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Unformatted text preview: , TV.” THE UNIVERSITY OF BRITISH COLUMBIA. Sessional Examinations December 2002 7 THE UNIVERSITY OF BRITISH COLUMBIA Counselling Services
Student Development & ServiCes 12001874 East Mall
MATHEMATICS 256. . Vancouver. 3.0. var 121 Differential Equations. Closed book examination Time: 2 1/2 hours. One standard size,double—sided formula sheet is permitted.
Calculators allowed. This examination patpér has 4579535. Make sure that you
have a complete paper. SHOW YOUR WORK TO J USTIFY YOUR ANSWER. PART A. DO ALL 4 QUESTIONS. [15]— 1.(i) Solve the initial value problem 3/  y = we“, y(0) = 1 (ii) Find the solution of the given initial value problem and determine
the interval in which it is deﬁned
7 2 I , y
=  0:1.
y Haﬁz/U The differential equation , $231” +2ccy' — 2y 2 O, :r > 0 has solutions y1(a:) = x and y2(a:) = 22’2 = 1/302
0 Show that yl and y2 are linearly independent. 0 Show how somebody would the solution yg given yl o Solve the initial value problem 2221/" + 2221/— 2y = 0; 31(1) = 1, y’(1)= 5 [20]2. Consider the system 8
,_.\
A
H
V
H (1271(t) + ——a:1(t) + am2(t) 8
N, \
A Oh
\_/  where a is a parameter. (i)Find the value of the parameter oz for which the solutions of the
system approach the origin like a spiral, at the same rate as 67:. With a as in part (i), solve the system m’la)  @1121“) + $203) + e“: = —a:1(t) + ax2(t) + e"t slut with 351(0) = 1, 932(0) =' —1. [20]—3. Consider a forced, damped mass—spring system described by
the initialvalue problem
u” + W’ + u = N) (*) where 'y > 0 is the damping constant. (i) Find all the values of '7 for which the system is critically damped.
(ii) If 7:0, f(t) =¢os<iat), solve with u(0) = 0 = u'(0), w 74 1
and then with u(0) = 0 =u'(0), w = 1. [15]4.1) Use the improved Euler method with step size h = 0.1 to
compute the approximate value of the solution of the following initial value problem at t = 0.2 y’ = t2+y2, y(0) = 1 2) If A is the approximation to y(0.2) found in part 1) and if B
is the approximate value of y(0.2) with step size 0.02, ﬁnd a linear _
combination of A and B which would give a better approximation to y(0.2).
Do not compute B. PART B. Sections 101102 (Fournier, Peterson)0nly [15]5)1) Solve the heat equation ut = 7ruw,0<>m<2,t>0
u(0,t) 0 = u(2,t)
Man, 0) = sin(7T—2$) + 3sin(3—727—:£) 2) Find liming; u(m, t) or explain why this limit does not exist. [15]—6 1) Solve the wave equation um = 7m”, 0<m<2,t>0
u(0,t) = 0=u(2,t)
u(:c,0) = sin(7r2—$)+3sin(§%£), 0<$<2
Ut(m,0) = I 0, O<$_<2 . 2) Find limt_+00 u(a:, t) or explain why this limit does not exist.
PART C. Section 103 (Bui)only [15]5.1) Solve the initial boundaryvalue problem ut(a:,t) = um(m,t) — u(x,t) , 0 < :1: < 1, t > 0 uz(0,t) = 0, u(1,t) = 10 u(:z: 0) = 106081100 + 300307—11) + 5cos(§—7r—m)
’ cosh(1) 2 2 Note cosh(:c) = {6m + e‘z}/2.
2) Find limt_,0° u(a:,t). [15]—6. Solve the initial boundary—value problem for the wave equap
tion utt(a:,t) = um($,t) —— u(m,t), 0 < :6 < 1, t > 0
u(0, t) = O =’u(1,t) u(:c,0) = 33(1 — m) 111(1), 0) = 0 THE END ...
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 Spring '08
 KEQINLIU
 Math, Differential Equations, Equations, Boundary value problem, Partial differential equation

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