Math450 - Quiz
July. 01
Name:
1
S/,*dA
1. Consider the second order ODE
a"+y:3sin2r*l-r.
(1) Find the general solution to this equation.
(2) Find the solution to this equation that satisfies the bound
where the din'iensionless parameter
B. ILL (30
i z 3
A
is known in heat transfer as the Biol number. If Bi > 1. then (35) implies that
"LZ ~ 1 :2: O or, returning to dimensional terms.
:72':l
n.(L.l)
Fall 2017
Math 450
G. Silva
Homework 07
To be handed in by Thursday, November 30th, 8:30am (Michigan time)
Name:
For continuously differentiable functions f (x), F (x) and constants > 0, c1 and c2 , w
Fall 2017
Math 450
G. Silva
Homework 06
To be handed in by Thursday, November 9th, 8:30am (Michigan time)
Name:
The Airy differential equation is the ODE
u00 (t) tu(t) = 0.
(1)
(a) (5 points) Verify t
Figure 7. Representative
equipotentials, (tom (24).
with general solution \l/(p) : .rl ~t B hip. [01; we could obtain the latter by setting the
coefficients oi all qi~tlepeiitleiii terms in (15) equal
Figure 3. Branch cut for log 2.
dz ' dz dz ,
~ ~ 2 w : 2m, (l4)
- ca 5 02 3 . C'i-l(-C'2) 3
according to the important little integral in Section 233 so
([2 ' (is
f : / T, (15)
Cl 0 ( C2 1.;
Sure en
EXERCISES 17.2
1. (a) Prove (311).
(c) Prove (3c).
(b) Prove (3b).
(d) Prove (3d).
2. Provide a proof of (4a) that is analogous to the proof of (4b)
given in (5).
3. Prove that
(a) f is both even and
932 Chapter
EXERCISES 17.10
I 7. Fourier Series, Fourier Integral, Fourier Trunsibrm
1. Using (1a) )de1ive the 1esult
1
(1+itu
Fcfw_H(;r)c :
il" Re a, > 0. (This case was worked in Example 2, bu
912
Chapter 17. Fourier Series, Fourier Integral. Fourier Trauslorrn
then how do we know which basis to use? As we shall see in Chapters l8~2(1, that
decision will be based on the mathematical context
880 Chapter 17. 7ourier Series. Fourier Integral. Fourier Transform
8 7G 1 cos III'
22mm W 21
T I; [12 ( )
where the second equality follows front Theorem 17.5.4. Further. the final riglit~hand side.
EXERCISES 18.3
1. Verify by direct substitution that (14) does indeed satisfy
the diffusion equation (la),
2. (On the sign of Sepmation mnrranr) Wiiting the separation
constant as H in (6) rather th
Computer software. To nd the roots of (24) using Maple we can use the fsolve
command, but it is best to include the option of Specifying the interval in which to
search. From Fig. 3, it is evident tha
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23.5. Cauchy Integral Fornm
f(z) : 'u(:r:, g) + ii;(:i;. y) are harmonic, we see that (26) is very close to providing
the solution to the classical Dirichlet problem, namely, V272, : 0 in the interior
14:5, (25)
122-1-9| r. (z 4 31;)(1 +211) 4. 1
2 (2)(2) L
where the second inequality in (24) follows from the fact that cosh y and sinhy are mono~
tonically increasing functions of '1 , and the last e
Name: SLn"
Math 450 - Quiz 2
July 07
1. Let
/(c)
be a periodic function whose value in one fundamental period interval is given by
rf(r\:Ir'
\-'
Sketch the graph of
Sh,be,
and find its Fourier series.
Math 45O - Quiz
July 15
Name:
3
9(*e
1. lFill in b]anks. In this problem you don't need to do any computations.
Consider the diffusion problem
(
cfw_
:4urr,
t) : 0, u,(tr,t) : 0,
"(0,
u1
]
(0<"(r,0(tc