December 2004 - McGill UNIVERSITY FACULTY OF SCIENCE FINAL...

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Unformatted text preview: McGill UNIVERSITY FACULTY OF SCIENCE FINAL EXAMINATION MATH 263 ORDINARX DIFFERENTIAL E UATIONS AND LINEAR ALGEBRA {LL/1 6A., / fut c t, Examiner: Professor P. Bartcllo 4 Date: Friday December 17, 2004 Associate Examiner: Professor J. X in 2:00 PM — 5:00 PM INSTRUCTIONS Please answer all 7 questions. Calculators are not permitted. Please answer in exam booklets provided. This is a closed book exam. Translation and Regular Dictionaries are permitted. 6. This exam consists of the cover page and 2 pages of 7 questions and 1 Page of a Table. EJ‘FPJN'.‘ MATH 263 Final Exam - 17 Dec. 2004 1. (15 marks) (it) Find the solution of the differential equation 20% + Wiy' = 1;. subject to the initial condition y(1) = 2. (b) Find an implicit expression for the the general solution of the differ— ential equation sin y (Zace‘r'zy'2 + e {Dds + (26%; + cosy 1n w)dy = 0. ((1) Find the solution of the differential. equation dy _ 3112 + 9.162 -— “ 0 six Elyse ’ (EL) ) subject to the initial condition y(l) : .1. 2. (10 marks) Use variation of parameters to solve 2:23;” + ziry’ — cly : 2s: (3: > 0). Note: marks will be awarded based on the method. 3. (15 marks) Consider the equation 1 y“) + at!” = 97(56)- (a) Set g(:r:) : U. i. Convert this to a second order ODE and find its general solution. ii. Using the Wronskian, verify that the solutions obtained are lin— early independent. (1)) Set g(:17) = 2m + 4. Find the solution to the original fourth order ODE using undetermined coefficients. 4. {15 marks) (10 marks) Use Laplace transforms to solve the initial value problem T,_2?_{t 05t<5, J J‘ 5 tZS ’ with initial condition y() : yo. MATH 263 Final Exam - 17 Dec. 2004 5. (10 marks) Find the inverse Laplace transform of (a) 1 (b) 5 (a) (5 marks) Solve the system of equations :13; — 2:32 + 3.713 = 7 —.’B1+.’1’32 — 23:3 = —5 23:1 —.’1’32 —::7;; : 4. (b) (5 marks) Find the eigenvalues and eigenvectors of the matrix 1—1 4 —2' 7. (10 marks} Find the general solution of the system 0f differential equations x :51 + 41'»; 36] I (17,2 2 E + ‘ ’fABLE 6.2.1 Elementary Laplace Transforms I. 10. 11. 13. 14. 16. 17. 18. 19. r”- f0): g_l{F(S)} 1 n 2 positive integer .t”, p > —I .sinat .cosar .sinhar .Coshar . 6‘” sin br 9‘“ cos 19! We“, n 2 positive integer . 11((1‘) ugflffi m C) €L‘If(r) . f (ct) j; f0 — agenda- 5(r b c) f"”(r) (—r)“f(r) - F0) 2 96mm 1 —, S>0 s s>a S>a S>fl (5 —a)' +b“ (S W (HOW-5'1a S > a —{'5 , S > 0 €_“5F(5) F(S—C) 1 s D EF(E)‘ F(S)G(S) C>0 —(.'S 9 SHF(S) _ SH—lf‘(0) ___ . _ _ _ f(Ir—I)(O) FWm ...
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