Semester 2 Assessment, 2022
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School of Mathematics and Statistics
MAST20030 Differential Equations
Reading time: 30 minutes — Writing time: 3 hours — Upload time: 30 minutes
This exam consists of 8 pages (including this page) with 7 questions and 105 total marks
Permitted Materials
• This exam and/or an offline electronic PDF reader and blank loose-leaf paper.
• One double sided A4 page of notes (handwritten only).
• No calculators are permitted. No headphones or earphones are permitted.
Instructions to Students
• Wave your hand right in front of your webcam if you wish to communicate with the
supervisor at any time (before, during or after the exam).
• You must not be out of webcam view at any time without supervisor permission.
• You must not write your answers on an iPad or other electronic device.
• Off-line PDF readers (i) must have the screen visible in Zoom; (ii) must only be used to
read exam questions (do not access other software or files); (iii) must be set in flight mode
or have both internet and Bluetooth disabled as soon as the exam paper is downloaded.
• Marks are awarded for
– Using appropriate mathematical techniques,
– Showing full working, including results used,
– Accuracy of the solution,
– Using correct mathematical notation.
• Write your answers on A4 paper. Page 1 should only have your student number, the
subject code and the subject name. Write on one side of each sheet only. Each question
should be on a new page. The question number must be written at the top of each page.
Scanning and Submitting
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question order and all the same way up. Use a scanning app to scan all pages to PDF.
Scan directly from above. Crop pages to A4.
• Submit your scanned exam as a single PDF file and carefully review the submission in
Gradescope. Scan again and resubmit if necessary. Do not leave Zoom supervision until
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c©University of Melbourne 2022 Page 1 of 8 pages Can be placed in Baillieu Library
MAST20030 Differential Equations Semester 2, 2022
Question 1 (8 marks)
Consider the ordinary differential equation (ODE)
x2y′′(x) + xy′(x)− n2y(x) = 0, n = 0, 1, 2, 3, …
(a) Given n > 0:
(i) Solve the ODE by trialling y(x) = xλ.
(ii) Use the Wronskian to prove that the solutions from part (i) are linearly independent.
(b) Given n = 0, use reduction of order to find the general solution to the ODE.
Question 2 (15 marks)
(a) Use the Weierstrass test to determine if the following series converges uniformly
2−n cos(1 + nπx)
(b) Consider
(1− x2)y′′(x)− xy′(x) + y(x) = 0.
(i) Solve the ODE using a power series expanded about x = 0
(ii) State the interval of convergence of any infinite series solutions you find.
(iii) Suppose y(0.2) = 1, y′(0.2) = 0. Without solving the initial value problem (IVP),
what is the largest interval in x in which a unique solution is guaranteed for the IVP?
Page 2 of 8 pages
MAST20030 Differential Equations Semester 2, 2022
Question 3 (15 marks)
Consider the following system of ODEs
x′(t) + 2x(t)− y(t)− 3 = 0, x(0) = 0,
y′(t) + 2y(t)− x(t) + 1 = 0, y(0) = 0.
(a) Convert the system into a vector-valued IVP.
(b) Solve the system by matrix diagonalisation.
Question 4 (18 marks)
(a) Find the inverse Laplace transform of
s2 + 2s+ 10
s(s2 + 2s+ 10)
(b) The displacement of a damped mass-spring system, y(t), is governed by the ODE
+ 10y = u(t)− u(t− 5)
subject to
y(0) = 0, y′(0) = 0.
(i) Give an interpretation of the forcing term and the initial conditions.
(ii) Use Laplace transform to solve for y(t). Express your solution in explicit form.
Page 3 of 8 pages
MAST20030 Differential Equations Semester 2, 2022
Question 5 (13 marks)
Consider the function defined over [−π,π].
−π ≤ x < 0 0 ≤ x ≤ π. (a) Sketch f(x). (b) Is f(x) even, odd or neither? (c) Find g(x), the Fourier series representation of f(x) = f(x+ 2π)? (d) Given your answer to part (c): (i) How do the Fourier coefficients in g(x) decay for large n? (ii) Would you expect g(x) to converge uniformly? Briefly explain your answer. (iii) For which x in [−π,π] does g(x) exhibit Gibbs phenomenon? What is the error of the truncated Fourier approximation at this x value? (e) Consider the function h(x) = cos , x ∈ [−π,π]. Without calculations, does the Fourier series of h(x) converge uniformly? Question 6 (8 marks) Consider the 1D wave equation , c ∈ R>0, x ∈ (−∞,∞), t ∈ [0,∞)
u(x, t) → 0 as x → ±∞
u(x, 0) = δ2(x) + δ−2(x) and ∂tu(x, 0) = 0.
(a) From first principles, find the Fourier transform of
(i) f(x) = δ2(x)
(ii) g(x) = δ−2(x)
(b) Use your results in part (a) to find the inverse Fourier transform of h(k) = cos 2k.
(c) Find the IVP for the Fourier transform of u.
(d) Without calculations, briefly describe the remaining steps to find u.
Page 4 of 8 pages
MAST20030 Differential Equations Semester 2, 2022
Question 7 (28 marks)
The temperature u(r, θ, t) of a metal semi-annulus plate is modelled by the heat equation
, 1 < r < 2, 0 ≤ θ ≤ π, t > 0.
(a) Consider the steady state temperature u(r, θ) subject to the boundary conditions
u(r, 0) = 0, 1 < r < 2, u(r,π) = 0, 1 < r < 2, u(1, θ) = 0, 0 ≤ θ ≤ π, sin 4θ − sin 7θ, 0 ≤ θ ≤ π. (i) Use separation of variables to show that the steady state heat equation reduces to: r2R′′(r) + rR′(r)− λR(r) = 0 S′′(θ) + λS(θ) = 0, where λ ∈ R is the separation constant. (ii) Solve for u(r, θ). Assume that the cases λ = 0 and λ < 0 lead to trivial solutions: do not work through these cases. Also you may quote any relevant parts of your solution to Question 1 to avoid repeating any working. (b) The condition = 0 leads to a time-evolving temperature u(r, t) governed by , 1 < r < 2, 0 ≤ θ ≤ π. Assume all four boundaries of the annulus plate are insulated and the initial condition is u(r, 0) = r, 1 ≤ r ≤ 2. (i) Find the total heat of the annulus plate. (ii) Without solving the governing PDE for u(r, t), sketch u(r, t) versus r, 1 ≤ r ≤ 2, 0 ≤ t < ∞ for the following four time states t: 0 < t1 < t2 < lim t → ∞. (c) Suppose the formal series solution u(θ, t) = un(θ, t) is a genuine solution to , 0 ≤ θ ≤ π, t > 0.
Which of the series
un(θ, t) and its derivatives converge uniformly in the given domain?
End of Exam — Total Available Marks = 105
Page 5 of 8 pages
MAST20030 Differential Equations Semester 2, 2022
MAST20030 Differential Equations Formulae Sheet
1) Laplace Transforms
1. f(t) (Lf)(s) = F (s) =
f(t)e−st dt (Definition of Transform)
5. sin(at)
6. cos(at)
7. sinh(at)
8. cosh(at)
9. δ(t− a) e−as (a ≥ 0)
10. f ′(t) sF (s)− f(0)
11. f ′′(t) s2F (s)− sf(0)− f ′(0)
12. f (n)(t) snF (s)−
sn−1−kf (k)(0)
14. e−atf(t) F (s+ a) (s-Shifting Theorem)
15. f(t− a)u(t− a) e−asF (s) (a > 0, t-Shifting Theorem)
f(τ) g(t− τ) dτ F (s)G(s) (Convolution)
Page 6 of 8 pages
MAST20030 Differential Equations Semester 2, 2022
2) Standard Limits
= 0 (p > 0) (ii) lim
rn = 0 (|r| < 1) a1/n = 1 (a > 0) (iv) lim
= 0 (all a) (vi) lim
= 0 (p > 0)
= ea (all a) (viii) lim
= 0 (all p, a > 1)
3) The Generalised Harmonic Series (p-Series)
convergent if p > 1
divergent if p ≤ 1
4) Geometric Series
convergent if |r| < 1 divergent if |r| ≥ 1 5) Fourier Series Formulae (an cos knx+ bn sin knx) , kn = f(x) cos knx dx f(x) sin knx dx 6) Fourier Transforms Formulae f (k) = (Ff)(k) = f(x)e−ikx dx f(x) = (F−1 f )(x) = f(k)eikx dk = (ik)n f(k), xnf(x) = in Page 7 of 8 pages MAST20030 Differential Equations Semester 2, 2022 7) Trigonometric and Hyperbolic Formulae cos2 x+ sin2 x = 1 cosh2 x− sinh2 x = 1 1 + tan2 x = sec2 x 1− tanh2 x = sech2 x cot2 x+ 1 = cosec2 x coth2 x− 1 = cosech2 x sin(x+ y) = sinx cos y + cosx sin y sinh(x+ y) = sinhx cosh y + coshx sinh y cos(x+ y) = cosx cos y − sinx sin y cosh(x+ y) = coshx cosh y + sinhx sinh y sinx sin y = [cos(x− y)− cos(x+ y)] sinhx sinh y = [cosh(x+ y)− cosh(x− y)] cosx cos y = [cos(x− y) + cos(x+ y)] coshx cosh y = [cosh(x+ y) + cosh(x− y)] sinx cos y = [sin(x− y) + sin(x+ y)] sinhx cosh y = [sinh(x+ y) + sinh(x− y)] e−ix − eix e−ix + eix Page 8 of 8 pages 程序代写 CS代考 加微信: powcoder QQ: 1823890830 Email: powcoder@163.com