Download Expression for Time - Math Tools for Engineering - Exam and more Exams Mathematics in PDF only on Docsity!
EGR 265, Math Tools for Engineering Problem Solving February 7, 2011, 50 minutes
Name (Print last name first):..........................................
Student ID Number:...........................
TEST I
Problem 1 Determine the order of the following ODEs.(4P+4P+4P+4P) Also, state if they are linear or non-linear.
(a) y′^ + x = cos y (b) y′^ + y ex= x (c) y(4)^ − y^5 = x^2 (d) y(3)^ + exxy = 1
Problem 2 (a) Which of the following functions are solutions of x^4 y′^ + 2xy^2 = 4x^5? (8P) y 1 = x^2 , y 2 = −x^2 , y 3 = x, y 4 = − 2 x^2.
(b) Which of the functions from part (a) solve the initial value problem y(0) = 0? (4P) x^4 y′^ + 2xy^2 = 4x^5 ,
(c)Uniqueness Theorem for first order ODEs? If yes, why? If no, why not? (5P∗^ (Bonus) Does your answer to part (b) agree with the content of the Existence and∗)
Problem 3 (a) In the 3direction field for × 3-grid of points y′ (^) = x (^2) (y − 1). (8P)x = 0, 1 , 2 and y = 0, 1 , 2 provided in the figure below draw a
(b) Without solving the DE, use the direction field to guess the solution of the IVP x (^2) (y − 1), y(1) = 1. Check that your guess is correct by verifying that it is a solution of the y′^ = IVP. (4P)
- Problem
- Solve the IVP (15P) y′ = 2xy 2 , y(0) = −
- Problem
- Solve the IVP (15P) y′ + y = x, y(0) =
- Problem
- Solve the IVP (15P) y′ − xe−y = 0, y(0) =
Problem 7 The number of bacteria in a culture is given byAn initial population of n(0) = 100 bacteria grows at a constant rate n(t), where the time t is measured in hours.k = n′(t)/n(t). After one hour 150 bacteria are present. (a) Find the rate of growth k by solving the differential equation for n(t). (10P) (b) Find an expression for the time n(td) = 200. (5P) td necessary for the number of bacteria to double:
Note: Your answers will contain natural logarithms which do not need to be evaluated. (a)
(b)
