MATH 241 MAKE-UP FINAL JANUARY 12, 2012 (LU, SHATZ) The examination consists of 15 problems, the rst
ten are worth 10 points each, the second ve are worth 20 points each. Answer all of them. The rst 10 are multiple choice, NO PARTIAL CREDIT given, but NO PENALTY FOR GUESSING. To answer, CIRCLE THE ENTIRE STATEMENT YOU DEEM CORRECT in the problem concerned. No work is required to be shown for these 10 problems, use your bluebooks for computations and scratch work. The next 5 are long answer questions, PARTIAL CREDIT GIVENYOU MUST SHOW YOUR WORK. SHOW THE WORK FOR THESE IN THE SPACE PROVIDED ON THE EXAM SHEETS. DO NOT USE BLUE BOOKS FOR THESE 5 QUESTIONS; DO NOT HAND IN YOUR BLUE BOOKS. No books, tables, notes, calculators, computers, phones or electronic equipment allowed; one 8 and 1/2 inch by 11 inch paper allowed, handwritten on both sides, in your own handwriting; no substitutions of this aid allowed. NB. In what follows, we write ux for problems concerning PDE. YOUR NAME (print please): YOUR PENN ID NUMBER: YOUR SIGNATURE: INSTRUCTORS NAME (CIRCLE ONE): LU SHATZ
u x
and uxx for
2u , x2
etc. in those
SCORE: Do not write belowfor grading purposes only. I II III IV V VI VII VIII IX X XI XII XIII XIV XV
TOTAL:
I)
If is the ellipse dz.
cos2 (iz) z 2 iz+2
x2 9
y2 2
= 1, traced counterclockwise, evaluate
a) b) c) d) e) II)
/3 2/3 0 4 4/3 If g(x) = 3 for x < 0 and g(x) = 0 for 0 x < and if
we extend g to be 2 periodic, then when we expand g(x) in a complex ikx , we nd the limit Fourier Series k= ck e limR equals a) b) c) d) e)
2 R k=R ck
0 2 1/2 3/2 The complex function f (z) = zcot2 z is dened for z = 0 near z = 0. z = 0 is a pole of order 2 with residue 0 z = 0 is a removable singularity with residue 0 z = 0 is an essential singularity with residue 1/6 z = 0 is a pole of order 3 with residue 1/6 z = 0 is a pole of order 1 with residue 1.
III) a) b) c) d) e)
IV)
For the Sturm-Liouville Problem y + y = 0, y (0) = y(2) = 0,
the eigenvalues are: a) b) c) d) e) V) = = = =
k 4 ,k k 8 ,k
odd odd odd odd
k2 2 4 ,k k2 2 16 , k
= k 2 2 , k odd. Consider the function f (x) = |x| for < x . We extend f
to be 2 periodic and compute its Fourier Series ao k=1 ak cos(kx) + bk sin(kx). 2 + Then the sum of the innite series a2 b2 is: k=1 k k a) b) c) d) e) VI) 1
2
0
4
2. Consider the heat equation ut = uxx with boundary values u(0, t) =
1 u(, t) = 0 and with initial condition u(x, 0) = sin(3x). Then u( , 2 ) is: 6
a) b) c) d) e)
exp(9/2) exp(7/2) exp(5/2) exp(3/2) exp(1/2)
VII)
Consider Laplaces PDE: uxx + uyy = 0, with three dierent sets
of boundary conditions: (BC1 ) : u(0, y) = 0, u(1, y) = y, u(x, 0) = 0, u(x, 1) = 2x, (BC2 ) : u(0, y) = 0, u(1, y) = y, u(x, 0) = u(x, 1) = 0, (BC3 ) : u(0, y) = u(1, y) = 0, u(x, 0) = 0, u(x, 1) = x. Write u2 for a solution with (BC2 ) and u3 for a solution with (BC3 ). Which combination below satises the DE and (BC1 )? a) b) c) d) e) 2u3 u2 2u2 u3 u2 + u3 u3 u2 2u2 + u3
VIII)
Consider three curves (all traced counter clockwise): 1 : |z| = 2, 2 : |z i| = 2, 3 : |z | = 2.
Dene Ik as a) b) c) d) e)
z 3 exp( 1 ) dz, with k = 1, 2, 3. Then: z2
I1 = I2 = I3 I2 = I3 = I1 I1 = I2 = I3 = I1 I1 = I3 = I2 I1 = I2 = I3 .
IX) We expand the function f (x) = 0 for < x < 0 and f (x) = 1 for 0 < x < in a real Fourier Series: ao + ak coskx + bk sinkx. Then the k=1 2 value of the series ao + a2 bk equals: k=1 k a) b) c) d) e) 3 2 2 1.
X)
Consider the Sturm-Liouville Problem on [1, 2] given by
x2 y + ( x3 )y = 0 and certain boundary conditions. The eigenfunctions, fj (x), satisfy an orthogonality relation
2 1 fj (x)fk (x)w(x) dx
= 0, j = k.
Then w(x) equals: a) b) c) d) e) x 1 1/x
1 x2 1 x3
THE NEXT 5 PROBLEMS ARE LONG ANSWERWORK IS REQUIRED TO BE SHOWN ON THE EXAM SHEETSNOT IN YOUR BLUE BOOKS.
XI)
Compute the integral
cos6x 0 x2 +9
dx.
XII) Let u(x, y) be a harmonic function (that is, it satises Laplaces DE: u = uxx + uyy = 0) in the innite strip: 0 < x < , y 0 and say it satises the partial boundary conditions: u(0, y) = u(, y) = 0 and u(x, y) is bounded as y . a) Find the general solution with just these partial boundary conditions.
b) Now assume the last boundary condition u(x, 0) = 5sin14x. What is the explicit solution with all boundary conditions satised?
c) What is the value of u(x, y) along the midline when x = /2?
XIII) Consider the modied wave equation utt = uxx + cosx on the interval [0, ]. It has a special solution u(x, t) = cosx. a) Solve this DE with the boundary conditions: u(0, t) = 1, u(, t) = 1.
b) If now initial conditions are given: u(x, 0) = cosx and ut (x, 0) = 1, explicitly nd the solution, u(x, t), of the problem.
XIV) Given f (x) = x on [0, ], we extend it to be an even function (f (x) = f (x)) on [, ]. a) Compute its Fourier Series on [, ].
1 1 1 b) What is the sum of the series 1 + 312 512 712 + 912 + 112 132 152 + ?
10
XV) Compute the complex Fourier Series for cosx on the interval [1, 1], where we extend cosx to be periodic with period 2.
END OF THE EXAM (EXTRA SHEET OF PAPER FOLLOWS)
