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Unformatted text preview: Be sure that this examination has 8 pages including this cover‘ The University of British Columbia
Sessional Examinations  April 2003 ’ Mathematics 256
Diﬁ’erential Equations Closed book examination ' Time; 2% hours Print Name ___.__._.._.___.___ Signature
'3 Student Number_______‘__;____. Instructor’s Name Section Number ‘ .UNlVERSlTY OF BRITISH COLUMBlA . . S
Counselling Servrce _
Student Development & Services 12001874 East Mall 1
Vancouver, 8.0. V6T 12 THE Special Instructions: Calculators may NOT be used. An 8.5” x ll” information sheet is
allowed. Show your work in the spaces provided.  Rules governing examinations 1. Each candidate should be prepared to produce his ilbrary/AMS card upon request.
2. Read and observe the following rules: . I No candidate shall be permitted to enter the examination room after the expiration of one half
hour. or to leave during the ﬁrst half hour of the examination. Candidates are not permitted to ask questions of the invigilators, except in cases of supposed
errors or ambiguities in examination questions. CAUTION  Candidates guilty of any of the following or similar practices shall be immediately
Missed from the examination and shall be liable to disciplinary action. (5) Making use of any books, papers or memoranda, other than those authorized by the
examiners. ('3) Speaking qr communicating with other candidates
(C) Purposely exposing written papers to the view of other candidates. The plea of accident or
1°rrgetrulness shall not be received. '3' sm°klng is not permitted during examinations. April 2003 Mathematics 256 Name [8] 1. Solve the following initial value problems. ‘
a) 29’ = (t +1)e”, 31(0) = 0
b) y’ = 2y + e‘”, 29(0) = 1 ,c) 2:] = 2m  y
y' = :2: — 2y,
3(0) = 1, y(0) = 1. d) y” +2y’ HI = 0, y(0) = 1, y’(0) = o. Page 2 of 8 pages Continued on page 3 April 2003 Mathematics 256 Name —____._._____._ Page ‘3 of 8 pages [7] 2, Water containing a salt concentration of 0.01 g/L is introduced into a beaker of volume 3 L at
the rateof 0.1 L/s, and the wellstirred mixture is removed at the rate of 0.05 L/s. Initially,
the beaker contains 2 L of pure water. a) Let A(t) denote the amount of salt in the beaker at time t. Find the initial value problem
satisﬁed by A(t). b) Solve this initial value problem to ﬁnd A(t). c) How much salt is in the beaker at the moment it overﬂows? Continued on page 4 m 3 April 2003 Mathematics 256 Name Page ‘4 of 8 pages The solution y(t) of a certain initial value problem is approximated using the Improved Euler
method. With a step size of h = 3 / 4, the approximate value for y(1) is found to be 1.20. With
a step size h = 1/4, the approximate value for y(1) is found to be 1.40. 3,) Using Richardson extrapolation, ﬁnd an improved approximation to y(1). b) Estimate how small it should be so that the approximation to y(1) is correct to Within
103. . .  Continued on page 5 April 2003 Mathematics 256 I Name __________________ Page‘s of 8 pages [7] 4. Consider the system
93/ = ‘2m + y + my yi=32$2y+m2. 3) Verify that (a: = 1, y = 1) is a critical point of this system,
b) Find the apprOJdmating linear system near this critical point. and use this to determine
the type of the critical point (circle all that apply):
node, saddle point, spiral point, centre, stable, asymptotically stable, unstable.
c) Sketch some typical trajectories in the rayplane near (an = 1, y = 1). Continued on page 6 Apm 2003 Mathematics 256 Name _.____'__._____ PageL6 of 8 pages [7] ' 5, The current I (t) through an RLC circuit is modeled by the initial value problem i [ll 21']
+ + C I = 5mm), [(0) = I’(0) = 0. Here 0 > O is the capacitance.
a) For which values of C does the transient response oscillate? b) Find the steadystate response when 0 = l. Continued on page 7 April 2003 Mathematics 256 Name ___—____._________ Page]: of 8 pages ‘
[7] 6, The temperature u(:1:,t) of a rod occupying the interval 0 S a: S 2 satisﬁes the heat equation
I ut=um, 0<$<2, t>O.
The rod is initiale held at temperature zero:
u(:1:,0) 20, 0952'.
At time t = 0 the ends are placed in contact With thermal reservoirs (and kept there) at temperatures 0 and 10: .
u(0, t) = O, u'(2,t) = 10, t > 0. Find u(:z:, :5). Continued on page 8 April 2003 Mathematics 256 Name ____,_____.____ PageaS of 8 pages
> [7] 7. The displacement u(a:, t) of a vibrating string satisﬁes the wave equation
uﬂ=u¢m O<a:<1, tl>0,,
and the boundary conditions
u(0,t) = 0, u3(1,t) = 0, t > 0 (one end is ﬁxed, the other is free). At t = 0, the string has a displacement of u(x,0) =
sin(57rx/2), and has no initial velocity. Use the method of separation of variables to ﬁnd u(m, t). ...
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 Spring '08
 KEQINLIU
 Math, Differential Equations, Equations, Boundary value problem

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