University of Toronto MAT234H1S midterm test Monday, March 5, 2012 Duration: 120 minutes

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1 University of Toronto MAT234H1S midterm test Monday, March 5, 2012 Duration: 120 minutes Only aids permitted: an 8.5 by 11 inch hand-written cheat sheet Instructions: Make sure this test contains 12 pages. Do not remove any pages from this test. Answer all questions. Present your solutions in the space provided. This means: show your work! Use the backs of the pages if you need more space. TOTAL MARKS: 70. The value for each question is indicated in parentheses beside the question number. NAME: (as on your T-card) STUDENT NUMBER: SIGNATURE: CHECK THE LECTURE SECTION YOU RE ENROLLED IN: Lec01 Mary Pugh Lec2 Daniel Egli CHECK THE TUTORIAL YOU LL PICK YOUR EXAM UP AT: TUT01 TUT02 TUT03 TUT04 Monday Monday Tuesday Tuesday SF3202 WB130 SF3202 GB405 Dave Yan Kimia Ghobadi Kimia Ghobadi Pradyumna Challa MARKER S REPORT: MARK Q.1 Q.2 Q.3 Q.4 Q.5 Q.6 Q.7 Q.8 Total

2 Formulas you may find useful. DO NOT TEAR THIS PAGE FROM THE TEST. 1. e u du = e u + C 2. u n du = un+1 + C, n 1 n du = ln u + C u 4. cos u du = sin u + C sin u du = cos u + C cos 2 u du = csc 2 u du = cot u + C sin u cos u du = 1 cos u du = sec 2 u du = tan u + C tan u du = ln sec u + C cot u du = ln csc u + C 1 sin u du = sec u tan u du = sec u + C csc u cot u du = csc u + C sec u du = ln sec u + tan u + C csc u du = ln csc u + cot u + C 1 a2 u 2 du = sin 1 u a + C = arcsin u a + C 1 a 2 + u du = 1 u 2 a tan 1 a + C = 1 a arctan u a + C 1 u u 2 a du = 1 u 2 a sec 1 + C = 1 a a arcsec u + C a cos n x dx = 1 n sin x cosn 1 x + n 1 cos n 2 x dx n 15. sin 2 θ + cos 2 θ = 1 tan 2 θ + 1 = sec 2 θ 16. cos(α + β) = cos(α) cos(β) sin(α) sin(β) sin(α + β) = cos(α) sin(β) + sin(α) cos(β) 17. sin(2θ) = 2 sin θ cos θ cos(2θ) = cos 2 θ sin 2 θ = 2 cos 2 θ 1 = 1 2 sin 2 θ 18. e iα = cos(α) + i sin(α) cos(α) = eiα +e iα sin(α) = eiα e iα 2 2i 19. cosh(α) = eα +e α sinh(α) = eα e α 2 2

3 1. [9 marks] For each of the following ODE, write the order of the ODE and circle whether it s linear or nonlinear, homogeneous or nonhomogeneous. (a) [3 marks] LINEAR or NONLINEAR (b) [3 marks] LINEAR or NONLINEAR (c) [3 marks] LINEAR or NONLINEAR xy + sin(x)y = y ORDER: HOMOGENEOUS or NONHOMOGENEOUS yy = 2 ORDER: HOMOGENEOUS or NONHOMOGENEOUS y + y y = 0 ORDER: HOMOGENEOUS or NONHOMOGENEOUS 2. [9 marks] (a) [3 marks] Consider x 2 y xy = y {y 1 (x), y 2 (x)} = {x, x ln(x)}. Is {y 1, y 2 } a basis of solutions?

4 (b) [3 marks] Consider Is {y 1, y 2 } a basis of solutions? xy x 2 y = 0 {y 1 (x), y 2 (x)} = {5, x 2 }. (c) [3 marks] Consider Is {y 1, y 2 } a basis of solutions? y + y = 0 {y 1 (x), y 2 (x)} = {1 + e x, 5e x + 5}.

5 3. [9 marks] Consider the differential equation y = 2 x y (1) (a) [3 marks] The differential equation (1) is separable. Use this fact to solve the initial value problem with y( 1) = 2. What is the domain of your solution? At what points in the domain is y (x) defined? (b) [4 marks] The differential equation (1) is exact. Use this fact to solve the initial value problem with initial data y(2) = 2. What is the domain of your solution? At what points in the domain is y (x) defined?

6 (c) [2 marks] Sketch the two solutions you found in parts (a) and (b)

7 4. [12 marks] Consider the differential equation y = cos(y) (2) (a) [3 marks] Using the graph below, sketch the direction field for this ODE.! (b) [4 marks] Find the general solution (either implicitly or explicitly).

8 (c) [2 marks] Solve the initial value problem with initial data y(0) = 0. (d) [3 marks] As x, what does your solution of the initial value problem do? That is, what is lim x y(x)? If your answer only uses the direction field, you ll only get partial credit. For full credit, your need to (correctly) take a limit in your answer to part (c).

9 5. [6 marks] Consider the ODE Find the general solution. y (4) y (3) 6y = 0.

10 6. [7 marks] Solve the following initial value problem: y = y + e 4x y(0) = 6 5

11 7. [8 marks] Solve the following initial value problem: e 2x + y + y 2y = 0 y(0) = 1 y (0) = 1

12 8. [10 marks] Private detective Philip Marlowe discovers the body of celebutante Paris Kardashian. It s a bracing 17 degree evening. At the time of death, her temperature was 37 degrees. When he found her body, its temperature was 27 degrees. An hour later, when the police arrive, the body s temperature is 22 degrees. Mysteriously, all of this is happening in a parallel universe, one in which our laws of physics do not apply. Normally, the simplest model one would use is that the change in temperature of the body in time is proportional to the difference between its temperature and the ambient 1 temperature. Instead, in Philip Marlowe s universe, the change in temperature of the body in time is proportional to the cube of the difference. How long had she been dead when Philip Marlowe found her? 1 Here, ambient temperature means the temperature of the air and ground around the body.

Math 223 Final. July 24, 2014

Math 223 Final. July 24, 2014 Math 223 Final July 24, 2014 Name Directions: 1 2 3 4 5 6 7 8 9 10 11 12 13 14 Total 1. No books, notes, or evil looks. You may use a calculator to do routine arithmetic computations. You may not use your