heaviside’s Unit Step Function Given the IVP: d^2 y / dt^2 = h ( t ), y (0) = - 1, v (0) =1 where v ( t ) = dy / dt and the graph of h ( t ) is as shown in the figure.
a) Express h ( t ) in terms of the unit step function. b) Express v ( t ) in terms of the unit step function c) Express y ( t ) in terms of the unit step function. d) Overlay a sketch of the graph of y ( t ) and v ( t ) onto the figure.
Mock Examination III
May 30, 2009 (^999) Kenneth KidoguchiKenneth Kidoguchi
- An RLC Circuit
Lq Rq q ( ) t C
(^) + + = ε
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.
Present the analysis to find an expression for i ( t ) in simplified form and sketch a graph of i ( t ) for 0 < t < 5π.
Switch (^) R
L C
ε( t )
4
1
( ) 2 1 Forcing Function in Volts
n n
t u t n
ε = − (^) ∑ − − π =
Given a system that is initially quiescent with: L = 1 Henry = inductance, R = 0 Ω = resistance, C = 1/4 Farads = capacitance
Mock Examination III
May 30, 2009 (^101010) Kenneth KidoguchiKenneth KidoguchiKenneth Kidoguchi
The motion of an ideal pendulum can be modelled by the ODE:
where in radians, is the angular displacement of the pendulum bob about its natural rest position and is the rate of change of the angular displacement with respect to time, t in seconds. With:
mL θ +^ cL θ + mg θ = f t ( )
m = 2 kg = mass of the pendulum bob m/s^2 = acceleration due to gravity c = 0 gram per second = damping coefficient metres = pendulum length Newtons = forcing function
, present the analysis to: a) Find θ( t ), the solution to this IVP. b) Plot θ( t ) and Ω( t ) in the t -domain on the interval 0 < t < 3π. c) Plot θ( t ) vs. Ω( t ) in the phase plane on the interval 0 < t < 3π.
7. A Swinging Pendulum
L
θ < 0 θ > 0
θ ( ) t
d θ / dt = Ω( ) t
g = π^2
L = π( / 2)^2
f t ( ) = π 2 ( δ ( t − π − δ) ( t − 2 π))
Given ICs: θ(0) = Ω(0) = 0 −
