Student 
Number 
Semester 2 Assessment, 2017 
School of Mathematics and Statistics 
MAST20030 Differential Equations 
Writing time: 3 hours 
Reading time: 15 minutes 
This is NOT an open book exam 
This paper consists of 4 pages (including this page) 
Authorised Materials 
• Mobile phones, smart watches and internet or communication devices are forbidden. 
• Calculators, tablet devices or computers must not be used. 
• One A4 sheet, handwritten only on both sides, is permitted. 
• No books, printed/photocopied materials or notes are permitted. 
Instructions to Students 
• You must NOT remove this question paper at the conclusion of the examination. 
• You should attempt all questions. There are six questions whose marks are indicated. 
• The total number of marks available is 100. 
Instructions to Invigilators 
• Students must NOT remove this question paper at the conclusion of the examination. 
This paper may be held in the Baillieu Library 
MAST20030 Semester 2, 2017 
Question 1 (15 marks) 
In this question, we consider the homogeneous third-order linear ODE 
t3 (1 − t ) d 
3y 
dt3 
+ t2 
d2y 
dt2 
− 2t dy 
dt 
+ 2y (t ) = 0. (1) 
(a) Show that the functions y1 (t ) = t and y2 (t ) = t2 are linearly independent solutions. (3 marks) 
(b) Use the wronskian 
wr[y1,y2,y3] = 

y1 y2 y3 
y ′1 y 
′ 
2 y 
′ 
3 
y ′′1 y 
′′ 
2 y 
′′ 
3 

to deduce that any solution y3 (t ) to the ODE in (1), such that {y1,y2,y3} is linearly independent, 
satisfies the following inhomogeneous second-order linear ODE: 
t2 
d2y3 
dt2 
− 2t dy3 
dt 
+ 2y3 (t ) = 
t − 1 
t 
. (5 marks) 
(c) Verify that t and t2 are complementary solutions to this ODE and then use reduction of order to find 
a solution y3 such that {y1,y2,y3} is linearly independent. (5 marks) 
(d) Write down the general solution to the ODE in (1). (2 marks) 
Question 2 (15 marks) 
This question concerns the following homogeneous second-order linear ODE: 
ty ′′(t ) + y ′(t ) + ty (t ) = 0. 
(a) By trialling a series solution of the form 
y1 (t ) = 
∞∑ 
j=0 
c jt 
j , 
show that c1 must be 0. Write down the formal solution y1 (t ) for which c0 = 1. (6 marks) 
(b) What is the radius of convergence of your formal solution? On which intervals of R does it converge 
uniformly? (3 marks) 
(c) By quoting any relevant results from lectures, state whether your formal solution y1 (t ) is a genuine 
solution or not. [You do not have to show that your solution is genuine.] (2 marks) 
(d) Compute the wronskian of y1 (t ) with another linearly independent solution y2 (t ). Use the result to 
show that y2 (t ) necessarily has a logarithmic singularity at t = 0. (4 marks) 
Page 2 of 4 pages 
MAST20030 Semester 2, 2017 
Question 3 (15 marks) 
Consider the second-order linear IVP 
4y ′′(t ) + 12y ′(t ) + 9y (t ) = 2δ+2 (t ), y (0) = 2, y 
′(0) = −2. 
(a) Use Laplace transforms to solve this IVP fory (t ). Write your answer as a piecewise-defined function 
(no Heavisides allowed). (8 marks) 
(b) Check explicitly if y (t ) is continuous at t = 2. Is it differentiable there? (5 marks) 
(c) Sketch a rough plot of y (t ) for 0 6 t 6 4, illustrating x- and y-intercepts as well as your results for 
the previous part. (2 marks) 
Question 4 (15 marks) 
Consider the following mixed BVP: 
y ′′(x ) + λy (x ) = 0, y (0) = y ′(L) = 0. 
(a) Show, from first principles, that the eigenvalues and eigenfunctions of this BVP are 
λn = 
pi 2 (n − 12 )2 
L2 
, yn (x ) = sin 
pi (n − 12 )x 
L 
, n = 1, 2, 3, . . . (5 marks) 
(b) Fill in the blanks: These eigenfunctions form for a 
of functions satisfying . (3 marks) 
(c) Use your answer to the previous part to prove that 
f (x ) = 
∞∑ 
n=1 
bn sin 
pi (n − 12 )x 
L 
⇒ bn = 2 
L 
∫ L 
0 
f (x ) sin 
pi (n − 12 )x 
L 
dx . (3 marks) 
(d) Show that the bn when f (x ) = x have the form bn = 
2L(−1)n−1 
pi 2 (n − 12 )2 
. (4 marks) 
Page 3 of 4 pages 
MAST20030 Semester 2, 2017 
Question 5 (25 marks) 
Marlene, Darlene and Sharlene are building a mega-extreme ultimate skipping machine (TM). Their first 
prototype has two motors spaced pi metres apart (they’re geeky like that) whose purpose is to control the 
vertical position and the angle the rope makes at each end. We model the height u (x , t ) of the rope using 
the wave equation and the following motor control instructions: 
u (0, t ) = 0, ∂xu (pi , t ) = 1. 
The ladies set their initial rope configuration so that u (x , 0) = x and ∂tu (x , 0) = −x . 
(a) Writeu (x , t ) = ulin. (x , t )+urest (x , t ), whereulin. (x , t ) is chosen so that ∂2xulin. (x , t ) = 0 and it satisfies 
the inhomogeneous boundary conditions above. Solve for ulin. and show that urest (x , t ) satisfies an 
IBVP with homogeneous boundary conditions. Write this IBVP out in full. (5 marks) 
(b) Find a formal solution to this homogeneous IBVP and thus write down a formal solution for u (x , t ). 
(10 marks) 
(c) What would you have to check in order to prove that this solution was a genuine solution of the wave 
equation? [You do not need to demonstrate it!] (4 marks) 
(d) Consider the energy functional 
E[v](t ) = 
1 
2 
∫ pi 
0 
[ 
(∂tv (x , t )) 
2 + c2 (∂xv (x , t )) 
2 
] 
dx . 
Show that E[v](t ) = 0 for all t > 0, where v is the difference of any two solutions to the original 
IBVP (with inhomogeneous boundary conditions). Then, explain why this proves that the solution 
u (x , t ) found above is unique. (6 marks) 
Question 6 (15 marks) 
Wewish to find the steady state temperature distributionu (x ,y) of a thin copper strip of effectively infinite 
length whose temperature is fixed to u (x , 0) = 0 along the boundary y = 0 and to u (x , 1) = (1 + x2)−1 
along the boundary y = 1. Modelling the steady state distribution as a solution of Laplace’s equation, 
compute the steady state solution as follows: 
(a) Take the complex Fourier transformFx of Laplace’s equation with respect to x (so y is treated as a 
constant) and obtain a differential equation in y only. (2 marks) 
(b) Find the general solution of this differential equation for (Fxu) (p,y). You should have two unknown 
functions of p in your answer. (3 marks) 
(c) Apply Fx to the two boundary conditions and thereby fix both functions of p. Write down the 
Fourier transform of the solution u (x ,y). (5 marks) 
(d) Write down the solution u (x ,y) as an integral over p. Don’t try to evaluate the integral. (2 marks) 
(e) Your result should be written in terms of complex-valued functions, but the solution must clearly 
be real. Show that the imaginary part of your result vanishes identically. (3 marks) 
End of Exam—Total Available Marks = 100 
Page 4 of 4 pages