







Study with the several resources on Docsity
Earn points by helping other students or get them with a premium plan
Prepare for your exams
Study with the several resources on Docsity
Earn points to download
Earn points by helping other students or get them with a premium plan
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: Order Differential Equations, Solutions, General Solution, Linearly Independent, Particular Solution, Inhomogeneous Problem, Initial Conditions, Oscillator, Damped Forced, Transform
Typology: Exams
1 / 13
This page cannot be seen from the preview
Don't miss anything!








Be sure that this examination has 13 pages including this cover
The University of British Columbia Sessional Examinations - April 2008
Mathematics 256 Differential Equations
Closed book examination Time: 2 12 hours
Name Signature
Student Number Instructor’s Name
Section Number
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
Total 100
Marks
[20] 1. Linear 2nd order differential equations. (a) Show that the equation t^2 y′′^ + 6ty′^ + 4y = 0,
has 2 solutions of form tn, i.e. find these solutions.
(b) Show that the 2 solutions in part (a) are linearly independent and hence write down the general solution of the above equation.
[20] 2. Consider the equation: y′′^ + βy′^ + 2y = g(t)
which models a damped forced oscillator for β > 0. (a) Writing x = (x 1 , x 2 )T^ where x 1 = y and x 2 = y′, transform this equation into a 2 x 2 linear system: x′^ = Ax + f ,
i.e. derive the expressions for the matrix A and vector function f.
(b) Assume that g(t) = 0. Find the general solution of the homogeneous system:
x′^ = Ax
(i) for β = 2; (ii) for β = 3.
(c) Sketch the phase plane close to x = 0 for the two above cases: (i) β = 2; (ii) β = 3. At what value of β does the phase plane change between these two qualitatively different solutions?
(d) For β = 2 and assuming g(t) = e−t, find the general solution of the inhomogeneous system: x′^ = Ax + f.
[15] 4. The function f (x) is defined for x ∈ [0, 1] by
f (x) = − 0 .5(x^2 + x)
(a) Sketch the odd and even extensions of f (x) to the interval x ∈ [− 1 , 1]. (b) Find the Fourier cosine series for f (x). To what value does the Fourier series converge to at x = 1?
(c) Find the Fourier sine series for f (x). To what value does the Fourier series converge to at x = 1?
(c) Find the transient solution, v(x, t) using separation of variables. Hint: you may save some time with the initial conditions by using the results from question 4.
(d) Using the 1st term in the series solution for v(x, t) estimate how long it takes for |v(x, t)| to decay to 1% of its initial size at x = 0.5.
The End