April 2009 MATH 256 Name Page 2 of 11 pages
Marks
[20] 1. Linear differential equations.
(a) Solve the initial value problem:
y0+ 3y= 2e−2t, y(0) = 1.
(b) Solve the initial value problem:
y0+ 2ty =e−t2
, y(0) = 3.
Continued on page 3
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Initial Value - Differential Equations - Exam, Exams of Differential Equations
This is the Exam of Differential Equations which includes Odd and Even, Odd and Even Extensions, Sine Series, Converge, Reasoning, Fourier Sine Series, Faster or Slower, Boundary Value etc. Key important points are: Initial Value, Solution, Maximum Value, Attained, General Solution, Homogeneous Linear System, Phase Plane Close, Odd or Even, Interval, Fourier Series
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Marks
[20] 1. Linear differential equations. (a) Solve the initial value problem: y′^ + 3y = 2e−^2 t, y(0) = 1. (b) Solve the initial value problem: y′^ + 2ty = e−t^2 , y(0) = 3.
(c) Solve the initial value problem: y′′^ + 2y′^ + y = 0, y(0) = 0, y′(0) = 2.
(d) Solve the initial value problem: y′′^ + 2y′^ + y = 2e−t, y(0) = 0, y′(0) = 0, and show that the solution y(t) has maximum value of 4e−^2. At what time is this value attained?
(b) Find the solution of the following initial value problem:
x′^ =
x +
( 12 t − 11 − 3
, x(0) =
[20] 3. The function f (t) is defined for t ∈ [− 1 , 1] by
f (t) =
{−t − 1 , , t ∈ [0, 1], 1 − t, t ∈ [0, 1], (a) Sketch the function f (t) over the interval t ∈ [− 1 , 1], and find the Fourier series for f (t). Is the function odd or even? (b) To what values should the Fourier series of part (a) converge to at: (i) t = 0.5; (ii) t = 2; (iii) t = 3; (iv) t = 3.5?
[20] 4. Consider the the following IBVP ut = uxx − cos 3πx, 0 < x < 1 , t ≥ 0 , subject to the boundary and initial conditions: ux(0, t) = 2, ux(1, t) = 2, u(x, 0) = 2x + cos πx (a) Writing u(x, t) = us(x) + v(x, t) state the problems that are satisfied by the steady state solution, us(x), and the transient solution v(x, t). (b) Find the steady state solution, us(x).
(c) Find the transient solution, v(x, t) using separation of variables, and hence write down the full solution, u(x, t). (d) Show that u(0. 25 , t) → 12 + √^12 , as t → ∞, and estimate how large t must be in order for ∣∣ ∣∣u(0. 25 , t) − 12 − √^12
The End
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The University of British Columbia Sessional Examinations - April 2009 Mathematics 256 Differential Equations
Closed book examination Time: 2 12 hours
Name Signature Student Number Instructor’s Name Section Number
Special Instructions:
Two 8 12 ” × 11” two sided cheat sheets are permitted. Non-programmable calculators allowed. Show your work in the spaces provided. You are encouraged to explain all steps in your solutions.
Rules Governing Formal Examinations
- Each candidate must be prepared to produce, upon request, a library/AMS card for identification.
- Candidates are not permitted to ask questions of the invigilators, except in cases of supposed errors or ambiguities in examination questions.
- No candidate shall be permitted to enter the examination room after the expiration of one half hour from the scheduled starting time, or to leave during the first half hour of the examination.
- Candidates suspected of any of the following, or similar, dishonest practices shall be immediately dismissed from the examination and shall be liable to disciplinary action. (a) Having at the place of writing any books, papers or memoranda, calculators, computers, audio or video cassette players or other memory aid devices, other than those authorized by the examiners. (b) Speaking or communicating with other candidates. (c) Purposely exposing written papers to the view of other candidates. The plea of accident or forgetfulness shall not be received.
- Candidates must not destroy or mutilate any examination material; must hand in all examination papers; and must not take any examination material from the examination room without permission of the invigilator.
Total 100