TITLE PAGE NAME: (Print in ink) STUDENT NUMBER: SEAT NUMBER: SIGNATURE: (in ink) (I understand that cheating is a serious offense) INSTRUCTIONS TO STUDENTS: This is a 3 hour exam. Please show your work clearly. No texts, notes, or other aids are permitted. There are no calculators, cellphones or electronic translators permitted. This exam has a title page, 8 pages of questions and two blank pages together with a formulas sheet. Please check that you have all the pages. You may remove the last two pages if you want, but be careful not to loosen the staple. The value of each question is indicated in the left hand margin beside the statement of the question. The total value of all questions is 100 points. Answer all questions on the exam paper in the space provided beneath the question. If you need more room, you may continue your work on the reverse side of the page, but CLEARLY INDICATE that your work is continued. Question Points Score 1 10 2 10 3 10 4 15 5 10 6 10 7 10 8 25 Total: 100
PAGE: 1 of 11 [10] 1. Evaluate the line integral of f (x, y, z) = xy along the curve C, where C is the first octant part of x2 4 + y2 = 1, x 2 + z 2 = 4 from (2, 0, 0) to (0, 1, 2).
PAGE: 2 of 11 2. [10] Let F(x, y) = (e x+y x 2 y) î+(e x+y + 3xy 2 ) ĵ be a force field. Find the work done by F on a particle that travels once counterclockwise around the triangle with vertices (0, 0), (2, 2) and (2, 2).
PAGE: 3 of 11 3. [10] Evaluate S F ˆn ds, where F(x, y, z)=(x+sin y) î+(y2 + cos z) ĵ+(e x z) ˆk and ˆn is the unit outer normal to the surface S enclosing the volume bounded by the parabolic cylinder z=1 x 2 and the planes z=0, y=0, y=2.
PAGE: 4 of 11 4. [15] Solve the differential equation x y + y + y = 0 using Maclaurin series. Simplify as much as possible and find the interval of convergence. Is your answer a general solution? Explain.
PAGE: 5 of 11 [10] 5. Let f (x)= x 4 where 0< x<4. Find coefficients a n so that f (x)= a 0 2 + a n cos nπx 4 n=1 for all x possible. in the interval 0< x<4. Explain your work and simplify as much as
1 x 2 if 1< x<0 [10] 6. Let f (x) = with f (x+2)= f (x). x if 0 x<1 (a) Draw the graph of f (x) for 3 x 3. PAGE: 6 of 11 (b) Describe the function g(x) to which the Fourier series of f (x) converges and draw the graph of g(x) for 3 x 3. (You are not asked to find the Fourier series.)
PAGE: 7 of 11 [10] 7. A string with constant linear density ρ is stretched tightly between the points x = 0 and x=10 on the x-axis. The tension in the string is a constantτ. The displacement of the string at time t=0 is shown in the figure below, and from this position, it is released. The left end of the string is fixed on the x-axis, but the right end is looped around a vertical rod, and can move vertically without friction. A damping force proportional to velocity, and also a restoring force proportional to displacement are taken into account. What is the initial-value problem for displacement y(x, t) of the string? Include the partial differential equation, and all boundary and initial conditions, and include intervals on which they must be satisfied. y 1 0.5 1 2 3 4 5 6 7 8 9 10 x
PAGE: 8 of 11 [25] 8. Solve the initial value problem for transverse vibration of a taut string with the following PDE, initial and boundary conditions. ( Hint: L 0 2 y y t 2= c2 2, 0< x<l, t>0 x 2 y x (0, t)=0, t>0 y(l, t)=0, t>0 y t (x, 0)=0 y(x, 0)= x(l x) (L 2x) sin (2n 1)πx 2L, 0< x<l, 0< x<l dx= 2L2 [(2n 1)π+4( 1) n ] ) (2n 1) 2 π 2
PAGE: 9 of 11 Extra space for solution of question 8
PAGE: 10 of 11 BLANK PAGE FOR ROUGH WORK
PAGE: 11 of 11 Sturm Liouville Systems of form y +λ y=0 Boundary Conditions Eigenvalues Eigenfunctions y(0)=0=y(l) λ n = n2 π 2 y (0)=0=y (L) λ n = n2 π 2 y(0)=0=y (L) λ n = (2n 1)2 π 2 y (0)=0=y(L) λ n = (2n 1)2 π 2, n 1 y L 2 n (x)=sin nπx L, n 0 y L 2 0 (x)=1, y n (x)=cos nπx L, n 1 y 4L 2 n (x)=sin (2n 1)πx 2L, n 1 y 4L 2 n (x)=cos (2n 1)πx 2L