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final2007

# final2007 - Final Exam MATH 150 Introduction to Ordinary...

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Final Exam MATH 150: Introduction to Ordinary Differential Equations J. R. Chasnov 13 December 2007 Answer ALL questions Full mark: 80; 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: To solve y + p ( x ) y = g ( x ) , y ( x 0 ) = y 0 , let μ = exp ( integraltext x x 0 pdx ). Then, y = 1 μ ( y 0 + integraltext x x 0 μgdx ). Question No. (mark) Marks 1 (10) 2 (10) 3 (10) 4 (10) 5 (10) 6 (10) 7 (10) 8 (10) Total

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Question 1 Score: (a) (8 pts) Find the solution of the following ode for y = y ( x ) that passes through the origin ( y (0) = 0): e x y y + e y x = 0 . (b) (2 pts) For what values of x does the solution exist? – 1 –
Question 2 Score: (a) (8 pts) Find the solution y = y ( x ) with x > 0 of xy + (1 + x ) y = xe x , y (1) = 0 . (b) (2 pts) Find the cubic equation whose positive root is the value of x at which y ( x ) is maximum. – 2 –

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Question 3 Score: On a level road, a car of mass m is travelling with velocity v 0 . Assume the driver shifts into neutral and coasts,
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