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Math 256 Section 202 Final Exam - Spring 2005, Exams of Differential Equations

Bharat Ratna Dr. B. R. Ambedkar UniversityDifferential Equations

The instructions and questions for the final exam of math 256 section 202, held in spring 2005. The exam covers various topics in mathematics, including calculus, differential equations, and integral calculus. Students are required to solve problems related to limits, derivatives, integrals, differential equations, and boundary value problems. The exam consists of 10 questions and lasts for 180 minutes. No calculators are allowed, and students are allowed to bring 3 handwritten cheat sheets.

Typology: Exams

2012/2013

Uploaded on 02/21/2013

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Math 256 Section 202 Final Exam
Spring 2005
Instructor: PC Chang
Last Name:
First Name:
Student Number:
Email Address:
INSTRUCTIONS:
• Write your last name, first name, student
number, and email address in the spaces above.
• No calculators allowed.
• 3 handwritten cheat sheets (both sides) allowed.
• This exam consists of 10 questions on 17 pages
(including this one).
• The maximum score on this exam is 100.
• You have 180 minutes to complete this exam.
• Good Luck!
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Download Math 256 Section 202 Final Exam - Spring 2005 and more Exams Differential Equations in PDF only on Docsity!

Math 256 Section 202 Final Exam

Spring 2005

Instructor: PC Chang

Last Name:

First Name:

Student Number:

Email Address:

INSTRUCTIONS:

  • Write your last name, first name, student

number, and email address in the spaces above.

  • No calculators allowed.
  • 3 handwritten cheat sheets (both sides) allowed.
  • This exam consists of 10 questions on 17 pages

(including this one).

  • The maximum score on this exam is 100.
  • You have 180 minutes to complete this exam.
  • Good Luck!
  1. Answer the following questions. You need not show work for this section.

A) What is the y -intercept of y = 3 x + 1? (1 mark)

B) True or False: 0 sin xdx

∞

∫ is^ an^ improper^ integral.

(1 mark)

C) Find the general solution of dydx = xy −^1. (1 mark)

D) What is the canonical form of the subcritical pitchfork bifurcation? (1 mark)

E) Find the steady-state solution of ut = uxx subject to the

boundary conditions u ( 0, t )= 0 , u ( 1, t )= 2. (1 mark)

  1. Solve the homogeneous equation xdy = (^2) ( x + y dx ).

(10 marks)

  1. Solve

2 y ′ + 2 xy = e − x , y (^) ( 0 )= 1. (10 marks)

  1. Using variation of parameters, find the general solution of

( )

1

y 3 y 2 y 1 e x

−

(10 marks)

5 Cont’d)

  1. Consider a very long (infinite) heat-conducting bar of rectangular cross-section, as shown below.

If the face y = b is kept at a constant temperature T 0 , and the other three faces are kept at zero temperature, compute the steady-state temperature distribution. (10 marks)

7 Cont’d)

8a) Consider the autonomous ODE ddt^ θ = r + sinθ. Sketch all the

qualitatively different phase portraits that occur as r is varied. Classify the bifurcations that occur, and find the bifurcation point. (7 marks)

8b) Suppose initially θ (^) ( 0 )= π. For what values of r does

lim t →∞ θ ( ) t = ∞? (3 marks)

  1. Consider the following BVP for the one dimensional heat equation u t = uxx

where 0 < x < 1 and t > 0 , with conditions

u (^) ( 0, t (^) ) = u (^) (1, t )= 0 for t > 0 , u x ( ,0 (^) ) = e − x for 0 < x < 1.

9a) Give a brief physical interpretation of this problem. (2 marks)

9b) Solve this BVP. (8 marks)

9b Cont’d)

  1. A stretched string of length L with its ends fixed at x = 0 and x = L has initial profile u x ( ,0) = f (^) ( x ) and is initially at rest. For t > 0 , it is subjected to forced vibrations described by the PDE u (^) xx − c −^2 utt = − g ′′ ( x ),

where g is a given function which satisfies g (^) ( 0 ) = g L ( )= 0. One method for solving this problem is to decompose the solution as u x t ( , (^) ) = v x t ( , (^) ) + w x ( ), where v and w each solve a modified version of this problem. Determine these modified problems. In particular, state the differential equations which v and w should solve, and the conditions which v and w should obey. Do not solve these problems.

(10 marks)