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Unformatted text preview: Final Exam MATH 150: Introduction to Ordinary Differential Equations J. R. Chasnov
21 December 2006 Answer ALL questions
Fall mark: 80; each question carries 10 marks. Time allowed 7 2 hours Directions 7 This is a closed book exam. You may write on the front and back of the exam papers. Student Name: Student Number: Question N0. (mark)
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—‘ —‘ O O Marks Question 1 Score: : (a) (8 pts) Solve the following differential equation for all values of :12: dy ydm +x, y() (b) (2 pts) Determine the value of 1; at which the solution attains its maximum value. What is the value of
y at its maximum? Question 2 Score: : (a) (8 pts) Find the solution y = y(:1:) of the following differential equation:
561/ + y = :62; 11(1) = C (b) (2 pts) Determine the value of c such that y(0) is ﬁnite. Question 3 Score: : A snowball melts in such a way that the rate of change in its volume is proportional to its surface area. 4 (Assume a spherical snowball with volume V : gwrg’ and surface area S : 47r7'2, with r the radius). Suppose the snowball was initially 4 cm in diameter and after 30 min its diameter is 3 cm. (a) (8 pts) Find a formula for the diameter (cm) of the snowball after time t (min).
(b) (2 pts) When will the snowball disappear? Question 4 Score: : (a) (8 pts) Determine the solution to the initial value problem i+2i=+m = e’t, 95(0) :i;(0) = 0. (b) (2 pts) Determine the time at which the solution attains its maximum. Question 5 Score: : Find the differential equation Whose general solution is
(a) (5 pts) :1:(t) = Aet —— Be‘2t.
(b) (5 pts) :1:(t) = Aet —— Be‘2t + 6%. Question 6 Score: : (a) (8 pts) Using a power series substitution, ﬁnd the general solution (with two free constants) of
(1 + m2)y" + 2my' — 2y : O in powers of :1: up to and including .726.
(b) (2 pts) Find the solution that satisﬁes y(0) = 0, y’(0) : 1. Question 7 Score: : Consider the Bessel equation near x = O: 3723/” + :zry’ —p2y = 0, p > 0. For a given 6 > 07 ﬁnd any initial conditions 34(6), y’(e) such that y(0) = 0. Question 8 Score: : Consider the system of equations
.i’l = —4$1 — 2.232, $2 = 5JJ1 + 21:2. (a) (6 pts) Find the general solution of the system of equations.
(b) (4 pts) Determine the solution satisfying 371(0) 2 1, 372(0) 2 0 . ...
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 Spring '09
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