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). 11 May 2016 1 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. May 11, 2016 2 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 →∞ 11 May 2016 3 Kenneth Kidoguchi Mock Examination II 11 May 2016 4 Kenneth Kidoguchi Mock Examination II 11 May 2016 5 Kenneth Kidoguchi Mock Examination II 11 May 2016 6 Kenneth Kidoguchi Mock Examination II 11 May 2016 7 Kenneth Kidoguchi Mock Examination II 11 May 2016 8 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 x1 + 8 x1 = 1+ e b) x2 + 6 x2 + 8 x2 = cos(t ) eit + e −it c) x3 + 6 x3 + 8 x3 = 85 2 11 May 2016 d ) x4 + 6= x4 + 8 x4 340 sin ( t + π / 2 ) e) x5 + 6 x5= + 8 x5 85 ( cos ( t ) + sin(t ) ) 9 9 Kenneth Kidoguchi Mock Examination II iπ a ) x1 + 6 x1 + 8 x1 = 1+ e 11 May 2016 10 Kenneth Kidoguchi Mock Examination II b) x2 + 6 x2 + 8 x2 = cos(t ) 11 May 2016 11 Kenneth Kidoguchi Mock Examination II eit + e −it c) x3 + 6 x3 + 8 x3 = 85 2 11 May 2016 12 Kenneth Kidoguchi Mock Examination II d ) x4 + 6= x4 + 8 x4 11 May 2016 340 sin ( t + π / 2 ) 13 Kenneth Kidoguchi Mock Examination II 11 May 2016 14 Kenneth Kidoguchi Mock Examination II e) x5 + 6 x5= + 8 x5 11 May 2016 85 ( cos ( t ) + sin(t ) ) 15 Kenneth Kidoguchi Mock Examination II 11 May 2016 16 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) 11 May 2016 17 t-Domain Phase Trajectory Kenneth Kidoguchi Mock Examination II 5. Forced Harmonic Oscillator System Response (2 of 3) 11 May 2016 Figure 1 Figure 2 Figure 3 Figure 5 18 Kenneth Kidoguchi Mock Examination II 5. Forced Harmonic Oscillator System Response (3 of 3) 11 May 2016 Phase Trajectory A Phase Trajectory B Phase Trajectory C Phase Trajectory D 19 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). 11 May 2016 20 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. 11 May 2016 21 Kenneth Kidoguchi Mock Examination II 11 May 2016 22 Kenneth Kidoguchi Mock Examination II May 11 May 11,2016 2016 23 Kenneth Kidoguchi 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 MayMay 11 11,2016 2016 24 Kenneth Kidoguchi