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The University of British Columbia
Final Exam Name (print): Math 316, Sect 201, April 12, 2006 Student No.:
Closed book examination. Time 2.5 hours. There are five questions worth a total of 100 marks. No notes allowed. Non-programming calculator is not needed (but is allowed). One self-made formula sheet can be used. In all questions, you must show work — i.e. display intermediate results — to get full credit. You may use one letter sized formula sheet and a calculator. Be neat! I will not attempt to decipher messy calculations.
Rules governing examination
- Each candidate should be prepared to produce his/her own library/AMS card upon re- quest.
- No candidates shall be permitted to enter the examination room after the expiration of one half hour, or to leave during the first half hour of examination.
- Candidates are not permitted to ask questions to invigilators, except in cases of supposed errors or ambiguities in examination questions.
- CAUTION — Candidates guilty of any of the following or similar practices shall be immediately dismissed from the examination and shall be liable to disciplinary action: (a) Making use of any books, papers or memoranda, other then those authorized by the examiner; (b) speaking or communicating with other candidates; (c) purposely exposing written papers to the view of other candidates.
- Smoking is not permitted during examination.
Question 1 2 3 4 5 6 Mark Total 8 14 16 16 22 24 100
Final Exam Name (print): Math 316, Sect 201, April 12, 2006 Student No.:
Problem 1. Find all singular points of the given equation and determine whether each one is regular or irregular.
(x^2 + x − 2)y′′^ + (x + 1)y′^ + 2y = 0.
Final Exam Name (print): Math 316, Sect 201, April 12, 2006 Student No.:
Problem 3. Consider a bar of length 1 whose left end is kept at zero degrees and whose right end is insulated, with some initial heat distribution f (x). State the initial-boundary value problem appropriate for this situation. Find the temperature distribution in the bar at any time t. (Take c^2 = 1.)
Final Exam Name (print): Math 316, Sect 201, April 12, 2006 Student No.:
Problem 4. Solve the initial boundary value for the wave equation utt = c^2 uxx, 0 < x < 1, t > 0 with boundary conditions u(0, t) = u(1, t) = 0 and initial conditions u(x, 0) = f (x), ut(x, 0) = g(x), where
f (x) = 2 sin 2πx + 3 sin 3πx, g(x) = sin πx.
- Final Exam Math 316, Sect 201, April 12,
- Final Exam Math 316, Sect 201, April 12,
- Final Exam Math 316, Sect 201, April 12,
- Final Exam Math 316, Sect 201, April 12,
Final Exam Name (print): Math 316, Sect 201, April 12, 2006 Student No.:
Problem 6. Solve the vibrating problem for circular membrane of radius 1 with c = 1, clamped along its circumference for the given initial data
u(r, θ, 0) = J 3 (α 32 r) sin 3θ, ut(r, θ, 0) = 0,
where α 32 is the second positive zero of function J 3.
