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**Unformatted text preview: **MCGILL UNIVERSITY FACULTY OF SCIENCE FINAL EXAMINATION MATH 315 ORDINARY DIFFERENTIAL EQUATIONS Examiner: Professor J. Hurtubise Date: Monday December 11, 2006
Associate Examiner: Professor G. Schmidt Time: 2:00PM — 5:00PM INSTRUCTIONS 1. Please answer all questions in the exam booklets provided. 2. This is a closed book exam.
3. Calculators are not permitted.
4. Use of a regular and/ or translation dictionary is not permitted. 5. This exam comprises the cover page, and 1 page of questions and 1 page of the
table of Elementary Laplace Transforms. MATH 315, FINAL EXAMINATION December 11th, 2006
1. Solve the following differential equations, with initial conditions, if given: a- (10 MS) 31’ = $23? 11(1) = 1
b. (10 pts) (3: + 2)sin(y)da: + a: cos(y)dy = 0 c. (10 pts) y” + 411’ + 4y = 1726—” d. (10 pts) I
(:2) = (i :3) (3:) e. (10 pts) y’” — y” — y’ + y = 26” + 3, given that one solution of the
homogeneous equation is y = e””. f. (10 pts) (353+ 1)y” — (9x+6)y’ +9y = 0, given that y1(:1:) = 63“? is a solution. 2. Solve in a series centred at :1: = 0. You must ﬁrst decide whether to use a regular power series or a Frobenius series. Discuss the radius of convergence of the
solutions. a. (10 pts) y" — ny’ + Ag 2 0, where /\ is a constant.
b. (10 pts) 29321;” + 3533/ +- (23:2 — 1)y = 0 3. a. (5 pts) Compute the inverse Laplace transform of
2(s — 1)e—29
32 — 2s + 2
b. (5 pts) Compute the Laplace transform of the square wave f f(m) = 1 + Z(—1>kuk(w)
k=1 c. (10 pts) Solve the initial value problem
11" + 21/ + 2y = 608(37) + 5(96 — 7r/2), 11(0) = 0, y'(0) = 0
Solve the same problem, but with initial conditions y(0) = 1, y’(0) = 0. MATH 319 Ordinary! DIHW+QJ 5%waﬁ0nM _ meet-”5‘7; -‘ TABLE 6.2.1 Elementary Laplace Transforms f(t) = £‘1{F(s)} F(s) = mm} Notes
1 .
1.1 —, s > 0 Sec. 6.1;Ex.4
s
1
2. e‘" , s > a Sec. 6.1; Ex. 5
s —- a
n' 3. t"; n 2 positive integer snPl , s > 0 Sec. 6.1; Prob. 27 I:
F 1 ': ,
4. tp, p > —1 b, s > 0 Sec. 6.1; Prob. 27
Sp+1 .'
. a
5. smat 2 2 , s > 0 Sec. 6.1; Ex. 6
s +a
6. cosat 2 s 2, s >,0 Sec. 6.1;Prob.6 E
s +a , .
7. sinhat 2 a 2, s > |a| Sec. 6.1; Prob. 8
s —a
8. coshat 2 s 2, s > [a] Sec. 6.1;Prob. 7 Iv
s ~‘a
at - b Fl.
9. e s1nbt —-§——2, s > a Sec. 6.1; Prob. 13 _
(s — a) + b 5-
s — a
10. e‘" cos bt 2 2, s > a Sec. 6.1; Prob. 14
(s — a) + b -
I 1".
11. t"e‘", n 2 positive integer n. 1, s > a Sec. 6.1; Prob. 18 -
(s — a)"+ -.
e—CS
12. L460?) , s > 0 Sec. 6.3
s
13. uc(t)f(t — c) e'”F(s) Sec. 6.3
14. ec‘f(t) F(s — c) Sec. 6.3 . I
If:'
1
15. f(ct) —F (5), c > 0 Sec. 6.3; Prob. 19;}.
c c ._-., .
t I‘ I
16. f0 fa — r)g(r)d1: F(s)G(s) Sec. 6.6 '
17. 6(t — c) e‘“ Sec. 6.5
18. f(")(t) s"F(s) —s"_1f(0)—---— f("‘1)(0) Sec. 6.2 19. (—t)"f(t) F01) (3) Sec. 6.2; Prob. 287$ ...

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