01 March 2018 Technology Allowed Examination Kidoguchi\m256_x2.1soln.docx MTH_256: Differential Equations Examination II Part 1 of 2 Technology Allowed Examination Student Name: STUDNAME Instructions: Read all instructions before beginning. The student s name shall be written where indicated on this examination cover page. This is part 1 of a two-part in-class examination. Part 1 of this examination consists of examination problems 1 through 4. Each solved problem carries up to 10 marks. This examination carries a total of 80 examination marks. The total time allowed for completion of both parts of this examination is 90 minutes. The total examination score will be the sum of marks from this examination and marks from the mock examination student group presentation. Part 2 of this examination may be collected after submission of Part 1. The student may choose to return to Part 1 of this examination upon completion and submission of Part 2. One 8½" 11" 6 mil sheet of notes may be used during both parts of this examination. Calculators and/or computers may be used during this part of the examination. Textbooks and student's personal lecture notes sind verboten! Analysis may be done on the back of examination sheet pages or on separate worksheets. If separate worksheets are used, the student s name and the problem being worked must be clearly identified in order to be considered during marking. Students should keep in mind that: Illegible work will remain unmarked. Undefined symbols have no meaning. Improper notation and/or incomplete mathematical sentences will be rejected. Graphs with unlabelled axes have no meaning. Ambiguous conclusions will be misinterpreted. Approximations must be so indicated. Solutions with incorrect units (dimensions) are incorrect solutions. Partial credit will be awarded for solutions that are coherent and logically presented. Marks will be deducted for scattered, disjointed analysis presentations. No credit will be awarded for ambiguous conclusions. Caveat: Any hint of cheating during the examination will result in an examination mark of zero. - 1 of 5 -
01 March 2018 Technology Allowed Examination Kidoguchi\m256_x2.1soln.docx 1. Forced Oscillations - Qualitative Analysis in the t-domain Given the IVP: m x cx kx F() t, where the system parameters m, c, k, forcing function, F(t), and initial conditions (ICs) are as listed in Table 1, complete this table by matching each IVP to the system response depicted in Figures 1, 2, 3, or 4. System m c k IC [x(0), v(0)] Table 1 Forcing Function t-domain Figure 1 1 1 1 [1/3, 0] F(t) = cos(3t) 4 2 1 5 3 [1/3, 0] F(t) = cos(t) 1 3 1 5 1 [1/3, 0] F(t) = cos(3t) 2 4 1 1 3 [1/3, 0] F(t) = cos(t) 3 Figure 1: x & dx/dt vs t Figure 2: x & dx/dt vs t Figure 3: x & dx/dt vs t Figure 4: x & dx/dt vs t - 2 of 5 -
01 March 2018 Technology Allowed Examination Kidoguchi\m256_x2.1soln.docx 2. Forced Oscillations - Qualitative Analysis in the Phase Plane Given the IVP: m x cx kx F() t, where the system parameters m, c, k, forcing function, F(t), and initial conditions (ICs) are as listed in Table 2, complete this table by matching each IVP to the system response depicted in Phase Trajectories A, B, C, or D. System m c k IC [x(0), v(0)] Table 2 Forcing Function Phase Trajectory 1 1 1 1 [1/3, 0] F(t) = cos(3t) D 2 1 5 3 [1/3, 0] F(t) = cos(t) A 3 1 5 1 [1/3, 0] F(t) = cos(3t) B 4 1 1 3 [1/3, 0] F(t) = cos(t) C Phase Trajectory A Phase Trajectory B Phase Trajectory C Phase Trajectory D - 3 of 5 -
01 March 2018 Technology Allowed Examination Kidoguchi\m256_x2.1soln.docx 3. Particle Motion and Uncle Heaviside s Unit Step Function A particle travels along the x-axis with velocity v(t). Its acceleration, a(t) = dv/dt is as shown in Figure 3. If the particle's initial velocity is v(0) = 2, find expressions for a(t) and v(t) in terms of the unit step function. Figure 3: Particle Acceleration a(t) - 4 of 5 -
01 March 2018 Technology Allowed Examination Kidoguchi\m256_x2.1soln.docx 4. Archimedes Buoy A frictionless floating cylindrical buoy has a radius r = e -ln(256) blands (where 1 bland = 40 rods), height h = 245 cm, and uniform mass density ρ B = 1/4 gram/cm 3. Let the surface of the water be at x = 0 and x(t) be the depth of the bottom of the buoy at time t in minutes. Assume that the mass density of water is ρ W gram/cm 3 and the acceleration due to gravity, g = 980 cm/s 2. At t = 0 the bottom of the buoy is at x(0) = 0 cm and moving at speed x (0) = v(0) = 0 cm/s. a) Write initial value problem (IVP) that describes the buoy motion in terms of x(t) and its derivatives. b) Solve the IVP. c) Sketch a properly labelled graph of x(t) on the interval 0 < t < where t is in seconds. - 5 of 5 -
01 March 2018 Traditional Non-Technology Examination Kidoguchi\m256_x2.2soln.docx MTH_256: Differential Equations Examination II Part 2 of 2 Traditional Non-Technology Examination Student Name: Instructions: Read all instructions before beginning. The student s name shall be written where indicated on this examination cover page. This is part 2 of a two-part in-class examination. Part 2 of this examination consists of examination problems 5 through 8. Each solved problem carries up to 10 marks. This examination carries a total of 80 examination marks. The total time allowed for completion of both parts of this examination is 90 minutes. The total examination score will be the sum of marks from this examination and marks from the mock examination student group presentation. Calculators and/or computers shall not be used to work problems in part 2 of this examination. The student may choose to return to Part 1 of this examination upon completion and submission of Part 2. Textbooks and student's personal lecture notes sind verboten! One 8½" 11" 6 mil sheet of notes may be used during both parts of this examination. Analysis may be done on the back of examination sheet pages or on separate worksheets. If separate worksheets are used, the student s name and the problem being worked must be clearly identified in order to be considered during marking. Students should keep in mind that: Illegible work will remain unmarked. Undefined symbols have no meaning. Improper notation and/or incomplete mathematical sentences will be rejected. Graphs with unlabelled axes have no meaning. Ambiguous conclusions will be misinterpreted. Approximations must be so indicated. Solutions with incorrect units (dimensions) are incorrect solutions. Partial credit will be awarded for solutions that are coherent and logically presented. Marks will be deducted for scattered, disjointed analysis presentations. No credit will be awarded for ambiguous conclusions. Caveat: Any hint of cheating during the examination will result in an examination mark of zero. - 1 of 5 -
01 March 2018 Traditional Non-Technology Examination Kidoguchi\m256_x2.2soln.docx 5. Plug-n-Chug Present the analysis to find a general solution in simplified form to each of the following ODEs. a) i x 6x 8 x 1 e 1 1 1 b) x 6x 8x cos t 2 sin t / 2 c) d) 2 2 2 x 6x 8x 85 cos( t / 2) 3 3 3 it it e e x 6x 8x 85 4 4 4 i2 e) x 6x 8 x 85sin t 5 5 5 N.B.: In all cases i 1 and the dot notation indicates that x is a function of t. - 2 of 5 -
01 March 2018 Traditional Non-Technology Examination Kidoguchi\m256_x2.2soln.docx 6. A Mass-Spring System A mass-spring system is described by the ODE m x cx kx F() t, where x is the position of the mass about its natural rest position in centimetres and dx/dt = v is the speed of the mass in cm/s. The mass, m = 1 gram, damping coefficient, c = 6 gram/s, and spring constant, k = 8 dynes/cm are constants. The forcing function is F(t) =85sin(t). Given initial condition x(0) = 6 and v(0) = present the analysis to: a) Find x(t), the solution to this initial value problem. b) Sketch a properly labelled phase trajectory of the system s steady state response in the phase plane (i.e., with x as the horizontal axis and v as the vertical axis. - 3 of 5 -
01 March 2018 Traditional Non-Technology Examination Kidoguchi\m256_x2.2soln.docx 7. Another Oscillating Pendulum The motion of an ideal pendulum is modelled by the ODE: ml cl mg F() t, where t is the angular position of the pendulum bob in radians about its natural rest position and is the rate of change of the angular t position with respect to t, time in seconds. Given: m = kg is the mass of the pendulum bob g = 2 m/s 2 is the acceleration due to gravity i c = 1 e kg/s is the damping coefficient L = ( /6) 2 metres is the length of the pendulum F(t) = 2 cos (4 t) is the forcing function in Newtons. With initial conditions: (0) = 0 and (0) = 0, present the analysis to: a) find (t), the solution to this initial value problem, and b) sketch a properly labelled graph of (t) in the t-domain on the interval 0 < t < 2 ensuring that all key points are clearly identified. - 4 of 5 -
01 March 2018 Traditional Non-Technology Examination Kidoguchi\m256_x2.2soln.docx 8. An Acceleration and Velocity Model A bullet passes through the mud wall of a rondavel. The acceleration of the bullet in terms of its velocity v while in the wall is described by the differential equation: dv kv dt 2 where k =ln(2). If the bullet enters the wall with a speed of 512 metres per second and exits the wall at a speed of 256 metres per second, present the analysis to find the distance travelled by the bullet while it is within the mud wall. Recall the Mock Exam II hint: dv dv dx. dt dx dt - 5 of 5 -