Math 257/316 Final Exam, April 21 2015
Last Name:
First Name:
Student Number:
Signature:
Instructions. The exam lasts 2.5 hours. No calculators or electronic devices of any kind are
permitted. A formula sheet is attached. There are 12 pages in this test including this cover page, blank
pages, and the formula sheet. Unless otherwise indicated, show all your work.
Rules governing examinations
• Each examination candidate must be prepared to produce, upon the request of
the invigilator or examiner, his or her UBCcard for identification.
• Candidates are not permitted to ask questions of the examiners or invigilators,
except in cases of supposed errors or ambiguities in examination questions, illegible
or missing material, or the like.
• No candidate shall be permitted to enter the examination room after the expiration
of one-half hour from the scheduled starting time, or to leave during the first half
hour of the examination. Should the examination run forty-five (45) minutes or less,
no candidate shall be permitted to enter the examination room once the examination
has begun.
• Candidates must conduct themselves honestly and in accordance with established
rules for a given examination, which will be articulated by the examiner or invigilator
prior to the examination commencing. Should dishonest behaviour be observed
by the examiner(s) or invigilator(s), pleas of accident or forgetfulness shall not be
received.
• Candidates suspected of any of the following, or any other similar practices, may
be immediately dismissed from the examination by the examiner/invigilator, and
may be subject to disciplinary action:
(a) speaking or communicating with other candidates, unless otherwise authorized;
(b) purposely exposing written papers to the view of other candidates or imaging
devices;
(c) purposely viewing the written papers of other candidates;
(d) using or having visible at the place of writing any books, papers or other
memory aid devices other than those authorized by the examiner(s); and,
(e) using or operating electronic devices including but not limited to telephones, calculators, computers, or similar devices other than those authorized by
the examiner(s)–(electronic devices other than those authorized by the examiner(s)
must be completely powered down if present at the place of writing).
• Candidates must not destroy or damage any examination material, must hand
in all examination papers, and must not take any examination material from the
examination room without permission of the examiner or invigilator.
• Notwithstanding the above, for any mode of examination that does not fall into
the traditional, paper-based method, examination candidates shall adhere to any
special rules for conduct as established and articulated by the examiner.
• Candidates must follow any additional examination rules or directions communicated by the examiner(s) or invigilator(s).
Problem #
Value
1
18
2
16
3
24
4
18
5
24
Total
100
Grade
1
1. Find the first four non-zero terms of each of two linearly independent series solutions of
00
0
2xy + 3y + y = 0
centred at x = 0.
2
(Blank page)
3
2. For the function
f (x) =
⇢
0 0x<1
1 1x2
(a) Sketch its even and odd 4-periodic extensions.
(b) Compute the first two terms in each of its Fourier cosine and sine series.
(c) Find the values of the Fourier cosine and sine series at x = 0, x = 1, and x = 2.
4
(Blank page)
5
3. For the wave equation problem
8
utt = 4uxx , 0 < x < 2
>
>
<
ux (0, t) = ux (2, t) = 0, ⇢
1 0x1
>
>
: u(x, 0) = 0, ut (x, 0) = 0 1 < x  2
(a) Find the solution.
(b) Describe how to find a finite-di↵erence approximate solution to this problem. Use
notation ukn = u(xn , tk ) to denote values of u at grid points (xn = n x, yk = k t).
6
(Blank page)
7
4. For the Laplace equation
8
< uxx + uyy = 0, 0 < x < ⇡, 0 < y < ⇡
uy (x, 0) = uy (x, ⇡) = 0
:
ux (0, y) = 0, ux (⇡, y) = 1 + a sin y
(a) For which value of a is there a solution?
(b) Find the solution.
8
(Blank page)
9
5. For the following heat equation with source term and non-zero, mixed boundary conditions:
8
< ut = 4uxx + e t sin( ⇡2 x), 0 < x < 1,
u(0, t) = 0, ux (1, t) = 1,
: u(x, 0) = 0
(a) Find the steady-state v(x) that the solution u(x, t) will converge to as t ! 1.
(b) Write u(x, t) = v(x)+ w(x, t) and find the problem (equation, boundary conditions,
and initial conditions) that w(x, t) solves.
(c) Use the method of eigenfunction expansion to find w(x, t).
10
(Blank page)
11
Formula sheet - final exam
constant coefficients
ay 00 + by 0 + cy = 0
ar2 + br + c = 0
y = Aer1 x + Ber2 x
y = Aerx + Bxerx
e x [A cos(µx) + B sin(µx)]
Euler eq
ax2 y 00 + bxy 0 + cy = 0
ar(r 1) + br + c = 0
y = Axr1 + Bxr2
y = Axr + Bxr ln |x|
x [A cos(µ ln |x|) + B sin(µ ln |x|)]
sin2 t + cos2 t = 1
sin t = 12 (1 cos(2t))
cosh2 t sinh2 t = 1
2
sinh t = 12 (cosh(2t) 1)
2
y1 (x) =
n=0
P1
an (x
x!x0
x0 )
n+r1
where a0 = 1.
r2 = 0:
bn (x
n=1
1
X
x0 ) +
n=0
1
X
bn (x
bn (x
x0 )n+r2 where b0 = 1.
x0 )n+r2 for some b1 , b2...
x0 )n+r2 where b0 = 1.
r2 is a positive integer:
x0 ) +
n=0
1
X
y2 (x) = ay1 (x) ln(x
Case 3: If r1
y2 (x) = y1 (x) ln(x
Case 2: If r1
y2 (x) =
The second linerly independent solution y2 is of the form:
Case 1: If r1 r2 is neither 0 nor a positive integer:
x!x0
Regular singular point x0 : Rearrange (?) as:
(x x0 )2 y 00 + [(x x0 )p(x)](x x0 )y 0 + [(x x0 )2 q(x)]y = 0
If r1 > r2 are roots of the indicial equation:
r(r 1) + br + c = 0 where
b = lim (x x0 )p(x) and c = lim (x x0 )2 q(x) then a solution of (?) is
Ordinary point x0 : Two linearly independent solutions of the form:
P1
y(x) = n=0 an (x x0 )n
Series solutions for y 00 + p(x)y 0 + q(x)y = 0 (?) around x = x0 .
ODE
indicial eq.
r1 6= r2 real
r1 = r 2 = r
r = ± iµ
Basic linear ODE’s with real coefficients
sin(↵ ± ) = sin ↵ cos ± sin cos ↵
cos(↵ ± ) = cos ↵ cos ⌥ sin sin ↵.
sinh(↵ ± ) = sinh ↵ cosh ± sinh cosh ↵
cosh(↵ ± ) = cosh ↵ cosh ± sinh sinh ↵.
Trigonometric and Hyperbolic Function identities
Math 257-316 PDE
1
n=1
1
X
bn sin(
n⇡x
),
L
bn =
2
L
0
Z
L
f (x) sin(
ODE: [p(x)y 0 ]0 q(x)y + r(x)y = 0, a < x < b.
0
BC:
↵1 y(a) + ↵2 y 0 (a) = 0,
1 y(b) + 2 y (b) = 0.
0
Hypothesis: p, p , q, r continuous on [a, b]. p(x) > 0 and r(x) > 0 for
x 2 [a, b]. ↵12 + ↵22 > 0. 12 + 22 > 0.
Properties (1) The di↵erential operator Ly = [p(x)y 0 ]0 q(x)y is symmetric
in the sense that (f, Lg) = (Lf, g) for all f, g satisfying the BC, where (f, g) =
Rb
f (x)g(x) dx. (2) All eigenvalues are real and can be ordered as 1 < 2 <
a
· · · < n < · · · with n ! 1 as n ! 1, and each eigenvalue admits a unique
(up to a scalar factor) eigenfunction n .
Rb
(3) Orthogonality: ( m , r n ) = a m (x) n (x)r(x) dx = 0 if m 6= n .
(4) Expansion: If f (x) : [a, b] ! R is square integrable, then
Rb
1
X
f (x) n (x)r(x) dx
f (x) =
cn n (x), a < x < b , cn = a R b
, n = 1, 2, . . .
2 (x)r(x) dx
n=1
a n
Sturm-Liouville Eigenvalue Problems
n⇡x
) dx.
L
PDE: utt = c2 uxx , 1 < x < 1, t > 0 IC: u(x, 0) = f (x), ut (x, 0) = g(x).
R x+ct
1
SOLUTION: u(x, t) = 12 [f (x + ct) + f (x ct)] + 2c
g(s)ds
x ct
D’Alembert’s solution to the wave equation
Sf (x) =
For f (x) defined in [0, L], its cosine and sine series are
Z
1
a0 X
n⇡x
2 L
n⇡x
Cf (x) =
+
an cos(
), an =
f (x) cos(
) dx,
2
L
L
L
0
n=1
Theorem (Pointwise convergence) If f (x) and f 0 (x) are piecewise continuous, then F f (x) converges for every x to 12 [f (x ) + f (x+)].
Parseval’s indentity
Z
1
1 L
|a0 |2 X
|f (x)|2 dx =
+
|an |2 + |bn |2 .
L L
2
n=1
Let f (x) be defined in [ L, L]then
a 2L-periodic
P1its Fourier series F f (x) isn⇡x
function on R: F f (x) = a20 + n=1 an cos( n⇡x
)
+
b
sin(
n
L )
R
RL
1 L
n⇡x
1 L
n⇡x
where an = L L f (x) cos( L ) dx and bn = L L f (x) sin( L ) dx
Fourier, sine and cosine series