Fall 2005

Fall 2005 - FACULTY OF ENGINEERING FINAL EXAMINATION MATHEMATICS 263 ‘l W ORDINARY DIFFERENTIAL EQUATIONS AND LINEAR ALGEBRA" Examiner

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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.

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Fall 2005 - FACULTY OF ENGINEERING FINAL EXAMINATION MATHEMATICS 263 ‘l W ORDINARY DIFFERENTIAL EQUATIONS AND LINEAR ALGEBRA" Examiner

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