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Unformatted text preview: NAME (underline family name): STUDENT NUMBER: SIGNATURE: FACULTY OF ENGINEERING
FINAL EXAMINATION MATH 263
ORDINARY DIFFERENTIAL EQUATIONS AND LINEAR ALGEBRA
Examiner: G. Schmidt Date: Thurs. December 14, 2006 Associate Examiner: B. Charbonneau Time: 9:00 AM  12:00 AM Instructions 1. Write your name and student number on this examination script.
2. No books, calculators or notes allowed. 3. This examination booklet consists of this cover, 9 pages of questions
and 2 blank pages (the cover page plus 11 numbered pages in all).
Please take a couple of minutes in the beginning of the examination
to scan the problems. (Please inform the invigilator if the booklet is
defective.) 4. Answer all questions. You are expected to show all your work. All
solutions are to be written on the page where the question is printed.
You may continue your solutions on the facing page. If that space is
exhausted you may continue on the blank pages at the end, clearly
indicating any continuation on the page where the question is printed. 5. Your answers may contain expressions that cannot be computed with
out a calculator. 6. Use of a regular and or translation dictionary is permitted. 7. Circle your answers where confusion could arise. GOOD LUCK! Score Table Final Examination December 14, 2006 MATH 263 1. (10 marks) Solve
'2 ex”, y(0) = 0 and ﬁnd the maximum interval on which the solution is valid. Final Examination December 14, 2006 MATH 263 2. (10 marks) Using the substitution 1) = y1/2, ﬁnd the solution y($) 0f xzy’ + 2y = el/xyl/Q, y(1) = 4 Final Examination December 14, 2006 MATH 263 3. (10 marks) Solve implicitly, d_y= y
dm x+x6’ 11(1): 1 (HINT: do not use separation of variables!) Final Examination December 14, 2006 MATH 263 4. (10 marks) Find the general solution y(m) of (143/ d3y dzy
— — 4— — = 2 .
dx4 dx3 + 8dx2 x Final Examination December 14, 2006 MATH 263 5. (12 marks) Find the solution y(x) of ea: 1/”  21/ + 2y = __ I _
cosma _ 2) y _ Final Examination December 14, 2006 MATH 263 6 6. (12 marks) Use Laplace transforms, and the table which follows, to solve y” + 231' + 2y = 2t + 364(t), y(0) = 1, y’(0) = —2. function f(t) Laplace transform F(s)
1/3 (3 > 0)
n! sn+1 ( ﬁ
3 (I) V O
_r sin at a/(s2 + a‘) (s > 0)
cos at
6—0705)
ua(t) = u(t — a) (a 2 0) e'M/s (s > 0
60(t) = 6(t — a) (a > O) u(t — a)f(t — a) or ua(t)f(t — a)
f Wt) s"F(s) — snwo) — st'm) . . . — f 7H (0)
f*9(t) = In f(r)g(tr)dr (s)G(s) ’11 Final Examination December 14, 2006 MATH 263 T 7. (12 marks in total) You are given the following matrix A with its row
reduced echelon form R: 2 0 —1 1 4 1 0 0 1 3
A: 2 0 3 5 12 , R: 0 0 1 1 2 .
1 0 1 2 5 0 0 0 0 0 (a) (4 marks) Write down a basis for the row space of A and express all rows
of A as linear combinations of the basis vectors. (b) (4 marks) Find a basis for the column space of A and express the ﬁfth
column of A in terms of that basis. (c) (4 marks) Find an orthogonal basis for the column space of A. Final Examination December 14, 2006 MATH 263 8 8. (12 marks in total) Consider the symmetric matrix A = < :11 g (a) (4 marks) Find the eigenvalues and the corresponding eigenspaces. (b) (3 marks) Find an orthogonal matrix P and a diagonal matrix D such
that A = PDPT. (c) (3 marks) Find the matrices of orthogonal projection onto each of the
two eigenspaces of A. ‘ (d) (2 marks) What is the relationship between A and the two projection
matrices you have found in Verify that your answer is correct. Final Examination December 14, 2006 MATH 263 9 2 —4 1 + z
is an eigenvector of A corresponding to the eigenvalue —1 + i. (a) (4 marks) Write down a basis, in vector form, of real solutions of the
system 9. (12 marks in total) Consider A = < 2 _5 The vector ( 1 +21 ) I
331
I
352 2x1  5x2
2x1 — 4:132. (b) (3 marks) Find the functions 321(75) and 552(t) which satisfy the above
system as well as the initial conditions 231(0) 2 1, 932(0) 2 0. (c) (3 marks) Write down an expression for 6““ involving the product of
speciﬁed real matrices and their inverses. (d) (2 marks) How do solutions of the system in (a) behave as t —> oo? Final Examination December 14, 2006 MATH 263 10 CONTINUATION PAGE FOR QUESTION NUMBER D You must refer to this continuation page on the page where the question
is printed. Final Examination December 14, 2006 MATH 263 11 CONTINUATION PAGE FOR QUESTION NUMBER D You must refer to this continuation page on the page where the question
is printed. ...
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 Summer '09
 SidneyTrudeau
 Differential Equations, Linear Algebra, Equations, real solutions

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