考试科目：Differential equation for engineering
专业：Civil & structural engineering
Examination in the School of Mathematical Sciences Semester 1, 2020
110041 MATHS 2106 Diﬀerential Equations for Engineers II
Instructions: • Refer to the Instructions page in the Exam module for instructions.
This exam has 6 question(s), for a total of 60 marks.
1. The interface y(x) between air and water in a time-independent open channel ﬂow can be approximated with the second order ODE
+ α2y = 0, x ≥ 0, (1) where the parameter α2 is a measure of the mean speed of the ﬂow. The ﬂow is in the positive x direction (i.e. from left to right).
(a)1 mark Re-write equation (1) as a system of ﬁrst-order ODEs by deﬁning y1 = y(x) and y2 = y0(x).
(b)2 marks Find the nullclines and equilibrium point for the system of ﬁrst-order ODEs.
(c)5 marks The ﬁrst order ODE that describes trajectories in the (y1,y2) phase plane is given by dy2 dy1 = −α2y1 y2 . Re-write the ODE in diﬀerential form and use the solution method for exact equations to ﬁnd the general solution. You MUST clearly show ALL the steps in the exact solution method as detailed in the lecture notes and lecture recordings.
(d)1 mark Determine the equation of the phase trajectory that satisﬁes the conditions y(0) = β and y0(0) = 0.
(e)3 marks Sketch the trajectory in the (y1,y2) phase plane that satisﬁes the conditions y(0) = β, with β > 0, and y0(0) = 0. Clearly label your sketch with: (i) the location of the conditions y(0) = β and y0(0) = 0, (ii) the intersection points of the nullclines with the phase trajectory, and (iii) the equilibrium point of the system.
2. The angular displacement θ(t) of a damped forced pendulum of length l swinging in a vertical plane under the inﬂuence of gravity can be modelled with the second order non-homogeneous ODE
θ00(t) + 2γθ0(t) + ω2θ(t) = f(t), (2)
where ω2 = g/l. The second term in the equation represents the damping force (e.g. air resistance) for the given constant γ > 0. The model can be used to approximate the motion of a magnetic pendulum bob being driven by the force, f(t), due to an electrical ﬁeld (e.g. robot arm).
(a)1 mark Write down the characteristic equation of the associated homogeneous equation, and determine an expression for the roots in terms of γ and ω.
(b)1 mark Write down the general solution to the associated homogeneous equation for γ < ω.
(c)1 mark Write down the general solution to the associated homogeneous equation for γ = ω.
(d)1 mark Write down the general solution to the associated homogeneous equation for γ > ω. (e)1 mark Determine the behaviour of the associated homogeneous equation as t →∞ for all three of your answers to (b), (c) and (d).
(f)5 marks Use the method of undetermined coeﬃcients to ﬁnd a particular solution to the non-homogeneous equation (2) for the given driving force f(t) = E cos(ωt), where E > 0 is a given constant.
(g)1 mark Determine the maximum and minimum values of the angular displacement, θ(t), as t →∞.
3. The second order variable coeﬃcient diﬀerential equation βxy00−αy = 0, (3) has a regular singular point at x = 0, where α > 0 and β > 0 are given constants. Therefore, equation (3) has at least one solution of the form
∞ X m=0
where r is chosen so that a0 6= 0. (a)6 marks Find the indicial equation and solve it for r.
(b)2 marks For the larger value of r from part (a), ﬁnd the corresponding recurrence relation for am.