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阿德莱德大学 110041 MATHS 2106

Differential equation for engineering

· 案例展示

考试科目:Differential equation for engineering
考试时间:21/07/2020 12:00-15:30
考试时长: 三个半小时
考试范围: 数学
国家:澳大利亚
学校: 阿德莱德大学
专业:Civil & structural engineering
年级: 大二

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
d2y dx2
+ 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
y(x) =
∞ X m=0
amxm+r
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.
 

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