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Unformatted text preview: Midterm Exam
MATH 150: Introduction to Ordinary Diﬀerential Equations J. R. Chasnov
24 October 2007 Answer ALL questions Full mark: 50; each question carries 10 marks. Time allowed – one 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) 5 (10) Total Marks Question 1 Score: Find the solution y = y(x) of the following initial value problem: y ′ = x2 y 2 , y(−1) = 2. Question 2 Score: Find the solution y = y(x) for x > 0 of xy′ − 2y = x3 cos x, that passes throught the point (x, y) = (π/2, 0). Question 3 Score: You borrow S0 = $100 000 from the bank at r = 12% interest for T = 3 years. Assume continuous compounding and repayment of the loan. (a) (4 pts) Determine the diﬀerential equation for the amount left on the loan S = S (t) at time t given a constant repayment rate of k and an interest rate r . (b) (4 pts) Assuming the loan is paid back after time T , solve the diﬀerential equation to determine k in terms of S0 , r and T . (c) (2 pts) For the speciﬁc loan above, what is the total amount of money paid back to the bank? Question 4 Score: Find the solution x = x(t) of x + x = 2e−t , ¨ x(0) = 0, x(0) = 0. ˙ Question 5 Score: Find a particular solution of x + 2x + x = 2e−t . ¨ ˙ ...
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This note was uploaded on 01/28/2011 for the course MATH 150 taught by Professor T.qian during the Spring '09 term at HKUST.
 Spring '09
 T.Qian
 Equations

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