This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
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 (HOW5'1a 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 ...
View
Full Document
 Summer '09
 SidneyTrudeau
 Differential Equations, Equations, Laplace, initial condition, Elementary Laplace

Click to edit the document details