Appendix D – Examination
Information
Formula Hint Sheet similar to that on the examination paper
Some questions may explicitly require derivations of some of these formulae. However any
formula below may be used without proof in any question. Such formulae may be cross-referenced
as HintSheet(1) or HS(1) etc.
cos(x − y) = cos x cos y + sin x sin y ,
sin(x − y) = sin x cos y − cos x sin y
cosh x = 21 (ex + e−x ) ,
sinh x = 21 (ex − e−x )
(uv)′′ = uv ′′ + 2u′ v ′ + vu′′
Z
∞
0
δ(t − a) f (t) dt = f (a) ,
(1)
(2)
(3)
(a ≥ 0)
P (D) (eαx f (x)) = eαx P (D + α)f (x)
Z ∞
f (t) e−st dt
F (s) = L {f } =
(4)
(5)
(6)
0
1
,
(s > a)
s−a
n!
(s > 0)
L {tn } = n+1 ,
s
s
ω
L {cos ωt} = 2
,
L {sin ωt} = 2
,
(s > 0)
2
s +ω
s + ω2
a
s
,
L {sinh at} = 2
,
(s > |a|)
L {cosh at} = 2
2
s −a
s − a2
½ ¾
½ 2 ¾
df
d f
L
= s2 F (s) − sf (0) − f ′ (0)
= sF (s) − f (0) ,
L
dt
dt2
½Z t
¾
1
(s > 0)
L
f (t) dt = F (s) ,
s
0
©
ª
L eat f (t) = F (s − a)
© ª
L eat =
L {H(t − a)f (t − a)} = e−as F (s) ,
(a ≥ 0)
L {δ(t − a)f (t)} = e−as f (a) ,
(a ≥ 0, s ≥ 0)
½
¾ Z ∞
f (t)
L
F (s) ds ,
L {tf (t)} = −F ′ (s)
=
t
s
µ
¶
µ ¶
Z L
cos nωt
L
an
1
f (t)
dt ,
Nn = (1 + δn0 )
=
sin nωt
Nn 0
2
bn
µ
¶
1
1
1
1
=
−
(s + a)(s + b)
b−a s+a s+b
µ
¶
1
1 1
s
=
−
s(s2 + a2 )
a2 s s2 + a2
µ
¶
1
1
1
1
= 2
−
s2 (s2 + a2 )
a
s2 s2 + a2
120
(7)
(8)
(9)
(10)
(11)
(12)
(13)
(14)
(15)
(16)
(17)
(18)
(19)
(20)
Appendix D:
121
EXAMINATION INFORMATION
MATH2067 Part 1, DEs
Examination Information
April 13, 2016
The Part 1 (Differential Equations) examination is nominally 1 hour long, comprises
4 questions worth approximately equal marks, and is held in the same 2 hour session as the
exam for Part 2 of the course. Past papers are not available, but typical exam questions are
indicated below. The best examination preparation is to work through the examples
given in lectures and Practice Classes, the relevant exercises using the Solution
Pointers, and sample exam questions starting on page 123. The Exercise Set questions
tagged A (Additional/Advanced) will generally strengthen your capabilities, but are optional
and should be left until last if attempted. A Hint Sheet like that one page back will be included
on the examination paper.
Pre-examination consultation: You may consult your lecturer at any time when he is available; or email ron.james@sydney.edu.au to get help. Formal pre-exam consultation times will
be advertised in the last week of semester, so check the Course Messages.
The examinable topics on which the 4 exam questions are based are given below. Each topic
is followed by a list of the main skills from which the question is likely to be composed. The
most-relevant Exercise-Set questions are shown in [...] after each exam question description.
Questions may be expressed in a physical or other context, but in such cases the governing
mathematical equations will be given either symbolically or verbally (e.g. boundary conditions
in Q4). The accent will be on problem solving, rather than stating or deriving theoretical results.
Mathematica: The DEs exam contains some small Mathematica parts, amounting to about
10-15% of the available marks. Typically you will be asked to supply commands (just a few
lines) associated with an earlier part of the same question. The commands will be selected from
the Mathematica section §0.5 of the printed notes, but you will only be tested on the syntax
and commands used in the example files and exercises.
Q1: Second order ODEs: Know how to solve DEs with constant coefficients, and obtain
solutions like those in Tables 2.3.1 on page 20, and 2.5.1 on page 23 (neither of which will
be given). Know how to write solutions in various forms, complex and real, using Euler’s
formula for complex exponentials and definitions of hyperbolic functions. The tables need not
be fully memorized, but it is advantageous to memorize the 2nd solution in the double root case,
i.e. x y1 . Know how to determine with reasoning whether a system is un-, over-, under- or
critically-damped, or none of these. For inhomogeneous DEs, know how to apply the Method of
Undetermined Coefficients using trial yp as in Table 2.9.1 on page 27 (which will not be given),
and the Exponential Shift formula (16) on page 26 (and in the Hint Sheet). Know how to
solve 2nd -order DEs with a driving force F0 cos ωt or F0 sin ωt. For forced systems, know how to
determine whether pure resonance occurs. Be able to solve DEs subject to boundary or initial
conditions. [Set2, QQ 2,4,5,6; PCs 1,2]
Typical exam question: Similar to Assignment Q1, but possibly involving other material,
e.g. something on resonance like Set2, Q6.
Mathematica part: main syntax & commands are =, ==, DSolve, y’[x], Plot, /.
See files SHM.m on page 19, SHMunitpulse.m and similar in Set6 Q4 and in Assignment Q1.
Appendix D:
EXAMINATION INFORMATION
122
Q2: Laplace transforms: There is no need to memorize the definition of the transform, or the
transform of any particular function. Know how to derive transforms and inverse transforms, as
in Set6 QQ 1,3. Know how to use transforms to solve IVPs, including ones with RHS involving
H(t) or δ(t). It is advantageous to be able to perform simple partial fractions decompositions
using 2 factors, but PFD formulae will be given either in the question or the Hint Sheet. [Set6:
QQ 1–6, PC3]
Typical exam question: Derive one or a few of the non-Advanced transform results in Set6
Q1. Using transform results like those in Set6 Q1, solve some IVP, possibly involving H(t) or
δ(t), and possibly a series similar to Assignment Q2, or Example 6.4.2 on page 43.
Mathematica part: main commands are those in Set6 Q2, plus the Apart and Sum commands,
e.g. as used in Set6 Q3(c), and Assignment Q2.
Q3: Fourier series: Know how to construct cosine-sine series and half-range series, following
the recipe given in Practice Class 5. Know how to sketch the appropriate periodic extensions if
relevant. Know which series to use based on the period and fundamental frequency, and odd or
even considerations. It is advantageous to memorize the Euler formulae for Fourier coefficients,
i.e. (7) on page 52, including the normalization factors as in the rule-of-thumb on page 60; but
these are also in the Hint Sheet. Be able to evaluate the coefficients, possibly using integration by
parts. At most one integration by parts, if any, will be required. Know the behaviour of Fourier
series near finite jump discontinuities – e.g. see Figure 10.1.1 on page 54 and Theorem 10.1.1
on page 54 (but you need not remember this theorem in all its detail. [Set10: QQ 1,2,4,5,6;
PC’s 3,4]
Typical exam question: Find a Fourier series for some simple function. To reduce integration
time, it is likely (a) that the given function will be even or odd (e.g. Set10, QQ 1(a),2), or
that a half-range expansion will be required (e.g. Set10, QQ 4,5,6); and (b) that functions
to be integrated will be linear (e.g. Example 10.5.2 on page 60), or piecewise linear (e.g.
Example 10.1.1 on page 53 and Set10 QQ 1,4).
Mathematica part: main syntax & commands are a[n ] to define functions, := to delay
evaluation, Integrate, Sum, Plot, Evaluate, FullSimplify, {...} and Table to form lists,
/.{Sin[n*Pi]->0,Cos[n*Pi]->(-1)^n} to apply rule lists. See sample file fourierHa.m on
page 55, and the Mathematica parts of Set10.
Q4: Separation of variables: Know how to solve the wave, diffusion (i.e. heat) and Laplace
equations with 2 independent variables (i.e. x, y or x, t). There is no need to memorize the
form of any of these PDEs. To answer such questions within the allotted time, you must be
able to give streamlined solutions as demonstrated in lectures for Example 11.5.3 on page 75
and later examples, and in the templates used for the Week7 Labs. Be able to clearly state the
equations defining the mathematical model, including boundary or initial conditions that might
be given in verbal form in the question. [Set11: QQ 1,3,4.]
Typical exam question: like the vibrating string Example 11.3.1 on page 73, or any one part
of Set11 QQ 1(a),3(a)–(d), 4(a)–(c), with an easy integral.
Mathematica part: Similar to that required by Exam Q3, and as in file stringvibes.m on
page 74, and Set11 Q1(c) (but not the A part).
Sample exam questions follow
Appendix D:
123
EXAMINATION INFORMATION
Sample Exam Questions
Q1 Samples
S1. Assignment Q1. [Ans: see Assignment solutions.]
S2. (a) Consider a forced oscillator with response y(t) satisfying
d2 y
+ 16y = 5 cos 4t .
dt2
(1)
State with reasoning (but before solving), whether or not you expect this system to
exhibit pure resonance.
(b) Find the general solution of (1).
(c) Find the solution of (1) if the system is initially at rest undisplaced , i.e.
y(0) = y ′ (0) = 0.
(IC1,2)
(d) Give Mathematica commands, about 5 lines, to find the solution y(t) of the IVP
comprising (1),(IC1,2), and to plot y(t) for 0 ≤ t ≤ e2 .
Ans: See Solution Pointer for Set2, Q6
(a) Pure resonance occurs because
√ driving frequency 4 (in cos 4t on the RHS of (1))
equals the natural frequency 16 (from 16y on the LHS of (1)).
(b) yc = c1 cos 4t + c2 sin 4t, yp = (5/8)t sin 4t. So GS is y = c1 cos 4t + (c2 + (5/8)t sin 4t)
(c) Applying (IC1,2) to GS yields y = (5/8)t sin 4t.
(d) Mathematica soln is similar to Assignment soln except
Plot[y[t] /. ans, {t,0,E^2}]
S3. Redo S2 for ÿ + 5ẏ + 4y = 5 cos 2t.
Ans: See Solution Pointer for Set2, Q6
(a) Pure resonance does not occur because damping 5ẏ is present.
(b) yc = c1 e−4t + c2 e−t , yp = (1/2) sin 2t. So GS is y = c1 e−4t + c2 e−t + (1/2) sin 2t.
(c) Applying (IC1,2) to GS yields y = (1/3)(e−4t − e−t ) + (1/2) sin 2t.
(d) Mathematica soln is similar to Assignment soln & S2 above.
Appendix D:
124
EXAMINATION INFORMATION
Q2 Samples
S1. Like Assignment Q2 but shorter.
S2. (a) Use the definition of the Laplace transform F (s) of f (t), to prove one or two results
(without the A ) from Table 1 on page 48.
(b) Consider a system that is undisturbed for t < 0, but subject to a sequence of 10
increasing magnitude impulses for t > 0 . The governing initial value problem for the
system response y(t) is
10
X
y ′′ + 2y ′ − 3y = t
n=1
δ(t − n) ,
y(0) = y ′ (0) = 0 .
(2)
Use Laplace Transforms and a partial fraction decomposition to find y(t) as a series
involving H(t) . If 2 ≤ t < 3 , give a formula for y not involving H(t) or Σ-notation.
(c) Give Mathematica commands, 2 or 3 lines, for (i) finding the transform of the RHS
of the d.e. in (2), and (ii) checking your PFD in part (b).
Ans:
(a) See Solution Pointers for Set6 Q1.
(b) Transform using Q1 on page 48 – (g) with n = 1, 2 and f, F replaced by y, Y ; and
(k) with a = n and f (t) ≡ t:
2
(s + 2s − 3)Y =
10 Z
X
∞
−st
te
n=1 0
Solve & use PFD:
Y = (1/4)
10
X
n=1
ne−ns
µ
δ(t − n) dt =
1
1
−
s−1 s+3
10
X
ne−ns
n=1
¶
Inverse transform using Q1 on page 48 – (j) with a = n; and (a) with a = 1, −3:
y = (1/4)
10
X
n=1
n H(t − n)(et−n − e−3(t−n) )
If 2 ≤ t < 3 , then H(t − n) = 0 for n ≥ 3 and
y = (1/4)
2
X
n=1
n H(t − n)(et−n − e−3(t−n) ) = (1/4)(et−1 − e3−3t + 2et−2 − 2e6−3t ).
(c) (i) r = t*Sum[DiracDelta[t-n],{n,1,10}];
LaplaceTransform[r,t,s]
(ii) Apart[1/(s^2+2s-3),s] or Apart[1/((s+3)(s-1)),s]
Appendix D:
EXAMINATION INFORMATION
125
Q3 Samples
S1. Consider the function f (t) = 1 + t , 0 ≤ t ≤ 2 .
(a) Sketch f (t) and the periodic extension F (t) relevant to the Fourier half-range sine
expansion, r(t) = Σn . . . say, of f (t).
Find r(t) and evaluate its Fourier coefficients, expressing your answer in trigonometricfree form. Let SN denote the partial sum of r(t) over terms n ≤ N . Write out
explicitly SN comprising the first 3 non-zero terms of r(t).
(b) State the values to which the series r(t) converges for all t. On the same graph as
in part (a), sketch approximately a partial sum, e.g. S10 . Indicate whether a Gibbs
phenomenon is present or not.
(c) Mathematica part might be similar to one of:
(i) Assume that the Fourier coefficients in part (a) have been defined as a Mathematica function b[n] of their index n. Give Mathematica commands (about 3
lines) for (i) finding S20 , and (ii) plotting S20 and f (t) on the same coordinate
frame over 0 ≤ t ≤ 2.
(ii) Assume that the Fourier coefficients in part (a) have been defined as a Mathematica function b[n] of their index n. Give Mathematica commands (about 3
lines) for (i) defining SN as a function of N , and (ii) plotting f (t) and SN , (N =
1, . . . , 5) on the same coordinate frame over 0 ≤ t ≤ 2.
(iii) Give Mathematica commands (about 2 lines) for finding b[n] by integration,
including simplifications of sin nπ and cos nπ.
Ans:
(a) Sketch f = 1 + t for 0 < t < 2, and continue that as an odd function for −2 < t < 0.
Repeat that shape with period p = 4, to produce the odd periodic extension F (t) of
f (t) as shown in Figure S1 on the next page. Here ω = 2π/p = 2π/4 = π/2. Since F
is odd, we write (with Thm 10.1.1 in mind)
F =
∞
X
bn sin
n=1
nπt
2
2
nπt
dt = · · · by parts
2
0
(
−4/(nπ), n even,
2
=
(1 − 3 cos nπ) =
nπ
8/(nπ),
n odd.
1
bn =
1
Z
(1 + t) sin
The 1st 3 non-zero terms are
µ
¶
3πt
2
πt
4
S3 =
− sin πt + 3 sin
4 sin
π
2
2
(3)
Appendix D:
126
EXAMINATION INFORMATION
(b) If r(t) is the Fourier series on the
RHS of (3), then, by Thm 10.1.1
on page 54,
(
F (t) , t 6= 0, ±2, ±4, ±6, . . .
r(t) =
0,
otherwise, i.e. at the discties
3
2
1
-2
-1
1
2
-1
A Gibbs phenomenon overshoot
is present in the partial sums
plots near the jump discontinuities, e.g. as shown by S10 in Figure S1.
-2
-3
Figure S1
(c) Mathematica part, one of:
(i) S20 = Sum[b[n]*Sin[n*Pi*t/2],{n,1,20}];
flist = {S20,1+t};
Plot[Evaluate[flist],{t,0,2}]
(ii) S[myN_] := Sum[b[n]*Sin[n*Pi*t/2],{n,1,myN}]
flist = {1+t, Table[S[n],{n,1,5}]};
Plot[Evaluate[flist],{t,0,2}]
(iii) b[n_] := Integrate[(1+t)Sin[n*Pi*t/2],{t,0,2}]
FullSimplify[b[n] /. {Sin[n*Pi] -> 0, Cos[n*Pi] -> (-1)^n}]
S2. Redo S1 for the half-range cosine expansion.
Ans:
(a) Sketch f = 1 + t for 0 ≤ t ≤ 2 and continue that as an even function for −2 ≤ t ≤ 0.
Repeat that shape with period p = 4, to produce the even periodic extension F (t) as
shown in Figure S2 on the next page. Here p = 4, ω = 2π/p = 2π/4 = π/2. Since F
is even, we write (with Thm 10.1.1 on page 54 in mind)
F =
∞
X
an cos
n=0
nπt
2
(4)
Z
1 2
(1 + t) cos 0 dt = 2 , by direct integration
a0 =
2 0
Z
nπt
1 2
(1 + t) cos
dt = · · · by parts
an>0 =
1 0
2
(
0,
n even,
4
=
(cos nπ − 1) =
2
2
(nπ)
−8/(nπ) , n odd.
The 1st 3 non-zero terms, since a2 = 0, are
8
S3 = 2 − 2
π
µ
πt 1
3πt
cos
+ 9 cos
2
2
¶
Appendix D:
127
EXAMINATION INFORMATION
3
(b) If r(t) is the Fourier series on the
RHS of (4), then by Thm 10.1.1,
2.5
r(t) = F (t) , ∀t .
2
There is no Gibbs phenomenon,
and the graph in Figure S2 shows
close agreement between F (t)
and S10 .
1.5
-2
-1
1
2
Figure S2
Q4 Samples
This question may be worded in a physical context, but the mathematical model will usually be
given symbolically. Typically it will be a simplified version of the vibrating string Example 11.3.1,
or any one part of Set11 Q’s 1(a),3(a)–(d), 4(a)–(c). It may involve completing a template as
in the Week7 Laboratory classes. The Mathematica part, if any, might involve Fourier material
as in Q3.
S1. Consider a uniform rod with ends at x = 0, 1, and with temperature distribution u(x, t)
that satisfies the diffusion equation ut = uxx . Suppose that at t = 0 the rod’s temperature
is a constant T0 for 0 ≤ x ≤ 1, and that for t > 0 the rod is fully insulated except that the
right end is kept at zero temperature. The governing mathematical model is:
∂u
∂2u
=
,
∂t
∂x2
ux (0, t) = u(1, t) = 0 ,
u(x, 0) = T0 ,
(5)
t>0,
0<x<1,
(BC1,2)
(IC)
where ux = ∂u/∂x etc. By trying fundamental solutions X(x)T (t), determine u(x, t) for
t > 0, as a series with coefficients An say. Evaluate An by integration, and write out
explicitly the first 2 non-zero terms of your series.
Ans: See Solution Pointer for Set11, Q3(c).
u(x, t) =
X
n=1,3,5,...
An cos
nπx −(nπ/2)2 t
e
,
2
where, since there is no n = 0 term,
Z 1
nπx
4T0
nπ
1
T0 cos
dx =
sin
.
An =
(1/2) 0
2
nπ
2
¶
µ
3πx −(3π/2)2 t
πx −(π/2)2 t 1
4T0
So u(x, t) = π cos
e
− cos
e
+ ...
2
3
2