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Unformatted text preview: FACULTY OF ENGINEERING FINAL EXAMINATION
» MATHEMATICS 263
‘l W ORDINARY DIFFERENTIAL EQUATIONS AND LINEAR ALGEBRA
" Examiner: Professor S. W. Drury Date: Tuesday, December 20, 2005 M Associate examiner: Professor G. Schmidt Time; 14:00 _ 17:00 Family Name: Given Names: W
' Student Number: m Instructions 1. Fill in the above clearly.
2. This is a closed book exam. Faculty standard calculators only are permitted. 3. This examination booklet consists of this cover, Pages 1 through 9 containing
questions; Pages 10 and 11, which are blank; and page 12 containing a Laplace
transform table. You are expected to show all your work. All solutions are to be written in the space
provided on the page where the question is printed. When that Space is exhausted,
you may write on the facing page. Any solution may be continued on the last
pages, or the back cover of the booklet, but you must indicate any continuation
clearly on the page where the question is printed! 4. You are advised to spend the ﬁrst few minutes scanning the problems. (Please
_ inform the invigilator if you ﬁnd that your booklet is defective.) PLEASE DO NOT WRITE INSIDE THIS BOX grand total /100 Final Examination — MATH263 December 2005 1. (10 points) Solve the initial value problem 3—:c0$($) + 33; sin(x) = (003(3)?) gm) z 1. Final Examination — MATH263 December 2005 2. (10 points) By making the substitution in = a: w y, solve the initial value problem 0!
Ezra—mks, y(o)=o. Final Examination  MATH263 December 2005 3 3. (10 points) By ﬁnding an integrating factor of the form :23pr solve the implicit initial
value problem (59:21; + 6y3)dac + (23:3 + Smy2)dy = 0, soiution passes through (m,y) = (I, 1). Leave your answer in implicit form. Final Examination — MATH263 December 2005 4. (10 points) Find the general solution of the equation 11'” + 2y” + y’ 2 64(2: + e”). Final Examination — MATH263 December 2005 5. (12 points) Find the general solution of the equation d2y dy
2— _ _ 2 _ 3 3:
35 x2 2m + y we Final Examination —— MATH263 December 2005 6 6. (12 points) Use Laplace transforms to ﬁnd the solution of the initial value problem d d
— + 6— + 13y : 1691: + 464:), y(0) : 0, alit—{(0) m 1
in t _>_ 0. Note that a table of elementary Laplace Transforms is provided at the end of this exam. There is no credit for other methods of solution. Final Examination — MAT H263 December 2005 7 1 1 a
7. {10 points) Let A = a a 1
1 ma, —1 (i) (2 points) Find the rank of A for each value of the scalar 0;.
(ii) (4 points) For each value of a such that A is a singular matrix, ﬁnd a basis of
the kernei of A1 1.6. of the solution space for Ax = O. (iii) (4 points) For each value of a such that A is a singular matrix, find a basis of
the column Space of A. Final Examination — MATH263 December 2005 8. (12 points) Consider the matrix 2) (i) (4 points) Find the eigenvalues of A.
(ii) (4 points) For each eigenvalue, ﬁnd a corresponding eigenvector.
(iii) (4 points) Find'an orthogonal matrix U such that U ’AU is diagonal. Final Examination —— MATH263 December 2005 9. (14 points) Consider the matrix B4; 5) ('1) (4 points) Find the characteristic polynomial and the eigenvalues of B.
(ii) (6 points) Compute the matrix exp(tB) where t is e scalar.
(iii) (4 points) Soive the initial value problem (22:94; was), (2302(1) Final Examination — MATH263 December 2005 10 CONTINUATION PAGE FOR PROBLEM NUMBER You must refer to this continuation page on the page Where the problem is printed! Final Examination m MATH263 December 2005 11 CONTINUATION PAGE FOR PROBLEM NUMBER You must refer to this continuation page on the page where the problem is printed! ...
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This note was uploaded on 01/22/2012 for the course MATH 263 taught by Professor Sidneytrudeau during the Fall '09 term at McGill.
 Fall '09
 SidneyTrudeau

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