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final2005 - Final Exam MATH 150 Introduction to Ordinary...

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Final Exam MATH 150: Introduction to Ordinary Differential Equations J. R. Chasnov 15 December 2005 Answer ALL questions Full mark: 90; each question carries 10 marks. Time allowed – 2 hours Directions – This is a closed book exam. You may write on the front and back of the exam papers. Student Name: Student Number: Question No. (mark) Marks 1 (10) 2 (10) 3 (10) 4 (10) 5 (10) 6 (10) 7 (10) 8 (10) 9 (10) Total
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Question 1 Score: (a) (8 pts) Solve the initial value problem for x 0: dy dx = 2 cos 2 x 3 + 2 y , y (0) = - 2 . (b) (2 pts) Determine where the solution attains its minimum value. What is the value of y at its minimum? – 1 –
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Question 2 Score: (a) (6 pts) With λ , a and b all positive, find the solution x = x ( t ) for t 0 of the following initial value problem: ˙ x + λx = a + be λt ; x (0) = 0 . (b) (2 pts) Determine the value of t at which x ( t ) is maximum. (c) (2 pts) Determine x as t → ∞ . – 2 –
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Question 3 Score: Newton’s law of cooling states that the temperature of an object changes at a rate proportional to the difference between its temperature and that of its surroundings. Suppose that the temperature of a cup of
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