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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 17, 2007
Associate Examiner: Professor G. Schmidt Time: 2:00 PM TO 5:00 PM INSTRUCTIONS Answer questions in the exam booklets provided.
Calculators are not permitted.
This is a closed book exam.
Use of a regular and or translation dictionary is not permitted. This exam consists of the cover page, one page of questions and 1 page of the
table of Elementary Laplace Transforms. MATH 315, FINAL EXAMINATION December 17th, 2007 1. Solve the following differential equations, ﬁnding the general solution, and
the particular solution corresponding to the initial conditions, if given: a. (10 pts) 3/ = 2y2 + my2,y(0) = 1 b. (10 pts) (1 + $2)y’ + 4131; = (1 + m2)‘2 c. (10 pts) (1 — m)y” + my’ — y = 2(w — 1)2e‘“, given that ez solves the
homogeneous equation. d. (10 pts) dw + y‘1(a: — sin(y))dy = 0 e- (10 MS) 11’” - y” - y’ + y = 0 f. (10 pts) y” — 431’ + 4y 2 2:1: + 4621 + sin(2ar:) 2. Solve in a series centred at :v = 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) (1 — 1:2)y” — 2xy’ + 2y 2 0,
b. (10 pts) 2m2y" -— my’ + (1 + 9;)y = 0, 3. a. (5 pts) Compute the inverse Laplace transform of
(s — 2)e‘s
s2 — 4s + 3 -
b. (5 pts) Compute the Laplace transform of the square wave f of period 2.
f(t)=l, OSt<1
= 0, 1 S t < 2
:f(t—2), 2st.
c. (10 pts) Solve the initial value problem
y” + 231' + 3y = sin(a:) + 6(93 — 37r), y(0) = 0, y'(0) = 0
Solve the same problem, but with initial conditions y(0) = 1, y' (0) = 0. TABLE 6.2.1 Elementary Laplace Transforms ft» = Y'mm 1%) = <2tfm}
1
I. 1 —, s > O
s
a! 1
2 e . s > a
s — a
n!
3. t”; n = positive integer "—H, s > 0
s
F 1 -
4.t”,p>—1 (pi—1 ), -s>0
s”T
5.sinat 2a ,, s>0
s +a‘
: S-
6. cosat 2 7, s > O
s +a'
7 sy'nh t a | |
. 1 a , - s > a
s2 — a2 .,
s
8. coshat 2 2, s > [a]
s —a
at ' b
9. e smbt ‘77, s > a
(s — a)“ + b'
10. e‘" cos bt #, s > a
(S — a)“ + b'
n!
11. t"e‘", n = ositive inteoer , s > a a
p A t: (S __a)n+l
‘ ‘ e—(‘S
12. uv(t) , s > 0
s
13. z.t(.(r)f(t —- c) ' e“"5F(s)
14. e"’f(t) F(s — c)
_ 1 .s .
13. f(ct) -—F <—) , c > O
c c
16. f' f(t — r)g(r)dr rF(s)G(s)
0
17.’ so — c)‘ r“
18 f'(")(f) ~ S”F(S) _ six—lf(0) _ . _ . _ fut—”(0) 19. (—t)"f(r) ' F<")<s> ...

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