{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

math256_december2002

# math256_december2002 - TV.” THE UNIVERSITY OF BRITISH...

This preview shows pages 1–4. Sign up to view the full content.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: , TV.” THE UNIVERSITY OF BRITISH COLUMBIA. Sessional Examinations December 2002 7 THE UNIVERSITY OF BRITISH COLUMBIA Counselling Services Student Development & ServiCes 1200-1874 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 6-7:. 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 initial-value 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 101-102 (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 boundary-value 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 ...
View Full Document

{[ snackBarMessage ]}

### Page1 / 4

math256_december2002 - TV.” THE UNIVERSITY OF BRITISH...

This preview shows document pages 1 - 4. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online