Midterm02Solutions

Midterm02Solutions - S OE.”{Ki UNIVERSITY OF WATERLOO...

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Unformatted text preview: S OE.” {Ki UNIVERSITY OF WATERLOO AMATH 250 MIDTERM 02, FALL 2010 VERSION A Student Name (Print Legibly) (FAMILY NAME) Signature (GIVEN NAME) Student ID Number Tutorial Section (or tutorial start time) COURSE NUMBER COURSE TITLE COURSE SECTION(S) DATE OF EXAM AMATH 250 Introduction to Differential Equations 001 Monday, November 15, 2010 DURATION OF EXAM 50 minutes NUMBER OF EXAM PAGES 6 (6 single—sided sheets) INSTRUCTOR G. Mayer EXAM TYPE Closed Book ADDITIONAL MATERIALS ALLOWED NONE Instructions Marking Scheme 1. Write your name, Signature, ID number and tuto— . rial section (or tutorial start time) on the cover. QueStlon Mark out Of 2. Answer all questions in the spaces provided. You cover 1 may also use the reverse sides of the exam pages to write your answers. Ask a proctor for extra 1 5 blank pages if necessary. 2 10 3. Show all your work for full marks. 4. Check that the examination has 6 single—sided 3 10 pages. ‘ 5. Your grade will be influenced by how clearly you 4 2 express your ideas, and how well you organize Total 28 your solutions. AMATH 25o - Midterm 02 Version A Fall 2010 Page 2 0% [5] 1. Consider the lift force F produced by the Wings of an airplane during flight. Suppose that F=f(p,v,S) where p is the density (mass per unit volume) of the air, 1) is the speed of the airplane, and S is the surface area of the wings. Use a suitable theorem to find the function f , up to a constant. Be sure to explain your logic. AMATH 250 - Midterm 02 Version A Fall 2010 Page 3 of 5 [3] 2. a) Determine the form of the particular solution of the following ODE d2 d—tg + 4y 2 cos(2t), y 2: y(t). Do not find the general solution. [7] b) Find the general solution to the following DE 5g” + 23/ + y := sint, and identify the transient and steady-state solutions (if any). e "5e _ ’W/ Ji‘ wxi’fi’XVQ-r) - ‘ X W ‘ ‘ ""‘* AMATH 250 ~ Midterm 02 Version A Fall 2010 Page 4 0&5? 3. Consider the nonlinear skydiver DE d m1 = mg — W, dt where m is the skydiver’s mass, 9 is the acceleration due to gravity, and ,3 is a drag coefficient. ‘ [3] a) Find [,8], the dimensions of ,8. Remember to Show your work. a L 2: {a W2 i it? 3“" Mfg [4] b) Introduce a characteristic velocity DC, and a characteristic time to. Use them to define a dimensionless velocity and a dimensionless time. AMATH 250 — Midterm 02 Version A Fall 2010 Page 5 of [3] 0) Write the nonlinear skydiver DE in terms of the dimensionless variables you introduced in question 3 part b). You do not need to solve the DE. [2] 4. Consider the following example of a Cauchy-Euler DE: t2y” — Zty’ - 4y 2 0, t # O, y = Solving this DE with the trial function y em ill not work. Find another way of solving this DE, and find its general Solution. ‘r :1 rarefied“ SOL-M3005 UNIVERSITY OF WATERLOO AMATH 250 MIDTERM 02, FALL 2010 VERSION B Student Name (Print Legibly) (FAMILY NAME) (GIVEN NAME) Signature Student ID Number Tutorial Section (or tutorial start time) COURSE NUMBER AMATH 250 COURSE TITLE Introduction to Differential Equations COURSE SECTION(s) 001 DATE OF EXAM Monday, November 15, 2010 DURATION OF EXAM 50 minutes NUMBER OF EXAM PAGES 6 (6 single—sided sheets) INSTRUCTOR _ G. Mayer EXAM TYPE Closed Book ADDITIONAL MATERIALS ALLOWED NONE Instructions Marking Scheme I. Write your name7 signature7 ID number and tuto— rial section (or tutorial start time) on the cover. QueStion 2. Answer all questions in the spaces provided. You Mark Out of cover 1 may also use the reverse sides of the exam pages to write your answers. Ask a proctor for extra 1 10 blank pages if necessary. 3. Show all your work for full marks. 2 5 4. Check that the examination has 6 single—sided 3 10 pages. 5. Your grade will be influenced by how clearly you 4 2 express your ideas, and how well you organize Total 28 your solutions. AMATH 250 - Midterm 02 Version B Fall 2010 Page 2 0f 6 [3] 1. a) Determine the form of the particular solution of the following ODE dgy w + 9y = sm(3t), y 2 11(15)- DO not find the general solution. f; [7] b) Find the general solution to the following DE 53/" + 23/ + y 2 cos t, and identify the transient and steady—state solutions (if any). AMATH 250 — Midterm 02 Version B Fall 2010 Page 3 of 6 [5] 2. Consider the lift force F produced by the Wings of an airplane during flight. Suppose that F=f(p,v,3) where p is the density (mass per unit volume) of the air, 2) is the speed of the airplane, and S is the surface area of the Wings. Use a suitable theorem to find the function f, up to a constant. Be sure to explain your logic. AMATH 250 - Midterm 02 Version B Fall 2010 Page 4 of 6 3. Consider the nonlinear skydiver DE d = mg ‘— 61127 Where m is the skydiver’s mass7 g is the acceleration due to gravity, and fl is a drag coefficient. a) Find [13}, the dimensions of 5. Remember to Show your work. b) Introduce a characteristic velocity vc, and a characteristic time tc. Use them to define a dimensionless velocity and a dimensionless time. fr AMATH 250 - Midterm 02 Version B Fall 2010 Page 5 of 6 [3] 0) Write the nonlinear skydiver DE in terms of the dimensionless variables you introduced in question 3 part b). You do not need to solve the DE. [2] 4. Consider the following example of a Cauchy-Euler DE: ’5— Li 1523/” + 2ty’ My; = 0, y = 3/05), 15% 0. Solving this DE with the trial function y = 6” will not work. Find another way of solving this DE, and find its general solution. ...
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This note was uploaded on 02/23/2012 for the course AMATH 250 taught by Professor Ducharme during the Fall '09 term at Waterloo.

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Midterm02Solutions - S OE.”{Ki UNIVERSITY OF WATERLOO...

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