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Exactly - Ordinary and Partial Differential Equations - Exam, Exams of Differential Equations

National Institute of Industrial EngineeringDifferential Equations

Main points of this exam paper are: Exactly, Autonomous Equation, Exactly, Solution, Nonzero Constants, Unique Solution, Initial Value Problem, Nonzero Constants, Unique Solution, Boundary

Typology: Exams

2012/2013

Uploaded on 03/21/2013

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MATH 251
Final Exam
May 7, 2004
Name:
Student Number:
Instructor:
Section:
This exam has 12 questions for a total of 150 points.
In order to obtain full credit for partial credit problems, all work must be shown. Credit will
not be given for an answer not supported by work.
THE USE OF CALCULATORS IS NOT PERMITTED IN THIS EXAMINATION.
The last page of this examination contains a table of Laplace transforms for your use.
Do not write in this box.
1:
2:
3:
4:
5:
6:
7:
8:
9:
10:
11:
12:
Total:
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MATH 251

Final Exam May 7, 2004

Name: Student Number: Instructor: Section: This exam has 12 questions for a total of 150 points. In order to obtain full credit for partial credit problems, all work must be shown.not be given for an answer not supported by work. Credit will

THE USE OF CALCULATORS IS NOT PERMITTED IN THIS EXAMINATION. The last page of this examination contains a table of Laplace transforms for your use.

Do not write in this box. 1: 2: 3: 4: 5: 6: 7: 8: 9: 10: 11: 12: Total:

  1. (12 points) Circle the correct answer to each of the following statements: (a) (2 points) True or False: The autonomous equationlibrium solution. y′^ = y^3 + y has exactly one real equi-

(b) (2 points) True or False: ay′′ (^) + by′ (^) + cy = 0, y(0) = 1 For nonzero constants, y′(0) = 1, always has a unique solution. a, b, and c, the initial value problem

(c) (2 points) True or False: For nonzero constants ay′′ (^) + by′ (^) + cy = 0, y(0) = 1, y(5) = 1, always has a unique solution. a, b, and c, the boundary value problem

(d) (2 points) True or False: If F (s) + G(s). F (s) = L{f (t)} and G(s) = L{g(t)}, then L{f (t) + g(t)} =

(e) (2 points) True or False: If a matrix x′ (^) = Ax has the general solution x(t) = A has a repeated real eigenvaluec r, then the system 1 ξert^ +^ c 2 ξtert^ where^ ξ^ is an eigenvector of^ r.

(f) (2 points) True or False: An odd periodic function can be represented by a Fourier sineseries.

  1. (9 points) Consider the mass-spring system described by the equation u′′^ + γu′^ + 9u = 0 . (a) (3 points) In the absence of damping (i.e. γ = 0), what is the system’s natural period?

(b) (3 points) If γ = 4, what is the system’s quasi-frequency?

(c) (3 points) If γ = 6, is the system underdamped, critically damped, or overdamped?

  1. (15 points) Solve the initial value problem y′′^ − 6 y′^ + 9y = δ(t − 1) − 2 δ(t − 3) , y(0) = 0, y′(0) = 0.
  1. (18 points) (a) (9 points) Find the general solution of x′^ =

[− 2 3

]

x.

(b) (3 points) Classify the type and stability of the critical point at (0, 0).

(c) (3 points) If x(0) =

[− 3

α

]

and lim t→∞ |x(t)| = 0, then what is the value of α?

(d) (3 points) Make a simple sketch of this system’s phase portrait.

  1. (15 points) (a) (12 points) Solve the initial value problem x′^ =

[ 3 − 1

]

x; x(0) =

[ 1

]

(b) (3 points) Classify the type and stability of the critical point at (0, 0).

  1. (10 points) Separate the partial differential equation uxx − utx + 5x^3 ut = 0 into a system of two equations of one independent variable each.
  1. (15 points) (a) (10 points) Find all positive eigenvalues, and their corresponding eigenfunctions, of thehomogeneous boundary value problem y′′^ + λy = 0; y′(0) = 0, y′(2π) = 0.

(b) (5 points) Is λ = 0 also an eigenvalue? If yes, give an example of its eigenfunction.

  1. (15 points) Find the particular solution of the homogeneous heat conduction problem: 5 uxx = ut, 0 < x < 4 , t > 0 , u(0, t) = 0, u(4, t) = 0, u(x, 0) = f (x) = 2 sin πx 4 − sin πx