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Unformatted text preview: Midterm Exam MATH 150: Introduction to Ordinary Differential Equations J. R. Chasnov
8 November 2006 Answer ALL questions
Fall mark: 60; each question carries 10 marks. Time allowed 7 55 minutes 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 No. (mark) Marks
10)
) ) )
10) 6 (10)
Total ._\
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OO Question 1 Score: : Find the solution 7" = r(t) of the following initial value problem: d7“
a = —4rt, 7“(0) = r0. Question 2 Score: : Find the solution y = y(:1:) of
y’ = 31? + 11 that passes throught the point ($41) = (—1, 1). Question 3 Score: : Mrs. Wong is buying an apartment and must borrow S0 dollars. She wants a Tyear mortgage with ﬁxed
annual payments of k: dollars per year, and can borrow money at an interest rate 1“ per year. Assume
continuous compounding and continuous mortgage payments. Determine Mrs. Wong’s annual payments k: in terms of So, 1“ and T. Question 4 Score: : Consider the damped harmonic oscillator equation
mi + 7$ + km 2 07 with m, *y, k > 0. Deﬁne the kinetic energy K, potential energy V and total energy E by 1 1 .
Kzimma, Vzikzmz, E:K+V.
Showthat dE
_:_ K
dt C ’ Where c is a constant. Determine c. Question 5 Score: : (a) (8 pts) Given 04, ﬁnd the solution of the following initial value problem:
i} +m' — 21; = e’t, 95(0) 2 0, :‘c(0) = a. (b) (2 pts) Determine a such that lin1,;_>oO :1:(t) = 0. Question 6 Score: : Solve the following differential equation, with M, F > 0, ..+ 2 _ Fcoswt, if0§t§27r/w;
‘17 “”3— 0, ift>27r/w. Assume initial conditions 1:(0) : $(0) : 0. (Solve the problem by requiring 1; and x' to be continuous at
t = 27T/w.) ...
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