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1 2M06 Xmas Test, page 1 of 16 MATHEMATICS 2M06 XMAS MIDTERM TEST Dr. P Sastry Dr. Z. Kovarik DAY CLASS DURATION OF TEST: 3 hours Maximum total: 110 points MCMASTER UNIVERSITY TERM TEST December 13, 2006 THIS TEST INCLUDES 16 PAGES AND 14 QUESTIONS. YOU ARE RESPONSILE FOR ENSURING THAT YOUR COPY OF THE TEST IS COMPLETE. RING ANY DISCREPANCY TO THE ATTENTION OF YOUR INVIGILATOR. Instructions: 1. Use of Casio FX-991 calculator only is allowed. 2. Put your name and student number at the top of each page. 3. In Part A, provide complete solutions on this exam paper in the spaces provided below each question. 4. In Part, PRINT the letter corresponding to the answer of your choice in the box beside the corresponding question number below. An incorrect answer is worth zero marks. 5. At the end of the paper, there are two blank pages for rough work, and a formula sheet. 6. Each question in Part A is worth 10 marks. Part marks can be given where appropriate. 7. Part each question is worth 5 marks. No part marks will be given in this section. Part A Part Question Grade Question Answer Total:

2 2M06 Xmas Test, page 2 of 16 Part A Provide complete solutions in the spaces below the questions. ww w 1. The differential equation a" bc C Cœ! has one solution C"ab œ Þ Find another solution C abß linearly independent of C abþ "

3 2M06 Xmas Test, page 3 of 16 Þ Use variation of parameters to find a particular solution of the differential ww w / equation C C C œ Þ "

4 2M06 Xmas Test, page 4 of Find the general solution of the differential equation ac " b.c œˆ.c.. Make sure that no solution is "lost" in the process of solving. Hint: You can start with?œ.c and Chain Rule gives. C.? œ? Þ...C

5 2M06 Xmas Test, page 5 of Use Laplace Transform to solve the system of differential equations with unknown ab >ßC> abà ww w %C œ! w ww 1 C œ $ˆ > % w w a b a b a b a b = a= + b œ = = + Þ with initial conditions C! œ!ßc! œ"ß! œ!ß! œ!þ + " = Hint: It may help to know that

6 2M06 Xmas Test, page 6 of Given the differential equation solved by power series centered at œ!à ww C C œ! _ (a) find the recurrence relation for the coefficients - in C œ! œ! (b) find the first five non-zero terms of the solution w conditions C a! b œ" and C a! b œ"þ C a b subject to initial

7 2M06 Xmas Test, page 7 of Find the form of a particular solution of ab $ ww w C %C %C œ / sin which contains just enough undetermined coefficients to give a solution.

8 2M06 Xmas Test, page 8 of For the differential equation ww w C C Cœ! (a) find a series solution using Frobenius method, centered at 0, starting with a recursion relation for the coefficients (b) explain why the second, linearly independent solution does not have a series expression centered at 0.

9 2M06 Xmas Test, page 9 of Find the general solution of the differential equation w C C œè C observing that the equation is first-order homogeneous.

10 2M06 Xmas Test, page 10 of 16 Part 9. (Multiple choice question: transfer your choice to front page! ) ww w One of the solutions of the differential equation C C a% % bcœ! is (a) N % ab (b) N a%b (c) N ab (d) N 4a"'b (e) none of the above choices

11 2M06 Xmas Test, page 11 of 16 1!.. (Multiple choice question: transfer your choice to front page! ) The inverse Laplace transform _ " 1=Î š % is (a) " > > (b) " sina b ˆ 1 sina> b ˆ 1 h h > (c) " (d) " cosa b a b cosa b ˆ 1 > h > 1 > h > (e) none of the above choices.

12 2M06 Xmas Test, page 12 of 16 1 ". (Multiple choice question: transfer your choice to front page! ) w The differential equation C œ Csin C has a critical point at C œ!. This critical point is (a) an attractor (b) a repeller (c) a singular point (d) a semi-stable critical point (e) none of the above choices. Hint. The series for the sine function is on the formula sheet. 12. (Multiple choice question: transfer your choice to front page! ) Choose the correct classification of the singular point = of the differential equation ww w a " bc a " bc a " bcœ! (a) 0 regular, 1 irregular, 1 regular (b) 0 irregular, 1 regular, " irregular (c) 0 irregular, 1 regular, " regular (d) 0 regular, 1 regular, " regular (e) none of the above choices

13 2M06 Xmas Test, page 13 of (Multiple choice question: transfer your choice to front page! ) ww w " " " The differential equation ab > ab > % ab > œ sinˆ > models one of the following situations: (a) an undamped free simple harmonic oscillator (b) a critically damped free simple harmonic oscillator (c) an underdamped driven simple harmonic oscillator (d) an overdamped driven simple harmonic oscillator (e) none of the above choices 14. (Multiple choice question: transfer your choice to front page! ) A fundamental system of solutions of the Cauchy-Euler equation ww w C $C Cœ!ß! is as follows: (a) " e ß f (b) elnacos bß lnasin bf (c) e lnacos bß lnasin bf (d) " " e lnacos bß lnasin bf " (e) e ß ln f Reminder: Did you transfer all your multiple choice answers to the front page?

14 2M06 Xmas Test, page 14 of 16 Rough Work Area Nothing here will be graded

15 2M06 Xmas Test, page 15 of 16 Rough Work Area Nothing here will be graded

16 2M06 Xmas Test, page 16 of 16 Formula Sheet Laplace Transform: _ Definition: _e ab f ' => 0 > œ! / 0 a> b.> for = sufficiently large " 8 8x Table: _ ef " œ = _ e> fœ = 8 " _ e/ " f œ _ e$ a> + bf œ / = + _ ecosa5> bf œ _ esina5> bf œ = 5 = 5 += / for _ eha> + bf œ where ha> + b œ œ = " for > + Transformation Rules: Denote _ e0 ab > fœ J ab = and _ e1 ab > fœ K ab = Then _e+0 ab >,1 ab > fœ +J ab =,K ab = (linearity) a8b _ a b 8 8 " a8 " b 0 > œ= J a= b = 0 a! b á 0 a! b +> _e/ 0 ab > f œ J a= + b (translation on the = -axis) += _ eha> + b0 a> + bf œ/ J a= b (translation on the > -axis) _ea0 1bÐ>Ñfœ J ab = K ab = > where a0 1 ba> b œ ' 0a7b1 a> 7b. 7(convolution of 0 and 1) > "! = " X ab ' " / =X!! _ š ' 0 a7b. 7 œ J a= b (special case, 0 " ) => _e0> fœ / >>.> ab ( 0> abperiodic, period X) Variation of Parameters: Given ww w C :C ;Cœ1ß a b a b a b :ß; a b a b, 1 a b continuous on c+ß, d and given C" and C C linearly independent solutions of the homogeneous equation, then a particular solution is given by : C: ab œ?" abc" ab? abc ab w w where [œcc " CC " is the Wronskian and w [ C 1 w [ C 1? ab œ a b œ a b a b? ab œ a b œ a b a b and " [ a b [ a b [ a b [ ab Reduction of Order: ww w Given C : abc ; abc œ! where C" is a non-zero solution, then another linearly independent solution is given by ' Ta b. C ab œ C" a b' /. C " ab $ Some power series: / œ " á "x x $x % ' x %x 'x $ & ( $x &x (x $ % $ % cos œ " á sin œ á lna" b œ áß ll " essel Differential Equation: ww w C C a / bcœ! with solutions N/ a b and ]/ ab Series for N/ab À N _ 8 / / ab œ! a " b ˆ 8 8x > a" / 8 b 8œ! THE END

MATHEMATICS 3D03 Instructor: Z.V. Kovarik DURATION OF EXAMINATION: 3 HOURS MCMASTER UNIVERSITY FINAL EXAMINATION April, 2010

MATHEMATICS 3D03 Instructor: Z.V. Kovarik DURATION OF EXAMINATION: 3 HOURS MCMASTER UNIVERSITY FINAL EXAMINATION April, 2010 Mathematics 303(2010), Final exam Answers, page 1 of * First Name: ANSWERS Last Name: Student Numer: MATHEMATICS 303 AY CLASS Instructor: Z.V. Kovarik URATION OF EXAMINATION: 3 HOURS MCMASTER UNIVERSITY