midterm - Midterm Exam MATH 150: Introduction to Ordinary...

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Unformatted text preview: Midterm Exam MATH 150: Introduction to Ordinary Differential Equations J. R. Chasnov 7 November 2008 Answer ALL questions Full mark: 40; each question carries 10 marks. Time allowed – 1 hour 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(x0 ) = y0 , let µ = exp ( x x0 pdx). Then, y = 1 µ (y0 + x x0 µgdx). Question No. (mark) 1 (10) 2 (10) 3 (10) 4 (10) Total Marks Question 1 Score: Find the solutions y = y(x) of the following initial value problems: (a) (5 pts) √ y′ = xy, y(0) = 1; (b) (5 pts) xy′ + 2y = e3x, y(1/3) = 0. Question 2 Score: John deposits $1000 in Bank I for a 10 year period with his investment compounded continuously at a 6% annual interest rate. Jane deposits $1000 in Bank II for a 10 year period with her investment compounded once every 5 years. (a) (4 pts) How much is John’s investment worth at the end of 10 years? (b) (4 pts) Determine the annual interest rate that Bank II must pay Jane if John and Jane are to have the same amount at the end of 10 years. (c) (2 pts) Sketch on the same graph the value of John’s and Jane’s bank accounts over the ten year period. Question 3 Score: Suppose x = x(t) is a solution of the equation x + 2x + x = 0; ¨ ˙ x(0) = x0 , x(0) = u0 . ˙ (a) (8 pts) Solve for x(t). (b) (2 pts) Assuming a fixed x0 > 0, determine the values of u0 (in terms of x0 ) such that x(t) will be negative for some t. Question 4 Score: Suppose x = x(t) is a solution of the equation x + 2x + 5x = 1 − e−t , ¨ ˙ (a) (8 pts) Solve for x(t). (b) (2 pts) Determine the solution as t → +∞. x(0) = 0, x(0) = 0. ˙ ...
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midterm - Midterm Exam MATH 150: Introduction to Ordinary...

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