c315exf06

# c315exf06 - MCGILL UNIVERSITY FACULTY OF SCIENCE FINAL...

• Test Prep
• 3

This preview shows pages 1–3. Sign up to view the full content.

This preview has intentionally blurred sections. Sign up to view the full version.

This is the end of the preview. Sign up to access the rest of the document.

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\$ ...
View Full Document

• Fall '09

{[ snackBarMessage ]}

### What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern