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Unformatted text preview: UNIVERSITY OF WATERLOO TERM TEST #1
SPRING TERM 2008 COURSE AM 250 COURSE TITLE INTRODUCTION TO DIFFEREN—
TIAL EQUATIONS SECTION(S) 001 HELD WITH COURSE(S) N/A SECTION(S) OF HELD WITH COURSE(S) N/A DATE OF EXAM June 9th, 2008 TIME PERIOD 4:30  5:20 pm DURATION OF EXAM 50 minutes NUMBER OF EXAM PAGES 6 pages (including this cover sheet) INSTRUCTOR D. Harmsworth EXAM TYPE Closed book ADDITIONAL MATERIALS ALLOWED Scientiﬁc Calculator Name (print) ID Number Signature Instructions: 1. Print your name and ID Number on this
page and sign it, 2. Answer the questions in the space pro MARKING SCHEME
Vided. Continue on the back of the page if necessary. Show all your work. Question Value 3. The last page may be removed and used
for rough work. 4. Your grade will be inﬂuenced by how
clearly you express your ideas and by how
well you organize your solutions. AM 250 — Term Test #1 Page 2 of 6 Spring Term 2008 d
[7] 1. (a) Find the general solution to the DE 232% + 3933; = 6. [8] (b) Produce a qualitative sketch of the family of solutions, using information from both the
DE and your result from part (a). Indicate as a dotted curve the set of points at which
the slope of the solution curves is zero, and include some typical solution curves for which
the constant of integration is positive, and some for which it is negative. Hint: as you should be able to see from your result to part (a), the solutions are all odd,
so you can consider just an > 0 to begin, and then use symmetry. AM 250 — Term Test #1 Page 3 of 6 Spring Term 2008 2. Consider a circuit consisting of a resistor of resistance R, an inductor of inductance L, and a
source voltage V(t). An inductor is a device which takes advantage of the fact that a current
passing through a wire induces a magnetic ﬁeld, which can be magniﬁed by wrapping the wire in a coil. However, for our purposes all you need to know is that the voltage drop across
di
such a device is given by VL(t) : Ld—t’ where i(t) is the current (a formula conﬁrmed by experiment). Also, the voltage across the resistor is VR(t) = Ri(t), and these voltage drops
must be balanced by the source voltage: di .
LEE + R2 — V(t). [2] a) Find the dimensions of R and L, given that the dimensions of voltage are M L2T‘2Q_1
(voltage is deﬁned as the work required to move a charge from a reference point (where V = 0) to the given location, so it has dimensions of Tim  Ifill—GEMlit—51351) charge} _ [charge]
[3] b) Suppose V(t) = WK“. Deﬁne dimensionless variables as 7' : 5L5 and 77 = $3, and
nondimensionalize the DE.
[9] c) Solve the nondimensionalized DE, assuming that 2(0) 2 O and that k 7E %. Then express your solution in terms of the original variables 1' and t. AM 250 — Term Test #1 Page 4 of 6 Spring Term 2008 [6] e) What happens if k = %? Find the solution for this case. (Use the nondimensionalized
DE, and then express your result in terms of the original variables.) [3] Bonus: Produce a rough sketch of the solutions for k: << 7%, k >> 1L3, and k : g AM 250  Term Test #1 Page 5 of 6 Spring Term 2008 [10] 3. For a celestial body in orbit around another (eg a moon orbiting a planet, or a satellite
orbiting the earth) the period P of the orbit depends on the masses of the bodies, M1 and
M2, the universal gravitational constant G (whose dimensions are [G] = M ‘1L3T_2), and the
semi—major axis of the ellipse of orbit, a (i.e. the distance between the objects, if the orbit is
circular). Use Buckingham’s H Theorem to investigate how the period of a satellite depends
on its distance from the earth. If one satellite is placed twice as high up as another, of the
same mass, how much longer will it take to orbit the earth? AM 250  Term Test #1 Page 6 of 6 Spring Term 2008 (This page is for rough work) ...
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 Summer '09
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