Physics 228 - Winter 2016
HW 9
Due Wednesday 3/9/16
Except as noted, all problems are in the class text, Boas.
Example problems: Not to be turned in solutions can be found in the back of the
text
13.2:
13.3:
13.4:
13.5:
1, 12
2, 3, 7
2, 5, 6
1, 2, 4, 8, 1
Physics 228, W2016
Solutions to problem set #1
8.1, #7: Solving dp/dt = F we get p(t) = F t + C; with the initial condition
p(0) = 0, we can set the constant C = 0. Then
p= q
m0 v
1 v 2 /c2
c
pc
=q
v= 2
.
2
2
p +m c
1 + m2 c2 /p2
=
(1)
Plugging in p = F t
Name:_
Midterm Exam II Physics 228 Winter 2009
3/4/09
This is a CLOSED book exam. Do NOT open the exam until 11:30 AM. Exams will
not be accepted after 12:25 PM. Be certain to indicate what techniques you have
used to solve each question (but do not write
Physics 228 - Winter 2016
HW 7
Due wednesday 2/24/16
All problems are in the class text by Boas, except as noted.
Example problems: Not to be turned in solutions can be found in the back of the
text, or in Appendix B. See also the Mathematica notebook.
12
Physics 228 - Winter 2016
HW 8
Due wednesday 3/2/16
Recall that Midterm II is thursday 3/3/16
All problems are in the class text by Boas, except as noted.
Example problems: Not to be turned in solutions can be found in the back of the
text. See also the M
Physics 228 - Winter 2016
HW 2
Due wednesday 1/20/16 in class or in Ann Nelsons mailbox by 12:30pm
All problems are in the class text by Boas, except as noted.
Example problems: Not to be turned in solutions can be found in the back of the
text
8.6:
14.2:
Z
S=
4x2
m
mgx2
=
dt (x 2 (1 + 2 ) +
2
a
a
Z
dxt0 (
m
4x2
mgx2
)
(1
+
)
+
2t02
a2
a
Note that x = 1/t0 , dt = t0 dx. Since the Integrand is dependent of t, Eulers equations give
d 0 m
4x2
mgx2
( 0 (t ( 02 (1 + 2 ) +
) = 0
dx t
2t
a
a
which implies that
0
Physics 228, winter 2016
Problem set #4
Due: Wed. 2/10/16
Example problems. Not to be turned in solutions given in the companion volume of solved problems:
9.1:
9.2:
9.3:
9.5:
9.8:
1
5
3, 13
11
4, 23, 25
Assigned problems. All the book problems are to be
Physics 228 - Winter 2016
HW 4
Due Wednesday 2/34/16
Midterm Exam I is Thursday 2/4/16
All problems are in the class text by Boas, except as noted.
Example problems: Not to be turned in solutions can be found in the back of the
text, or in Appendix B. See
Physics 228 - Winter 2016
HW 3
Due Monday 1/27/16
All problems are in the class text by Boas, except as noted.
Example problems: Not to be turned in solutions can be found in the back of the
text. See also the Mathematica notebooks on Laplace transforms a
Physics 228 - Winter 2016
HW 6
Due 2/17/16
All problems are in the class text by Boas, except as noted.
Example problems: Not to be turned in solutions can be found in the back of the
text, or in Appendix B.
12.1:
12.2:
12.4:
12.5:
12.7:
2, 9
4
3
3, 9, 12
Physics 228, Winter 2016
Problem set #1
Due: Wed. Jan 13 in lecture or in Ann Nelsons mail box before 12:30 pm.
May be turned in up until 12:30 pm on Jan 20 for up to 1/2 credit. Note: If
turning in late, please do not put with other HWs but give to Ann N
HW 1
Ex 8.3 : 1
DSolve[y '[x] + y[x] = Exp[x], y[x], x]
y[x]
x
2
+ -x C[1]
Ex 8.4 : 12
DSolve[y '[x] y[x] / x - Tan[y[x] / x], y[x], x]
Solve:ifun : Inverse functions are being used by Solve, so some solutions may not be found; use Reduce for complete so
Name:_
Final Exam Solutions Physics 228 Winter 2009 3/18/09
This is a CLOSED book exam. Do NOT open the exam until 2:30 PM. Exams will
not be accepted after 4:30 PM. Be certain to indicate what techniques you have used
to solve each question (but do not w
Name:_
Final Exam Physics 228 Winter 2009 3/18/09
This is a CLOSED book exam. Do NOT open the exam until 2:30 PM. Exams will
not be accepted after 4:30 PM. Be certain to indicate what techniques you have used
to solve each question (but do not write a the
Name and student number:
Physics 228, Winter 2016
Second Midterm Exam
Thursday 2/4/16
This exam is worth 30 points, and consists of 3 problems. You may use but will not need a
calculator. You may not use books, notes or laptop computers. Possibly useful e
Name and student number:
Physics 228, Winter 2016
First Midterm Exam
SOLUTIONS
1
Thursday 2/4/16
Problem 1. (6 points) Use the residue theorem to evaluate the integral
Z 2
1
d
,
(5 + 4 cos )
0
Hint: To turn this into a contour integral start by writing z
Lecture 21 Power Series Method at Singular Points Frobenius
Theory
10/28/2011
Review.
P
n
The usual power series method, that is setting y =
n=0 an (x x0) , breaks down if x0 is a singular
point. Here breaks down means cannot find all solutions.
Its poss
2u
t
2
= c2 2 u
The Wave Equation
Waves are all around us. We are all familiar with water waves and
sound waves and know that waves on guitar strings produce music.
Also, most of us have heard that electromagnetic radiation can be
thought of as traveling
Physics 116C
Fall 2012
The Spherical Harmonics
1. Solution to Laplaces equation in spherical coordinates
In spherical coordinates, the Laplacian is given by
2
1
1
1
2
2
~
= 2
r
+ 2 2
sin
+ 2 2
.
r r
r
r sin
r sin 2
(1)
We shall solve Laplaces equatio
Physics 228 - Winter 2016
HW 2 Solutions
8.6: 39 Find solution of LRC circuit with applied voltage V=A sin( t )
Use sin( t )=(1/(2 i)(exp(i t )-exp(-i t ) (see mathmatica notebook)
8.6: 42 [And find the specific solution for the initial conditions
!
" y (
Physics 228 Winter 2016
HW 3 Solutions
*14.7: 58
Solution: We want to find an inverse transform for
!
p3
p3
p3
G (p)= 4
=
=
,
p 16 (p 2 + 4 )(p 2 4 ) ( p + 2i )( p 2i )( p + 2 )( p 2 )
using the Bromwich integral (Eq. 14.7.16 in Boas), i.e., using complex
Physics 228
HW 7 Solutions
Thursday 2/26/09
12.9: 2
Solution: Finally we get to expand functions in series of Legendre polynomials
using the full technology. We consider the function
With the definition
we have
For low orders we can just do the integral
P
. First-order ODE.
Determine the general solution to the ODE
2133/(16) + W?) = 563/2-
Determine the value of any integration constants if the boundary condition y(1) = 1
is imposed.
3 ~ 5- 3 e 1:
1' C
an, \QEJ
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1
$0 3. 9 g? Km; : Q
Possibly Useful Information
Integrals:
Z
dx
ax + b
=
Z
1
ln | ax + b |
a
Z
d sin (t ) sin
=
dx
x (ax + b)
=
x
1
ln
b
ax +
b
2 cos t + sin (t 2 )
4
Dirac -function: the Dirac -function (which is only meaningful inside an integral) satisfies
(
Z b
f (x0
CROW Pcztf V
uQ\
S QJ
L 2.8 QMCLWL E Q 19%;? a. ckiB Pk Cu.) K O
uLmk m u inf G I 1 4.
\LL\ upmssk u r Q9 JV. gun, AC3 can 01 smwu . 2% 85k LZRb-gx (imvl
1" EPQKM (LOAN: uLX gwm a A
4\
u w 9-4 4 1 k
g 369an 5 (9&0 5 can: can 1
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\
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